Download pdf - Italian Journal of Pure and Applied Mathematics ISSN

N° 32 – August 2014

Italian Journal of Pure andApplied Mathematics

ISSN 2239-0227

EDITOR-IN-CHIEFPiergiulio Corsini

Editorial BoardSaeid Abbasbandy

Reza AmeriLuisa Arlotti

Krassimir AtanassovMalvina Baica

Federico BartolozziRajabali Borzooei

Carlo CecchiniGui-Yun Chen

Domenico Nico ChillemiStephen Comer

Irina CristeaMohammad Reza Darafsheh

Bal Kishan DassBijan Davvaz

Mario De SalvoAlberto Felice De Toni

Franco EugeniGiovanni Falcone

Yuming FengAntonino Giambruno

Furio HonsellLuca Iseppi

James JantosciakTomas Kepka

David KinderlehrerAndrzej Lasota

Violeta LeoreanuMaria Antonietta Lepellere

Mario MarchiDonatella MariniAngelo MarzolloAntonio Maturo

M. Reza MoghadamPetr Nemec

Vasile OproiuLivio C. PiccininiGoffredo PieroniFlavio Pressacco

Vito RobertoIvo RosenbergGaetano Russo

Paolo SalmonMaria Scafati Tallini

Kar Ping ShumAlessandro Silva

Florentin SmarandacheSergio Spagnolo

Stefanos SpartalisHari M. SrivastavaMarzio Strassoldo

Yves SureauCarlo TassoIoan TofanAldo Ventre

Thomas VougiouklisHans Weber

Xiao-Jun YangYunqiang Yin

Mohammad Mehdi ZahediFabio ZanolinPaolo Zellini

Jianming Zhan

F O R U M

EDITOR-IN-CHIEF

Piergiulio Corsini Department of Civil Engineering and Architecture Via delle Scienze 206 - 33100 Udine, Italy [email protected]

VICE-CHIEFS


MANAGING BOARD

Domenico Chillemi, CHIEFPiergiulio CorsiniIrina CristeaAlberto Felice De ToniFurio HonsellVioleta LeoreanuMaria Antonietta LepellereElena MocanuLivio PiccininiFlavio PressaccoNorma Zamparo

EDITORIAL BOARD

Saeid Abbasbandy Dept. of Mathematics, Imam Khomeini International University, Ghazvin, 34149-16818, Iran [email protected]

Reza Ameri Department of Mathematics University of Tehran, Tehran, Iran [email protected]

Luisa Arlotti Department of Civil Engineering and Architecture Via delle Scienze 206 - 33100 Udine, Italy [email protected]

Krassimir Atanassov Centre of Biomedical Engineering, Bulgarian Academy of Science BL 105 Acad. G. Bontchev Str. 1113 Sofia, Bulgaria [email protected]

Malvina Baica University of Wisconsin-Whitewater Dept. of Mathematical and Computer Sciences Whitewater, W.I. 53190, U.S.A. [email protected]

Federico Bartolozzi Dipartimento di Matematica e Applicazioni via Archirafi 34 - 90123 Palermo, Italy [email protected]

Rajabali Borzooei Department of Mathematics Shahid Beheshti University, Tehran, Iran [email protected]

Carlo Cecchini Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy [email protected]

Gui-Yun Chen School of Mathematics and Statistics, Southwest University, 400715, Chongqing, China [email protected]

Domenico (Nico) Chillemi Executive IT Specialist, IBM Software Group IBM Italy SpA Via Sciangai 53 – 00144 Roma, Italy [email protected]

Stephen Comer Department of Mathematics and Computer Science The Citadel, Charleston S. C. 29409, USA [email protected]

Irina Cristea CSIT, Centre for Systems and Information Technologies University of Nova Gorica Vipavska 13, Rožna Dolina, SI-5000 Nova Gorica, Slovenia [email protected]

Mohammad Reza Darafsheh School of Mathematics, College of Science University of Tehran, Tehran, Iran [email protected]

Bal Kishan Dass Department of Mathematics University of Delhi, Delhi - 110007, India [email protected]

Bijan Davvaz Department of Mathematics, Yazd University, Yazd, Iran [email protected]

Mario De Salvo Dipartimento di Matematica e Informatica Viale Ferdinando Stagno d'Alcontres 31, Contrada Papardo 98166 Messina [email protected]

Alberto Felice De Toni Udine University, Rector Via Palladio 8 - 33100 Udine, Italy [email protected]

Franco Eugeni Dipartimento di Metodi Quantitativi per l'Economia del Territorio Università di Teramo, Italy [email protected]

Giovanni Falcone Dipartimento di Metodi e Modelli Matematici viale delle Scienze Ed. 8 90128 Palermo, Italy [email protected]

Yuming Feng College of Math. and Comp. Science, Chongqing Three-Gorges University, Wanzhou, Chongqing, 404000, P.R.China [email protected]

Antonino Giambruno Dipartimento di Matematica e Applicazioni via Archirafi 34 - 90123 Palermo, Italy [email protected]

Furio Honsell Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy [email protected]

Luca Iseppi Department of Civil Engineering and Architecture, section of Economics and Landscape Via delle Scienze 206 - 33100 Udine, Italy [email protected]

James Jantosciak Department of Mathematics Brooklyn College (CUNY) Brooklyn, New York 11210, USA [email protected]

Tomas Kepka MFF-UK Sokolovská 83 18600 Praha 8,Czech Republic [email protected]

David Kinderlehrer Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA15213-3890, USA [email protected]

Andrzej Lasota Silesian University Institute of Mathematics Bankova 14 40-007 Katowice, Poland [email protected]

Violeta Leoreanu-Fotea Faculty of Mathematics Al. I. Cuza University 6600 Iasi, Romania [email protected]

Maria Antonietta Lepellere Department of Civil Engineering and Architecture Via delle Scienze 206 - 33100 Udine, Italy [email protected]

Mario Marchi Università Cattolica del Sacro Cuore via Trieste 17, 25121 Brescia, Italy [email protected]

Donatella Marini Dipartimento di Matematica Via Ferrata 1- 27100 Pavia, Italy [email protected]

Angelo Marzollo Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy [email protected]

Antonio Maturo University of Chieti-Pescara, Department of Social Sciences, Via dei Vestini, 31 66013 Chieti, Italy [email protected]

M. Reza Moghadam Faculty of Mathematical Science Ferdowsi University of Mashhadh P.O.Box 1159 - 91775 Mashhad, Iran [email protected] Petr Nemec Czech University of Life Sciences, Kamycka’ 129 16521 Praha 6, Czech Republic [email protected]

Vasile Oproiu Faculty of Mathematics Al. I. Cuza University 6600 Iasi, Romania [email protected]

Livio C. Piccinini Department of Civil Engineering and Architecture Via delle Scienze 206 - 33100 Udine, Italy [email protected]

Goffredo Pieroni Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy [email protected]

Flavio Pressacco Dept. of Economy and Statistics Via Tomadini 30 33100, Udine, Italy [email protected]

Vito Roberto Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy [email protected]

Ivo Rosenberg Departement de Mathematique et de Statistique Université de Montreal C.P. 6128 Succursale Centre-Ville Montreal, Quebec H3C 3J7 - Canada [email protected]

Gaetano Russo Department of Civil Engineering and Architecture Via delle Scienze 206 33100 Udine, Italy [email protected]

Paolo Salmon Dipartimento di Matematica Università di Bologna Piazza di Porta S. Donato 5 40126 Bologna, Italy [email protected]

Maria Scafati Tallini Dipartimento di Matematica "Guido Castelnuovo" Università La Sapienza Piazzale Aldo Moro 2 - 00185 Roma, Italy [email protected]

Kar Ping Shum Faculty of Science The Chinese University of Hong Kong Hong Kong, China (SAR) [email protected]

Alessandro Silva Dipartimento di Matematica "Guido Castelnuovo" Università La Sapienza Piazzale Aldo Moro 2 - 00185 Roma, Italy [email protected]

Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA [email protected]

Sergio Spagnolo Scuola Normale Superiore Piazza dei Cavalieri 7 - 56100 Pisa, Italy [email protected]

Stefanos Spartalis Department of Production Engineering and Management, School of Engineering Democritus University of Thrace V.Sofias 12, Prokat, Bdg A1, Office 308 67100 Xanthi, Greece [email protected]

Hari M. Srivastava Department of Mathematics and Statistics University of Victoria Victoria, British Columbia V8W3P4, Canada [email protected]

Marzio Strassoldo Department of Statistical Sciences Via delle Scienze 206 - 33100 Udine, Italy [email protected]

Yves Sureau 27, rue d'Aubiere 63170 Perignat, Les Sarlieve - France [email protected]

Carlo Tasso Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy [email protected]

Ioan Tofan Faculty of Mathematics Al. I. Cuza University 6600 Iasi, Romania [email protected]

Aldo Ventre Seconda Università di Napoli, Fac. Architettura, Dip. Cultura del Progetto Via San Lorenzo s/n 81031 Aversa (NA), Italy [email protected]

Thomas Vougiouklis Democritus University of Thrace, School of Education, 681 00 Alexandroupolis. Greece [email protected]

Hans Weber Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy [email protected]

Xiao-Jun Yang Honorary Professor, High Engineer Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu, 221008, China [email protected]

Yunqiang Yin School of Mathematics and Information Sciences, East China Institute of Technology, Fuzhou, Jiangxi 344000, P.R. China [email protected]

Mohammad Mehdi Zahedi Department of Mathematics, Faculty of Science Shahid Bahonar, University of Kerman Kerman, Iran [email protected]

Fabio Zanolin Dipartimento di Matematica e Informatica Via delle Scienze 206 - 33100 Udine, Italy [email protected]

Paolo Zellini Dipartimento di Matematica Università degli Studi Tor Vergata, via Orazio Raimondo (loc. La Romanina) - 00173 Roma, Italy [email protected]

Jianming Zhan Department of Mathematics, Hubei Institute for Nationalities Enshi, Hubei Province,445000, China [email protected]

mailto:[email protected]




ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 32-2014 i

ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 32-2014 ii

ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 32-2014

Italian Journal of Pure and Applied MathematicsISSN 2239-0227

Web Sitehttp://ijpam.uniud.it/journal/home.html

Twitter@ijpamitaly

https://twitter.com/ijpamitaly

EDITOR-IN-CHIEFPiergiulio Corsini

Department of Civil Engineering and ArchitectureVia delle Scienze 206 - 33100 Udine, Italy

[email protected]

Vice-CHIEFS Violeta Leoreanu

Maria Antonietta Lepellere

Managing Board Domenico Chillemi, CHIEF

Piergiulio CorsiniIrina Cristea

Alberto Felice De ToniFurio Honsell


Elena MocanuLivio Piccinini

Flavio PressaccoNorma Zamparo

Editorial BoardSaeid Abbasbandy

Reza AmeriLuisa Arlotti

Krassimir AtanassovMalvina Baica

Federico BartolozziRajabali Borzooei

Carlo CecchiniGui-Yun Chen

Domenico Nico ChillemiStephen Comer

Irina CristeaMohammad Reza Darafsheh

Bal Kishan DassBijan Davvaz

Mario De SalvoAlberto Felice De Toni

Franco EugeniGiovanni Falcone

Yuming FengAntonino Giambruno

Furio HonsellLuca Iseppi

James JantosciakTomas Kepka

David KinderlehrerAndrzej Lasota


Mario MarchiDonatella MariniAngelo MarzolloAntonio Maturo

M. Reza MoghadamPetr Nemec

Vasile OproiuLivio C. PiccininiGoffredo PieroniFlavio Pressacco

Vito RobertoIvo RosenbergGaetano Russo

Paolo SalmonMaria Scafati Tallini

Kar Ping ShumAlessandro Silva

Florentin SmarandacheSergio Spagnolo

Stefanos SpartalisHari M. SrivastavaMarzio Strassoldo

Yves SureauCarlo TassoIoan Tofan

Aldo VentreThomas Vougiouklis

Hans WeberXiao-Jun YangYunqiang Yin

Mohammad Mehdi ZahediFabio ZanolinPaolo Zellini

Jianming Zhan

Forum Editrice Universitaria Udinese SrlVia Larga 38 - 33100 Udine

Tel: +39-0432-26001, Fax: [email protected]

1

http://ijpam.uniud.it/journal/home.html



http://ijpam.uniud.it/journal/home.html


Italian Journal of Pure and Applied Mathematics – N. 32–2014

On OD-characterizability of a certain alternating and symmetric groupA. Mahmiani, M.R. Darafsheh pp. 7-14Some uniqueness results on meromorphic functions sharing two setsAbhijit Banerjee, Pranab Bhattacharjee pp. 15-32A note on P-regular semiringsM.K. Dubey pp. 33-36Some classes of generalized Minmax polynomialsH.M. Srivastava, Gospava B. Djordjević pp. 37-48On γ-s-Urysohn closed and γ-s-regular closed spacesSabir Hussain, Ennis Rosas pp. 49-56(0,2; 0)-Interpolation on the unit circleSwarnima Bahadur pp. 57-66

On laplace transforms of generalized whittaker function of multi-variables ( ), ... 11

, ...,kk

M x xλ µ µ

M. Kamarujjama, Waseem A. Khan pp. 67-72Fekete-Szegö problem for concave univalent functions defined by Sălăgean operatorAlawiah Ibrahim, Maslina Darus, Sever S. Dragomir pp. 73-86r-Weak cb spacesD. Bhattacharya, L. Dey pp. 87-102More properties on flexible graded modulesFida Moh'd, Mashhoor Refai pp. 103-114On (N(k),ξ)-semi-Riemannian 3-manifoldsD.G. Prakasha, H.G. Nagaraja, G. Somashekhara pp. 115-124A new signing algorithm based on elliptic curve discrete logarithms and quadratic residue problemsNedal Tahat, Emad E. Abdallah pp. 125-132Eccentricity, space bending, dimensionM. Nitu, Fl. Smarandache, M.E. Selariu pp. 133-142Strongly duo and duo right S-ActsMohammad Roueentan, Majid Ershad pp. 143-154On automatic boundedness of linear operators on convex bornological spacesAbdelaziz Tajmouati pp. 155-164Stability solutions of nonlinear nabla fractional difference equations using fixed point theoremsJ. Jagan Mohan, N. Shobanadevi, G.V.S.R. Deekshitulu pp. 165-184Exp-function method using modified Riemann-Liouville derivative for singularly perturbed Boussinesq equations of fractional-orderQazi Mahmood Ul Hassan, Syed Tauseef Mohyud-Din pp. 185-192Common fixed point for self and nonself-maps through an implicit relationT. Phaneendra, D. Surekha pp. 193-202A characterization of projective special linear group L3(5) by nseShitian Liu pp. 203-212New inequalities of Hermite-Hadamard type for functions whose first derivatives absolute values are s-convexFeixiang Chen, Yuming Feng pp. 213-222On Thompson’s conjecture for Aut (J2 ) and Aut (McL)Yanheng Chen, Yuming Feng, Guiyun Chen pp. 223-234On some growth properties of differential polynomials in the light of relative orderSanjib Kumar Datta, Tanmay Biswas, M.D. Azizul Hoque pp. 235-246Jensen type weighted inequalities for functions of selfadjoint and unitary operatorsS.S. Dragomir pp. 247-264φJ-multipliers and φJ-multipliers quadratic on Jordan Banach algebrasAbdelaziz Tajmouati pp. 265-276A note on Boolean subsets of orthomodular posetsDietmar Dorninger, Helmut Länger pp. 277-282Some derivations on the bounds for the zeros of entire functions based on slowly changing functionsSanjib Kumar Datta, Dilip Chandra Pramanik pp. 283-300Soft intersection h-ideals of hemirings and its applications Xueling Ma, Jianming Zhan pp. 301-308Characterizations of regular Abel-Grassmann's groupoids by the properties of their ( ,∈ ∈∨qk)-fuzzy idealsXueling Ma, Jianming Zhan, Madad Khan, Tariq Aziz pp. 309-324Secret key distribution technique using theory of numbersS. Srinivasan, P. Muralikrishna, N. Chandramowliswaran pp. 325-328Some characterizations of intra-regular Abel Grassmann's groupoidsXueling Ma, Jianming Zhan, Madad Khan, Tariq Aziz pp. 329-346Some double lacunary sequence spacesKuldip Raj, Sunil K. Sharma, Seema Jamwal pp. 347-358A new approach to a certain generalized integral transform for certain space of BoehmiansS.K.Q. Al-Omari pp. 359-368On solitary ware solutions of nonlinear time-fractional Fornberg-Whitham equationQazi Mahmood Ul Hassan, Syed Tauseef Mohyud-Din pp. 369-378Global exponential stability of impulsive hybrid dynamical systems with any time delayXingjie Wu, Yang Liu, Zongmin Qiao pp. 379-392

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From Newton to Kepler. One simple derivation of Kepler's laws from Newton's ones.František Mošna pp. 393-400Hopf Modules in the weak Yetter-Drinfeld categoriesYin Yanmin pp. 401-414Right alternative rings with x(yz)−y(xz) in the center K. Madhusudhan Reddy, K. Suvarna pp. 415-418Hyers-Ulam stability of linear differential equations of second order with constant coefficientJianming Xue pp. 419-424A theorem about finite groups with special conjugacy classesXianglin Du, Yuming Feng, Jinkui Liu pp. 425-430Energy of an intuitionistic fuzzy graphB. Praba, V.M. Chandrasekaran, G. Deepa pp. 431-444On upper and lower almost contra-ω-continuous multifunctionsC. Carpintero, N. Rajesh, E. Rosas, S. Saranyasri pp. 445-460Irreducible ideals in ringsB. Venkateswarlu, R. Vasu Babu, E. Yohannes, T. Embiale pp. 461-466Inclusion results on subclasses of starlike and convex functions associated with struve functionsT. Janani, G. Murugusundaramoorthy pp. 467-476The modified (w=g)-expansion method and its applications for solving the modified generalized Vakhnenko equationElsayed M.E. Zayed, Ahmed H. Arnous pp. 477-492Rough neutrosophic setsSaid Broumi, Florentin Smarandache, Mamoni Dhar pp. 493-502Neutrosophic parametrized soft set theory and its decision makingSaid Broumi, Irfan Deli, Florentin Smarandache pp. 503-514Young type inequalities for matricesYang Peng pp. 515-518Bounds for the eigenvalues of matricesLimin Zou, Youyi Jiang pp. 519-524On thompson's conjecture for alternating groupShitian Liu, Yanhua Huang pp. 525-532Application of bipolar fuzzy soft sets in K-algebrasM. Akram, N.O. Alsherei, K.P. Shum, A. Farooq pp. 533-546Semigroup distances of finite groupoidsBarbora Batίková, Šárka Dvořáková, Milan Trch pp. 547-560On how to construct left semimodules from the right onesBarbora Batίková, TomášKepka, Petr Němec pp. 561-578Toward a new algorithm for systems of fractional differential-algebraic equationsH.M. Jaradat, M. Zurigat, Safwan Al-Shara', Qutaibeh Katatbeh pp. 579-594A characterization of higher derivations on Banach algebrasT.L. Shatery, S. Hejazian pp. 595-602

ISSN 2239-0227

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Exchanges

Up to December 2013 this journal is exchanged with the following periodicals:

1. Acta Cybernetica - Szeged H2. Acta Mathematica et Informatica Universitatis Ostraviensis CZ3. Acta Mathematica Vietnamica – Hanoi VN4. Acta Mathematica Sinica, New Series – Beijing RC5. Acta Scientiarum Mathematicarum – Szeged H6. Acta Universitatis Lodziensis – Lodz PL7. Acta Universitatis Palackianae Olomucensis, Mathematica – Olomouc CZ8. Actas del tercer Congreso Dr. Antonio A.R. Monteiro - Universidad Nacional del Sur Bahía Blanca AR9. AKCE International Journal of Graphs and Combinatorics - Kalasalingam IND10. Algebra Colloquium - Chinese Academy of Sciences, Beijing PRC11. Alxebra - Santiago de Compostela E12. Analele Ştiinţifice ale Universităţii “Al. I Cuza” - Iaşi RO13. Analele Universităţii din Timişoara - Universitatea din Timişoara RO14. Annales Academiae Scientiarum Fennicae Mathematica - Helsinki SW15. Annales de la Fondation Louis de Broglie - Paris F16. Annales Mathematicae Silesianae – Katowice PL17. Annales Scientif. Université Blaise Pascal - Clermont II F18. Annales sect. A/Mathematica – Lublin PL19. Annali dell’Università di Ferrara, Sez. Matematica I20. Annals of Mathematics - Princeton - New Jersey USA21. Applied Mathematics and Computer Science -Technical University of Zielona Góra PL22. Archivium Mathematicum - Brnö CZ23. Atti del Seminario di Matematica e Fisica dell’Università di Modena I24. Atti dell’Accademia delle Scienze di Ferrara I25. Automatika i Telemekhanika - Moscow RU26. Boletim de la Sociedade Paranaense de Matematica - San Paulo BR27. Bolétin de la Sociedad Matemática Mexicana - Mexico City MEX28. Bollettino di Storia delle Scienze Matematiche - Firenze I29. Buletinul Academiei de Stiinte - Seria Matem. - Kishinev, Moldova CSI30. Buletinul Ştiinţific al Universităţii din Baia Mare - Baia Mare RO31. Buletinul Ştiinţific şi Tecnic-Univ. Math. et Phyis. Series Techn. Univ. - Timişoara RO32. Buletinul Universităţii din Braşov, Seria C - Braşov RO33. Bulletin de la Classe de Sciences - Acad. Royale de Belgique B34. Bulletin de la Societé des Mathematiciens et des Informaticiens de Macedoine MK35. Bulletin de la Société des Sciences et des Lettres de Lodz - Lodz PL36. Bulletin de la Societé Royale des Sciences - Liege B37. Bulletin for Applied Mathematics - Technical University Budapest H38. Bulletin Mathematics and Physics - Assiut ET39. Bulletin Mathématique - Skopje Macedonia MK40. Bulletin Mathématique de la S.S.M.R. - Bucharest RO41. Bulletin of the Australian Mathematical Society - St. Lucia - Queensland AUS42. Bulletin of the Faculty of Science - Assiut University ET43. Bulletin of the Faculty of Science - Mito, Ibaraki J44. Bulletin of the Greek Mathematical Society - Athens GR45. Bulletin of the Iranian Mathematical Society - Tehran IR46. Bulletin of the Korean Mathematical Society - Seoul ROK47. Bulletin of the Malaysian Mathematical Sciences Society - Pulau Pinang MAL48. Bulletin of Society of Mathematicians Banja Luka - Banja Luka BiH49. Bulletin of the Transilvania University of Braşov - Braşov RO50. Bulletin of the USSR Academy of Sciences - San Pietroburgo RU51. Busefal - Université P. Sabatier - Toulouse F52. Calculus CNR - Pisa I53. Chinese Annals of Mathematics - Fudan University – Shanghai PRC54. Chinese Quarterly Journal of Mathematics - Henan University PRC55. Classification of Commutative FPF Ring - Universidad de Murcia E56. Collectanea Mathematica - Barcelona E57. Collegium Logicum - Institut für Computersprachen Technische Universität Wien A58. Colloquium - Cape Town SA59. Colloquium Mathematicum - Instytut Matematyczny - Warszawa PL60. Commentationes Mathematicae Universitatis Carolinae - Praha CZ61. Computer Science Journal of Moldova CSI62. Contributi - Università di Pescara I63. Cuadernos - Universidad Nacional de Rosario AR64. Czechoslovak Mathematical Journal - Praha CZ65. Demonstratio Mathematica - Warsawa PL66. Discussiones Mathematicae - Zielona Gora PL67. Divulgaciones Matemáticas - Universidad del Zulia YV68. Doctoral Thesis - Department of Mathematics Umea University SW69. Extracta Mathematicae - Badajoz E70. Fasciculi Mathematici - Poznan PL71. Filomat - University of Nis SRB72. Forum Mathematicum - Mathematisches Institut der Universität Erlangen D73. Functiones et Approximatio Commentarii Mathematici - Adam Mickiewicz University L74. Funkcialaj Ekvaciaj - Kobe University J75. Fuzzy Systems & A.I. Reports and Letters - Iaşi University RO76. General Mathematics - Sibiu RO77. Geometria - Fasciculi Mathematici - Poznan PL78. Glasnik Matematicki - Zagreb CRO79. Grazer Mathematische Berichte – Graz A80. Hiroshima Mathematical Journal - Hiroshima J81. Hokkaido Mathematical Journal - Sapporo J

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82. Houston Journal of Mathematics - Houston - Texas USA83. IJMSI - Iranian Journal of Mathematical Sciences & Informatics, Tarbiat Modares University, Tehran IR84. Illinois Journal of Mathematics - University of Illinois Library - Urbana USA85. Informatica - The Slovene Society Informatika - Ljubljana SLO86. Internal Reports - University of Natal - Durban SA87. International Journal of Computational and Applied Mathematics – University of Qiongzhou, Hainan PRC88. International Journal of Science of Kashan University - University of Kashan IR89. Iranian Journal of Science and Technology - Shiraz University IR90. Irish Mathematical Society Bulletin - Department of Mathematics - Dublin IRL91. IRMAR - Inst. of Math. de Rennes - Rennes F92. Israel Mathematical Conference Proceedings - Bar-Ilan University - Ramat -Gan IL93. Izvestiya: Mathematics - Russian Academy of Sciences and London Mathematical Society RU94. Journal of Applied Mathematics and Computing – Dankook University, Cheonan – Chungnam ROK95. Journal of Basic Science - University of Mazandaran – Babolsar IR96. Journal of Beijing Normal University (Natural Science) - Beijing PRC97. Journal of Dynamical Systems and Geometric Theory - New Delhi IND98. Journal Egyptian Mathematical Society – Cairo ET99. Journal of Mathematical Analysis and Applications - San Diego California USA100. Journal of Mathematics of Kyoto University - Kyoto J101. Journal of Science - Ferdowsi University of Mashhad IR102. Journal of the Bihar Mathematical Society - Bhangalpur IND103. Journal of the Faculty of Science – Tokyo J104. Journal of the Korean Mathematical Society - Seoul ROK105. Journal of the Ramanujan Mathematical Society - Mysore University IND106. Journal of the RMS - Madras IND107. Kumamoto Journal of Mathematics - Kumamoto J108. Kyungpook Mathematical Journal - Taegu ROK109. L’Enseignement Mathématique - Genève CH110. La Gazette des Sciences Mathématiques du Québec - Université de Montréal CAN111. Le Matematiche - Università di Catania I112. Lecturas Matematicas, Soc. Colombiana de Matematica - Bogotà C113. Lectures and Proceedings International Centre for Theorical Phisics - Trieste I114. Lucrările Seminarului Matematic – Iaşi RO115. m-M Calculus - Matematicki Institut Beograd SRB116. Matematicna Knjiznica - Ljubljana SLO117. Mathematica Balcanica – Sofia BG118. Mathematica Bohemica - Academy of Sciences of the Czech Republic Praha CZ119. Mathematica Macedonica, St. Cyril and Methodius University, Faculty of Natural Sciences and Mathematics - Skopje MK120. Mathematica Montisnigri - University of Montenegro - Podgorica MNE121. Mathematica Moravica - Cacak SRB122. Mathematica Pannonica - Miskolc - Egyetemvaros H123. Mathematica Scandinavica - Aarhus - Copenhagen DK124. Mathematica Slovaca - Bratislava CS125. Mathematicae Notae - Universidad Nacional de Rosario AR126. Mathematical Chronicle - Auckland NZ127. Mathematical Journal - Academy of Sciences - Uzbekistan CSI128. Mathematical Journal of Okayama University - Okayama J129. Mathematical Preprint - Dep. of Math., Computer Science, Physics – University of Amsterdam NL130. Mathematical Reports - Kyushu University - Fukuoka J131. Mathematics Applied in Science and Technology – Sangyo University, Kyoto J132. Mathematics Reports Toyama University - Gofuku J133. Mathematics for Applications - Institute of Mathematics of Brnö University of Technology, Brnö CZ134. MAT - Prepublicacions - Universidad Austral AR135. Mediterranean Journal of Mathematics – Università di Bari I136. Memoirs of the Faculty of Science - Kochi University - Kochi J137. Memorias de Mathematica da UFRJ - Istituto de Matematica - Rio de Janeiro BR138. Memorie linceee - Matematica e applicazioni - Accademia Nazionale dei Lincei I139. Mitteilungen der Naturforschenden Gesellschaften beider Basel CH140. Monografii Matematice - Universitatea din Timişoara RO141. Monthly Bulletin of the Mathematical Sciences Library – Abuja WAN142. Nagoya Mathematical Journal - Nagoya University,Tokyo J143. Neujahrsblatt der Naturforschenden Gesellschaft - Zürich CH144. New Zealand Journal of Mathematics - University of Auckland NZ145. Niew Archief voor Wiskunde - Stichting Mathematicae Centrum – Amsterdam NL146. Nihonkai Mathematical Journal - Niigata J147. Notas de Algebra y Analisis - Bahia Blanca AR148. Notas de Logica Matematica - Bahia Blanca AR149. Notas de Matematica Discreta - Bahia Blanca AR150. Notas de Matematica - Universidad de los Andes, Merida YV151. Notas de Matematicas - Murcia E152. Note di Matematica - Lecce I153. Novi Sad Journal of Mathematics - University of Novi Sad SRB154. Obzonik za Matematiko in Fiziko - Ljubljana SLO155. Octogon Mathematical Magazine - Braşov RO156. Osaka Journal of Mathematics - Osaka J157. Periodica Matematica Hungarica - Budapest H158. 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italian journal of pure and applied mathematics – n. 32−2014 (7−14) 7

ON OD-CHARACTERIZABILITY OF A CERTAIN ALTERNATINGAND SYMMETRIC GROUP

A. Mahmiani

Islamic Azad UniversityAliabad-e-KatoolIrane-mail: mahmiani [email protected]

M.R. Darafsheh

School of MathematicsCollege of ScienceUniversity of Tehran, TehranIrane-mails: [email protected]

Abstract. In this paper we will show that if G is a finite group with the same order

and degree pattern as the alternating group on 27 letters, then G is isomorphic to A27.

Furthermore we will show that there are three non-isomorphic finite groups with the

same order and degree pattern as the symmetric group on 27 letters.

Keywords: prime graph, characterization, alternating groups, symmetric groups.

2000 Mathematics Subject Classification: 20D05.

1. Introduction and preliminaries

Let G be a finite group. The set of all the prime divisors of G is denoted by π (G)and the set of elements order in G is denoted by πe (G). The prime graph or theGruenberg–Kegel graph of G is a simple graph denoted by Γ (G) or GK (G) and isdefines as follows. Vertices of Γ (G) are elements of π (G)and two distinct verticesp and q are joined by an edge if G possesses an element of order pq. In this case,we will write p ∼ q, otherwise p q means that G does not have an element oforder pq. The number of connected components of Γ (G) is denoted by t(G) andwe write πi = πi(G), 1 ≤ i ≤ t(G) for these connected components. In the casethat 2 ∈ π (G) we choose π1(G) for the component containing 2. For a naturalnumber n we write π (n) for the set of all primes dividing n. With this notation|G| can be expressed as a product of the numbers m1,m2, ...,mt(G) where mis arepositive integers with π (mi) = πi. The numbers mi, 1 ≤ i ≤ t(G), are called theorder components of G. We write OC (G) = m1,m2, ...,mt(G) and call it the setof order components of G. The set of prime graph components of G is denoted byT (G) =

π1 (G) , π2 (G) , ..., πt(G) (G)

. Throughout this paper we use standard

notation, in particular P denotes the set of all the prime numbers. All groupsconsidered in this paper are finite.

8 a. mahmiani, m.r. darafsheh

Definition 1. Let G be a finite group with |G| = pα11 pα2

2 ...pαkk , where pi ∈ P

and αi ∈ N . Let Γ(G) be the prime graph of G. For p ∈ π(G), the degree ofthe vertex p in Γ(G) is the number of vertices adjacent to p and it is denotedby deg(p), i.e., deg(p) = |q ∈ π(G) | p ∼ q|. We choose an ordering such asp1 < p2 < ... < pk in π(G) and define D(G) = (deg(p1), deg(p2), ..., deg(pk)). Wecall D(G) the degree pattern of G.

Definition 2. Let G be a natural number. A group M is called k-fold OD-charac-terizable if there are exactly k non-isomorphic groups G such that |G| = |M | andD(G) = D(M). A 1−fold OD-characterizable group is called an OD-characteri-zable group. In particular, an OD-cha-racterizable group is uniquely determinedby its order and the degrees of its vertices.

Definition 3. Let n ∈ N. A finite non-abelian simple group G is called a simpleKn−group if |π(G)| = n.

The Gruenberg–Kegel graph (prime graph) of a finite group G for the firsttime was defined in [5] and its significance can be found in many recent researches.Characterization of finite groups by their orders and degree pattern will helpus to know the properties of almost simple groups. In [1] it is shown that thealternating group Ap, where p and p − 2 are primes, and all sporadic simplegroups are OD-characterizable. It is also shown that certain groups of Lie typeare OD-characterizable, but the projective symplectic group SP6(3) is 2−foldOD-characterizable. In [4] it is proved that all the simple K4−groups exceptA10 are OD-characterizable whereas A10 is 2−fold OD-characterizable. In [2] itis proved that all simple groups of order less than 108 except A10 and U4(2) areOD-characterizable while A10 and U4(2) are 2−fold OD-characterizable. In [3]some almost simple groups related to L2(49) are considered and it is proved thatif G = L2(49) : V , where V is isomorphic to the Klein fourgroup, is the fullautomorphism group of L2(49), then G is 9−fold OD-characterizable, while if Wis a proper subgroup of V , then H = L2(49) : W is OD-characterizable. In [6]the authors proved that the alternating group of degree 16 is OD-characterizable.In this article our aim is to consider the OD-characterizability of the alternatingand symmetric groups A27 and S27. The importance of these groups is that bothof them have connected prime graph. Figure 1 shows the prime graph of A27.

Figure 1. The prime graph of A27

on od-characterizability of a certain alternating ... 9

For the proof we need to know about finite non-abelian simple groups of orderat most 19. In general for a given prime p, we denote by Sp the set of all non-abelian simple groups G such that p ∈ π(G) ⊆ 1, 2, ..., p. Therefore Table 1 listsall S19 groups. These are given in [7] and we reproduce them in Table 1 below.

Table 1: Non-abelian simple groups G with π(G) ⊆ 1, 2, 3, ..., 19

G |G| |Out(G)|S5 π(G) ⊆ 2, 3, 5A5

∼= L2(4) ∼= L2(5) 22.3.5 2A6

∼= L2(9) 23.32.5 4S4(3) ∼= U4(2) 26.34.5 2S7 7 ∈ π(G) ⊆ 2, 3, 5, 7L2(7) ∼= L3(2) 23.3.7 2L2 (8) 23.32.7 3U3(3) 25.33.7 2A7 23.32.5.7 2L2(49) 24.3.5.72 4U3(5) 24.32.53.7 6L3(4) 26.32.5.7 12A8

∼= L4(2) 26.32.5.7 2A9 26.34.5.7 2J2 27.33.52.7 2A10 27.34.52.7 2U4(3) 27.36.5.7 8S4(7) 28.32.52.74 2S6(2) 29.34.5.7 1O+

8 (2) 212.35.52.7 6S11 11 ∈ π(G) ⊆ 2, 3, 5, 7, 11 |Out(G)|L2(11) 22.3.5.11 2M11 24.32.5.11 1M12 26.33.5.11 2U5(2) 210.35.5.11 2M22 27.32.5.7.11 2A11 27.34.52.7.11 2McL 27.36.53.7.11 2HS 29.32.53.7.11 2A12 29.35.52.7.11 2U6(2) 215.36.5.7.11 6S13 13 ∈ π(G) ⊆ 2, 3, 5, ..., 13 |Out(G)|L3(3) 24.33.13 2L2(25) 23.3.52.13 4U3(4) 26.3.52.13 4S4(5) 26.32.54.13 2L4(3) 27.36.5.13 4


2F4(2)′ 211.33.52.13 2

L2(13) 22.3.7.13 2L2(27) 22.33.7.13 6G2(3) 26.36.7.13 23D4(2) 212.34.72.13 3Sz(8) 26.5.7.13 3L2(64) 26.32.5.7.13 6U4(5) 27.34.56.7.13 4L3(9) 27.36.5.7.13 4S6(3) 29.39.5.7.13 2O7(3) 29.39.5.7.13 2G2(4) 212.33.52.7.13 2S4(8) 212.34.5.72.13 6O+

8 (3) 212.312.52.7.13 24L5(3) 29.310.5.112.13 2A13 29.35.52.7.11.13 2A14 210.35.52.72.11.13 2A15 210.36.53.72.11.13 2L6(3) 211.315.5.7.112.132 4Suz 213.37.52.7.11.13 2A16 214.36.53.72.11.13 2Fi22 217.39.52.7.11.13 2S17 17 ∈ π(G) ⊆ 2, 3, 5, ..., 17 |Out(G)|L2(17) 24.32.17 2L2(16) 24.3.5.17 4S4(4) 28.32.52.17 4He 210.33.5273.17 2O−

8 (2) 212.34.5.7.17 2L4(4) 212.34.52.7.17 4S8(2) 216.35.52.7.17 1U4(4) 212.32.53.13.17 4U3(17) 26.34.7.13.173 6O−

10(2) 220.36.52.7.11.17 2L2(13

2) 23.3.5.7.132.17 4S4(13) 26.32.5.72.134.17 2L3(16) 212.32.52.7.13.17 24S6(4) 218.34.53.7.13.17 2O+

8 (4) 224.35.54.7.13.172 12F4(2) 224.36.52.72.13.17 2A17 214.36.53.72.11.13.17 2A18 215.38.53.72.11.13.17 2S19 19 ∈ π(G) ⊆ 2, 3, 5, ..., 19 Out(G)


L2(19) 22.32.5.19 2L3(7) 25.32.53.19 6U3(8) 29.34.7.19 18U3(19) 25.32.52.73.193 2L4(7) 29.34.52.7.19 4J3 27.35.5.17.19 2J4 23.3.5.7.11.19 1L3(11) 24.3.52.7.113.19 2HN 214.36.56.7.11.19 2U4(8) 218.37.5.72.13.19 6A19 215.38.53.72.11.13.17.19 2A20 217.38.54.72.11.13.17.19 2A21 217.39.54.73.11.13.17.19 2A22 218.39.54.73.112.13.17.19 22E6(2) 236.39.52.72.11.13.17.19 6

2. Main result

At first, we consider the alternating group on 27 letters, A27, and will show it isOD-characterizable. It is easy to verify that |A27| = 222.313.56.73.112.132.17.19.23andD (A27) = (8, 8, 7, 7, 5, 5, 4, 4, 2). Let G be a finite group such that |G| = |A27|,π(G) = 2, 3, 5, 7, 11, 13, 17, 19, 23, D (G) = (8, 8, 7, 7, 5, 5, 4, 4, 2), where D (G)is the degree pattern of G in its prime graph. Since deg(2) = 8 we deduce that theprime 2 is joined to all vertices in π(G)− 2, hence Γ(G) is a connected graph.Now it is easy to see that Γ(G) = Γ(A27) and that

πe(G) ⊇

2, 3, 5, 7, 11, 13, 17, 19, 23, 6, 10, 14, 22, 26, 34, 38, 46, 15, 27,33, 39, 51, 57, 69, 35, 55, 65, 85, 95, 77, 91, 119, 133, 143

.

Lemma 1. Let K be the maximal normal solvable subgroup of G. Then K is a2, 3−group. In particular, G is a non-solvable group.

Proof. Let p be a prime divisor of the order of K and Sp be a Sylow p−subgroupof K. Since KG, by the Frattini argument we obtain G = KNG(Sp). Next wedistinguish several cases:

Case (1) p = 23. In this case, S23 is a cyclic group of order 23, hence N = NG(S23)CG(S23)

is isomorphic to a subgroup of Z22. Therefore,∣∣N ∣∣ is a divisor of 22. But deg(23)

in Γ(G) is 2 and 23 is joined to 2 and 3 only, hence CG(S23) is a 2, 3, 23−group.Thus, |NG(S23)| = 2α.3β.11γ.23, where 0 ≤ α ≤ 22, 0 ≤ β ≤ 13, 0 ≤ γ ≤ 2and from G = KNG(S23) we deduce that 19 | |K|. Since K is assumed to besolvable, we can consider a 19− 23−Hall subgroup of K which has order 19.23


and should be cyclic. This implies 19 ∼ 23 which is a contradiction because A27

has no element of order 19.23. We conclude that 23 - |K|.

Case (2) p = 19 or 17. This case is treated with the same technique as in case (1)and a contradiction is obtained.

Case (3) p = 13 or 11. First, we consider p = 13, the case p = 11 is treated simi-larly. If a Sylow 13−subgroup of K has order 13, then with the same techniquesas in case (1) we obtain a contradiction, therefore we may assume that a Sylow

13−subgroup of K has order |S13| = 132. We have N = NG(S23)CG(S23)

≤ Aut(S13). If

S13 is a cyclic group, then |Aut(S13)| = 13.12 and if S13 is elementary abelian,then |Aut(S13)| = |GL2(13)| = 25.32.7.13. Therefore, in any case we deduce thatN is a 2, 3, 7, 13−group. Since deg(13) = 5 and 13 ∼ 2, 13 ∼ 3, 13 ∼ 5,13 ∼ 7, 13 ∼ 11, we obtain that NG(S23) is a 2, 3, 5, 7, 11, 13−group, hence19 | |K|. Again considering a 19, 13−Hall subgroup in K results in a contra-diction. Therefore, |K| cannot be divisible by 13 or 11.

Case (4) p = 7. In this case, a Sylow 7−subgroup of K has order 7, 72 or 73.If |S7| = 7 or 72, then using the same techniques as in case (1) and case (3) wewill obtain a contradiction. Hence, suppose |S7| = 73. From G = KNG(S7) andthe fact that 23 - |K|, we deduce that 23 | |NG(S7)|, so S7 is normalized by anelement σ of G with order 23. Since G has no element of order 23.7, ⟨σ⟩ shouldact fixed-point-freely on S7, implying 23 | 73 − 1, a contradiction.

Case (5) p = 5. In this case, |S5| = 5α, 1 ≤ α ≤ 6. By case (1), we have 23 - |K|,and from G = KNG(S5) we deduce that S5 is normalized by an element of order23, hence 23 | 5α − 1. But considering all the numbers 1 ≤ α ≤ 6 impossibility ofthe last divisibility is proved.

Therefore, we have proved that |K| can only be divisible by 2 and 3 and partof the lemma is proved. If G is a solvable group, then it has a 23, 19−Hallsubgroup of order 23.19 implying that 23 ∼ 19, a contradiction.

In the following lemma, we keep fixed the notation used in Lemma1.

Lemma 2. GK

is an almost simple group, i.e. S ≤ GK

≤ Aut(S) where S is asimple group isomorphic to A27.

Proof. Let us put G = GK

and S = Soc(G) where Soc(G) denotes the socle of the

group G, i.e., the subgroup of G generated by the set of all the minimal normalsubgroups of G. Then, S ∼= S1 × S2 × ... × Sn where S,

is are finite non-abeliansimple groups. We assume n ≥ 2 and derive a contradiction.

If 23 |∣∣S∣∣, then the order of only one Si is divisible by 23. We assume

23 | |S1|. Because only 23 ∼ 2 and 23 ∼ 3 hold in the prime graph of G, Si, i ≥ 2,must be 2, 3−groups contradicting simplicity of Si. Hence, 23 -

∣∣S∣∣.Since K is a 2, 3−group, we have 23|

∣∣G∣∣. From NG(S)

CG(S)= G

CG(S)≤ Aut(S),

we deduce that 23 ∈ π(G) ⊆ π(Aut(S)). But∣∣Aut(S)∣∣ = ∣∣S∣∣ . ∣∣Out(S)

∣∣, hence23 |

∣∣Out(S)∣∣. Let P1, P2, ..., Pk be non-isomorphic simple groups among S1, ..., Sn


such that S ∼= P t11 × P t2

2 × ... × P tkk , where t1 + t2 + ... + tk = n. We have

Out(S) = Out(P t11 ) × Out(P t2

2 ) × ... × Out(P tkk ), and from 23 -

∣∣S∣∣ we have23 - |Pi|, for all 1 ≤ i ≤ k, so Pis are simple 2, 3, 5, 7, 11, 13, 17, 19−group andby Table 1 we obtain 23 - |Out(Pi)|, 1 ≤ i ≤ k. But Aut(P ti

i )∼= PiWrSti , and

from 23∣∣Aut(P t1

1 )∣∣ = |P1|t1 .t1! we obtain 23 | t1! which implies t1 ≥ 23. Therefore,

(23!)2 × 223 = 242 must divide the order of G, a contradiction. Therefore, n = 1and S is a simple group.

Next, we consider the fact that∣∣G∣∣ | ∣∣Aut(S)∣∣, ∣∣G∣∣ = 222.313.56.73.112.132.17.19.23,

and S an S19-group. Now, by Table 1, we obtain S ∼= A27 and S ≤ GK

≤ Aut(S).

Theorem 1. G is isomorphic to A27.

Proof. By Lemma 2, A27 ≤ GK

≤ Aut(A27) = S27, thereforeGK

∼= A27 or S27.If G

K∼= A27, then |K| = 1, because |G| = |A27|, so G ∼= A27. If G

K∼= S27, then

2 |K| = 1, which is impossible.

Next, we consider OD-characterizability of the symmetric group S27. Thedegree pattern of S27 is the same as the degree pattern of A27, i.e., D(S27) =(8, 8, 7, 7, 5, 5, 4, 4, 2), but |S27| = 223.313.56.73.112.132.17.19.23. We assume thatG is a finite group with |G| = |S27| and D(G) = D(S27). As in the case of A27,if K is the maximal normal solvable subgroup of G, then we can prove that K isa 2, 3−group. Also, if we set G = G

Kand S = Soc(G), then similar techniques

yields S ∼= A27 and S ≤ G ≤ Aut(S).

Theorem 2. The symmetric group S27 is 3−fold OD-characterizable.

Proof. Since A27 ≤ GK

≤ Aut(A27) = S27 we will obtain GK

= A27 or S27. IfGK

= S27, then we obtain |K| = 1 and G = S27. If GK

= A27, then |K| = 2,hence K ≤ Z(G). Since K is the maximal normal solvable subgroup of G weobtain K = Z(G) and G is the central extension of Z2 by A27. Therefore, we havetwo possibilities for G , i.e., G = Z2 × A27 or G = Z2.A27 the covering group ofA27. Both groups have the same degree pattern and orders as S27 and theorem isproved.

References

[1] Darafsheh, M.R., Moghaddamfar,A.R. Zokayi, A.R., A characteri-zation of finite simple groups by the degrees of vertices of their prime graphs,Alg. Colloq., 12 (2005), 431-442.

[2] Zhang, L., Shi, W., OD-characterization of all simple groups whose ordersare less than 108, Front. Math. China, 3 (2008), 461-494.


[3] Zhang, L., Shi, W., OD-characterization of almost simple groups related toL2(49), Arch. Math. (Brno), 44 (2008), 191-199.

[4] Zhang, L., Shi, W., OD-characterization of simple K4-groups, Alg. Colloq.,16 (2009), 275-282.

[5] Williams, J.S., Prime graph components of finite groups, J. Algebra, 69(1981), 487-513.

[6] Zhang, L., Shi, W., Wang, L., Shao, C., OD-characterization of A16,J. Suzhou Univ., 24, 2008, 7-10.

[7] Zavarnitsine, A., Finite simple groups with narrow prime spectrum,Siberian Electronic Math. Reports., 6 (2009), 1-12.

Accepted: 19.04.2010


SOMEUNIQUENESS RESULTS ON MEROMORPHIC FUNCTIONSSHARING TWO SETS

Abhijit Banerjee

Department of MathematicsWest Bengal State UniversityKolkata-700126IndiaPresent Address: Department of MathematicsUniversity of KalyaniWest Bengal 741235Indiae-mail: abanerjee [email protected], [email protected]

Pranab Bhattacharjee

Department of MathematicsHooghly Mohsin CollegeChinsurah, Hooghly, West Bengal, 712101IndiaPresent Address: Department of MathematicsDarjeeling Government CollegeDarjeeling, West Bengal, 734101Indiae-mail: [email protected]

Abstract. Using the notion of weighted sharing, we prove some uniqueness theorems

of meromorphic functions that share two sets. The results in this paper improve and

supplement some recent ones of the first author and consequently provide a better

answer to the famous question of Gross than that was given previously.

Keywords: meromorphic functions, uniqueness, weighted sharing, shared set.


1. Introduction, definitions and results

In this paper, by meromorphic functions we will always mean meromorphic func-tions in the complex plane C. It will be convenient to let E denote any set ofpositive real numbers of finite linear measure, not necessarily the same at each oc-currence. For any non-constant meromorphic function h(z) we denote by S(r, h)any quantity satisfying

S(r, h) = o(T (r, h)) (r −→ ∞, r ∈ E).

16 abhijit banerjee, pranab bhattacharjee

Let f and g be two non-constant meromorphic functions and let a be a finitecomplex number. We say that f and g share a CM, provided that f − a andg − a have the same zeros with the same multiplicities. Similarly, we say that fand g share a IM, provided that f − a and g − a have the same zeros ignoringmultiplicities. In addition, we say that f and g share ∞ CM, if 1/f and 1/g share0 CM, and we say that f and g share ∞ IM, if 1/f and 1/g share 0 IM.

Let S be a set of distinct elements of C∪∞ and Ef (S)=∪a∈S

z:f(z)−a=0,

where each zero is counted according to its multiplicity. If we do not count the

multiplicity, the set∪a∈S

z : f(z)−a = 0 is denoted by Ef (S). If Ef (S) = Eg(S),

we say that f and g share the set S CM. On the other hand, if Ef (S) = Eg(S),we say that f and g share the set S IM. Evidently, if S contains only one element,then it coincides with the usual definition of CM (respectively, IM) shared values.

Let m be a positive integer or infinity and a ∈ C ∪ ∞. We denote byEm)(a; f) the set of all a-points of f with multiplicities not exceeding m, wherean a-point is counted according to its multiplicity. If, for some a ∈ C ∪ ∞,E∞)(a; f)=E∞)(a; g), we say that f , g share the value a CM. For a set S of

distinct elements of C, we define Em)(S, f) =∪a∈S

Em)(a, f). The condition

Em)(S, f) = Em)(S, g) obviously implies Ej)(S, f) = Ej)(S, g) for all 1 ≤ j ≤ m.

Inspired by the Nevanlinna’s three and four values theorems, in 1970s F. Grossand C.C. Yang started to study the similar but more general questions of twofunctions that share sets of distinct elements instead of values. For instance, theyproved that if f and g are two non-constant entire functions and S1, S2 and S3 arethree distinct finite sets such that f−1(Si) = g−1(Si) for i = 1, 2, 3, then f ≡ g.In 1976, F. Gross proposed the following question in [8]:

Question A Can one find two finite sets Sj (j = 1, 2) such that any twonon-constant entire functions f and g satisfying Ef (Sj) = Eg(Sj) for j = 1, 2must be identical ?

In [8], Gross wrote: If the answer of Question A is affirmative it would beinteresting to know how large both sets would have to be ?

Yi [21] and independently Fang and Xu [7] gave the same answer in thisdirection.

In 2003, Lin and Yi posed the following question.

Question B ([19]) Can one find two finite sets Sj (j = 1, 2) such that anytwo non-constant meromorphic functions f and g satisfying Ef (Sj) = Eg(Sj) forj = 1, 2 must be identical ?

Gradually, the research on Question B gained pace and today it has becomeone of the most prominent branches of the uniqueness theory. For the last twodecades several attempts have been made by different authors to consider theshared value problems relative to a meromorphic function sharing two sets andat the same time give affirmative answers to Question B under weaker hypothesissee [1]-[7], [10], [14]-[21], [28]-[28], [29]-[30].

some uniqueness results on meromorphic functions ... 17

In 1994, Yi [20] gave an affirmative answer to Question B and proved thatthere exist two finite sets S1 (with 2 elements) and S2 (with 9 elements) such thatany two non-constant meromorphic functions f and g satisfying Ef (Sj) = Eg(Sj)for j = 1, 2 must be identical.

Inspired by Question B, we will consider the uniqueness of two non-constantmeromorphic functions satisfying Ef (S) = Eg(S) and Ef (∞) = Eg(∞).According to what we know by now, perhaps this type of results were first ob-tained by Li and Yang in [17], where they proved that there exists one finite setS with 15 elements such that any two non-constant meromorphic functions satis-fying Ef (S) = Eg(S) and Ef (∞) = Eg(∞) must be identical. In 1995, Yi [21]and independently Li and Yang [17] proved that there exists a set S of 11 elementssuch that any two non-constant meromorphic functions with Ef (S) = Eg(S) andEf (∞) = Eg(∞) must be identical. In 1997 Fang and Guo in [6] exhibited aset S of nine elements with this property.

In 2002, Yi [25] proved the following result in which he not only reduced thecardinalities of the set S but also relaxed the sharing of the poles from CM to IM.

Theorem A. ([25]; see also [29]) Let n be a positive integer such that n ≥ 8,and let a, b be two nonzero complex numbers satisfying abn−2 = 2. Then thepolynomial

(1.1) P (w) = awn − n(n− 1)w2 + 2n(n− 2)bw − (n− 1)(n− 2)b2

has only simple zeros. Let S = w | P (w) = 0. If f and g are two non-constantmeromorphic functions satisfying Ef (S) = Eg(S) and Ef (∞) = Eg(∞) thenf ≡ g.

Dealing with the question of Gross in [6], Fang and Lahiri obtained a uniquerange set S with smaller cardinalities than that obtained previously imposingsome restrictions on the poles of f and g.

Theorem B. ([6]) Let S = z : zn + azn−1 + b = 0 where n (≥ 7) is an integerand a and b are two nonzero constants such that zn+azn−1+b = 0 has no multipleroot. If f and g are two non-constant meromorphic functions having no simplepoles such that Ef (S) = Eg(S) and Ef (∞) = Eg(∞) then f ≡ g.

Let S = z : z7 − z6 − 1 = 0 and

f =ez + e2z + . . .+ e6z

1 + ez + . . .+ e6z, g =

1 + ez + . . .+ e5z

1 + ez + . . .+ e6z

Obviously, f = ezg, Ef (S) = Eg(S) and Ef (∞) = Eg(∞) but f ≡ g. So, forthe validity of Theorem B, f and g must not have any simple pole.

In 2001, an idea of gradation of sharing known as weighted sharing has beenintroduced by I. Lahiri in [12], [13] which measures how close a shared value isto being shared CM or to being shared IM. In the following definition we explainthe notion.


Definition 1.1 [12], [13] Let k be a nonnegative integer or infinity. Fora ∈ C ∪ ∞ we denote by Ek(a; f) the set of all a-points of f , where ana-point of multiplicity m is counted m times if m ≤ k and k + 1 times if m > k.If Ek(a; f) = Ek(a; g), we say that f, g share the value a with weight k.

We write f , g share (a, k) to mean that f , g share the value a with weight k.Clearly if f , g share (a, k) then f, g share (a, p) for any integer p, 0 ≤ p < k. Alsowe note that f, g share a value a IM or CM if and only if f, g share (a, 0) or (a,∞)respectively.

Definition 1.2 [12] Let S be a set of distinct elements of C ∪ ∞ and k be a

nonnegative integer or ∞. We denote by Ef (S, k) the set∪a∈S

Ek(a; f).

Clearly, Ef (S) = Ef (S,∞) and Ef (S) = Ef (S, 0).

Using the notion of weighted sharing of sets, Lahiri [14] proved the followingtheorem, which improved Theorem B.

Theorem C. ([14]) Let S be defined as in Theorem B and n (≥ 7) be an integer.If for two non-constant meromorphic functions f and g, Θ(∞; f) +Θ(∞; g) > 1,Ef (S, 2) = Eg(S, 2) and Ef (∞,∞) = Eg(∞,∞) then f ≡ g.

Recently, the first author [3] has generalized Theorem C by investigating theproblem of further relaxation of the nature of sharing the set ∞ and obtainedthe following result.

Theorem D. ([3]) Let S be defined as in Theorem B and n (≥ 7) be an integer.If, for two non-constant meromorphic functions f and g, Θ(∞; f) + Θ(∞; g) >

1 +29

6nk + 6n− 5, Ef (S, 2) = Eg(S, 2) and Ef (∞, k) = Eg(∞, k), where

0 ≤ k < ∞, then f ≡ g.

In the mean time, the first author [1] has improved Theorem B by relaxingthe nature of sharing the set S and proved the following result.

Theorem E. Let S be defined as in Theorem B. If for two non-constant mero-

morphic functions f and g, Θ(∞; f) >1

2, Θ(∞; g) >

1

2and E3)(S, f) = E3)(S, g),

Ef (∞,∞) = Eg(∞,∞), then f ≡ g.

In the paper, we consider a new range set different from those mentionedearlier and with the help of the same we will improve and supplement TheoremsD and E.

The following theorems are the main results of the paper.


Theorem 1.1 Let

S =

z :

(n− 1)(n− 2)

4zn − n(n− 2)

2zn−1 +

n(n− 1)

4zn−2 − 1 = 0

,

where n (≥ 7) is an integer. If, for two non-constant meromorphic functions fand g, Ef (S, 2) = Eg(S, 2) and Ef (∞, k) = Eg(∞, k), where 0 ≤ k < ∞

and minΘf ,Θg > 4 − n

2+

3

nk + n− 1, then f ≡ g, where Θf = 2 Θ(0; f) +

2 Θ(1; f) + Θ(∞; f) and Θg can be similarly defined.

Corollary 1.1 Let S be given as in Theorem 1.1. If, for two non-constant mero-morphic functions f and g, Ef (S, 2) = Eg(S, 2), Ef (∞,∞) = Eg(∞,∞) and

minΘf ,Θg > 4 − n

2, where Θf and Θg have the same meaning as in Theorem

1.1, then f ≡ g.

Theorem 1.2 Let

S =

z :

(n− 1)(n− 2)

4zn − n(n− 2)

2zn−1 +

n(n− 1)

4zn−2 − 1 = 0

,

where n (≥ 8) is an integer. If for two non-constant meromorphic functions fand g Ef (S, 2) = Eg(S, 2) and Ef (∞, 2) = Eg(∞, 2), then f ≡ g.

Theorem 1.3 Let S be defined as in Theorem 1.1. If for two non-constant mero-morphic functions f and g, E3)(S, f) = E3)(S, g), Ef (∞,∞) = Eg(∞,∞)

and minΘf ,Θg > 4− n

2, where Θf and Θg have the same meaning as in Theo-

rem 1.1, then f ≡ g.

Theorem 1.4 Let S be defined as in Theorem 1.1. If for two non-constant mero-morphic functions f and g, Em)(S, f) = Em)(S, g), Ef (∞, k) = Eg(∞, k),

where 0 ≤ k < ∞, m ≥ 4 and minΘf ,Θg > 4− n

2+

3

nk + n− 1, where Θf and

Θg have the same meaning as in Theorem 1.1 then f ≡ g.

It is assumed that the readers are familiar with the standard definitions andnotations of the value distribution theory as those are available in [9]. We are stillgoing to explain some notations as these are used in the paper.

Definition 1.3 [11] For a value a in the extended complex plane, we denote byN(r, a; f |= 1) the reduced counting function of simple a points of f in |z| < r.Denote by N(r, a; f |≤ m) (N(r, a; f |≥ m), respectively) the counting functionof those a-points of f in |z| < r, where the multiplicity of each point is notgreater (not less, respectively) than m, m is a positive integer, and each point iscounted according to its multiplicity. Denote by N(r, a; f |< m) (N(r, a; f |> m),respectively) the counting function of those a-points of f in |z| < r, where themultiplicity of each point is less (greater, respectively) than m, and each point iscounted according to its multiplicity. Denote by N(r, a; f |≤ m), N(r, a; f |≥ m),N(r, a; f |< m), N(r, a; f |> m) the reduced forms of N(r, a; f |≤ m),N(r, a; f |≥ m), N(r, a; f |< m), N(r, a; f |> m) respectively.


Definition 1.4 Let f and g be two non-constant meromorphic functions suchthat f and g share a value a IM where a ∈ C ∪ ∞. Let z0 be an a-point of fwith multiplicity p, an a-point of g with multiplicity q. We denote by NL(r, a; f)(NL(r, a; g)) the counting function of those a-points of f and g where p > q(q > p), each a-point is counted only once.

Definition 1.5 Let f and g be two non-constant meromorphic functions and mbe a positive integer such that Em)(a; f) = Em)(a; g) where a ∈ C ∪ ∞. Let z0be an a-point of f with multiplicity p > 0, an a-point of g with multiplicity q > 0.

We denote by Nm)

L (r, a; f) (Nm)

L (r, a; g)) the counting function of those commona-points of f and g, where p > q (q > p), each a-point is counted only once.

Definition 1.6 [13] We denote by N2(r, a; f) = N(r, a; f) +N(r, a; f |≥ 2).

Definition 1.7 Let m be a positive integer. Also let z0 be a zero of f(z)− a ofmultiplicity p and a zero of g(z)−a of multiplicity q. We denote byN f≥m+1(r, a; f |g = a) (N g≥m+1(r, a; g | f = a)) the reduced counting functions of those a-pointsof f and g, where p ≥ m+ 1 and q = 0 (q ≥ m+ 1 and p = 0).

Definition 1.8 [8], [9] Let f , g share (a, 0). We denote by N∗(r, a; f, g) thereduced counting function of those a-points of f whose multiplicities differ fromthe multiplicities of the corresponding a-points of g.

Clearly, N∗(r, a; f, g) = N∗(r, a; g, f) and N∗(r, a; f, g) = NL(r, a; f)+NL(r, a; g).For Em)(a; f) = Em)(a; g) we can define N∗(r, a; f, g) in a similar manner and we

note that here N∗(r, a; f, g) = Nm)

L (r, a; f)+Nm)

L (r, a; g)+N f≥m+1(r, a; f | g = a)+N g≥m+1(r, a; g | f = a).

2. Lemmas

In this section, we present some lemmas which will be needed in the sequel. LetF and G be two non-constant meromorphic functions defined in C. Henceforthwe shall denote by H and V the following two functions

H =

(F

′′

F ′ − 2F′

F − 1

)−

(G

′′

G′ − 2G′

G− 1

)and

V =

(F

′

F − 1− F ′

F

)−(

G′

G− 1− G′

G

)=

F ′

F (F − 1)− G′

G(G− 1).

Lemma 2.1 [13] If F , G be two non-constant meromorphic functions such thatthey share (1, 1) and H ≡ 0, then

N(r, 1;F |= 1) = N(r, 1;G |= 1) ≤ N(r,H) + S(r, F ) + S(r,G).


Lemma 2.2 [19] If F , G be two non-constant meromorphic functions such thatE1)(1;F ) = E1)(1;G) and H ≡ 0 then

N(r, 1;F |= 1) ≤ N(r,H) + S(r, F ) + S(r,G).

Lemma 2.3 Let f and g be two non-constant meromorphic functions sharing(1,m), where 1 ≤ m < ∞. Then

N(r, 1; f) +N(r, 1; g)−N(r, 1; f |= 1) +

(m− 1

2

)N∗(r, 1; f, g)

≤ 1

2[N(r, 1; f) +N(r, 1; g)]

Proof. Let z0 be a 1- point of f of multiplicity p and a 1-point of g of multiplicityq. Since f , g share (1,m), we note that the 1-points of f and g up to multiplicitym are same. When p = q = 1, z0 is counted once, both in left and right hand sideof the above inequality but when 2 ≤ p = q ≤ m, z0 is counted 2 times in theleft hand side of the above inequality whereas it is counted p times in the righthand side of the above inequality. If p = m+ 1, then the possible values of q areas follows. (i) q = m + 1, (ii) q ≥ m + 2. When p = m + 2, then q can take thefollowing possible values (i) q = m + 1, (ii) q = m + 2, (iii) q ≥ m + 3. Similarexplanations hold if we interchange p and q. Clearly, when p = q ≥ m + 1, z0 iscounted 2 times in the left hand side and p ≥ m+ 1 times in the right hand sideof the above inequality. If p > q ≥ m+ 1, in view of Definition 1.8 we know z0 is

counted m +3

2times in the left hand side of the inequality and

p+ q

2≥ m +

3

2times in the right hand side of the above inequality. If q > p, we can explainsimilarly. Hence the lemma follows.

Lemma 2.4 Let f and g be two non-constant meromorphic functions such thatEm)(1; f) = Em)(1; g), where 1 ≤ m < ∞. Then

N(r, 1; f) +N(r, 1; g)−N(r, 1; f |= 1) +

(m

2− 1

2

)N f≥m+1(r, 1; f | g = 1)

+N g≥m+1(r, 1; g | f = 1)+

(m− 1

2

)N

m)

L (r, 1; f) +Nm)

L (r, 1; g)

≤ 1

2[N(r, 1; f) +N(r, 1; g)]

Proof. By the condition that Em)(1; f) = Em)(1; g), we note that every commonzero of f − 1 and g − 1 up to multiplicity m has the same multiplicities relatedto f and g. Let z0 be a 1-point of f with multiplicity p and a 1-point of gwith multiplicity q. If p = m + 1, then the possible values of q are as follows:(i) q = m + 1; (ii) q ≥ m + 2; (iii) q = 0. Similarly, if p = m + 2, the possiblevalues of q are as follows: (i) q = m + 1; (ii) q = m + 2; (iii) q ≥ m + 3;(iv) q = 0. If p ≥ m + 3, we can find the possible values of q similarly. Now,Lemma 2.4 follows from the above explanation.


Lemma 2.5 Suppose that F , G share (1, 0), (∞, 0) and a is a complex numbersatisfying a = 0, 1. If H ≡ 0 then

N(r,H) ≤ N(r, 0;F |≥ 2) +N(r, 0;G |≥ 2) +N(r, a;F |≥ 2) +N(r, a;G |≥ 2)

+N∗(r, 1;F,G) +N∗(r,∞;F,G) +N0(r, 0;F′) +N0(r, 0;G

′),

where N0(r, 0;F′) is the reduced counting function of those zeros of F

′which are

not the zeros of F (F − 1)(F − a) and N0(r, 0;G′) is similarly defined.

Proof. By the definition of H we verify that the possible poles of H resultfrom the following six cases: (i) The multiple zeros of F and G. (ii) The multiplea- points of F and G. (iii) Those common poles of F and G, where each such poleof F and G has different multiplicities related to F and G. (iv) Those common1-points of F and G, where each such point has different multiplicities related toF and G. (v) The zeros of F ′ which are not zeros of F (F − 1)(F − a). (vi) Thezeros of G′ which are not zeros of G(G − 1)(G − a). Now proceeding as in theproof of Lemma 2.4, we can get the result of the lemma.

Lemma 2.6 Let Em)(1;F ) = Em)(1;G), let F , G share (∞, 0) and let a be acomplex number satisfying a = 0, 1. If H ≡ 0, then

N(r,H) ≤ N(r, 0;F |≥ 2) +N(r, 0;G |≥ 2) +N(r, a;F |≥ 2) +N(r, a;G |≥ 2)

+Nm)

L (r, 1;F ) +Nm)

L (r, 1;G) +NF≥m+1(r, 1;F | G = 1)

+NG≥m+1(r, 1;G | F = 1) +N∗(r,∞;F,G) +N0(r, 0;F′) +N0(r, 0;G

′),

where N0(r, 0;F′) and N0(r, 0;G

′) has the same meaning as in Lemma 2.5.

Proof. The proof is obvious.

Lemma 2.7 [18] Let f be a non-constant meromorphic function and P (f) =a0 + a1f + a2f

2 + . . . + anfn, where a0, a1, a2 . . . , an are constants and an = 0.

Then T (r, P (f)) = nT (r, f) +O(1).

Next, we set

F =(n− 1)(n− 2)

4fn − n(n− 2)

2fn−1 +

n(n− 1)

4fn−2,(2.1)

G =(n− 1)(n− 2)

4gn − n(n− 2)

2gn−1 +

n(n− 1)

4gn−2,(2.2)

where f , g are two non-constant meromorphic functions and n ≥ 3 is an integer.

Lemma 2.8 Let F , G be given by (2.1) and (2.2), where n ≥ 7 is an integer andH ≡ 0. Suppose α1 and α2 are the roots of the equation

(n− 1)(n− 2)

4z2 − n(n− 2)

2z +

n(n− 1)

4= 0.


If F , G share (1,m) and f , g share (∞, k), where 2 ≤ m < ∞. Then, for acomplex number a(= 0, 1),

3n

2T (r, f) + T (r, g) ≤ 2N(r, 0; f) +N(r, 0; g)+N2(r, α1; f)

+N2(r, α1; g) +N2(r, α2; f) +N2(r, α2; g)

+N2(r, a;F ) +N2(r, a;G) +N(r,∞; f)

+N(r,∞; g) +N∗(r,∞; f, g)−(m− 3

2

)N∗(r, 1;F,G) + S(r, f) + S(r, g).

Proof. By the second fundamental theorem we get

(2.3)

2T (r, F ) + T (r,G) ≤ N(r, 1;F ) +N(r, 0;F ) +N(r, a;F )

+N(r,∞;F ) +N(r, 1;G) +N(r, 0;G) +N(r, a;G) +N(r,∞;G)

−N0(r, 0;F′)−N0(r, 0;G

′) + S(r, F ) + S(r,G).

Using Lemmas 2.1, 2.3, 2.5 and 2.7 we see that

N(r, 1;F )+N(r, 1;G) ≤ 1

2[N(r, 1;F ) +N(r, 1;G)](2.4)

+N(r, 1;F |= 1)−(m− 1

2

)N∗(r, 1;F,G)

≤ n

2T (r, f) + T (r, g)+N(r, 0; f) +N(r, 0; g)

+N(r, α1; f |≥ 2) +N(r, α2; f |≥ 2)

+N(r, α1; g |≥ 2) +N(r, α2; g |≥ 2)

+N(r, a;F |≥ 2) +N(r, a;G |≥ 2)

+N∗(r,∞; f, g)−(m− 3

2

)N∗(r, 1;F,G)

+N0(r, 0;F′) +N0(r, 0;G

′) + S(r, f) + S(r, g).

Using (2.4) in (2.3) the lemma follows in view of Definition 1.6.

Lemma 2.9 Let F , G be given by (2.1) and (2.2), where n ≥ 7 is an integer andH ≡ 0. Suppose αi, i = 1, 2 has the same meaning as given in Lemma 2.8. IfEm)(1;F ) = Em)(1;G) and f , g share (∞, k), where m, k are integers such that1 ≤ m < ∞ and k ≥ 0. Then for a complex number a(= 0, 1)

3n

2T (r, f) + T (r, g) ≤ 2N(r, 0; f) +N(r, 0; g)+N2(r, α1; f)

+N2(r, α1; g) +N2(r, α2; f) +N2(r, α2; g)

+N2(r, a;F ) +N2(r, a;G) +N(r,∞; f)

+N(r,∞; g) +N∗(r,∞; f, g)


−(m

2− 3

2

)NF≥m+1(r, 1;F | G = 1) +NG≥m+1(r, 1;G | F = 1)

−(m− 3

2

)N

m)

L (r, 1;F ) +Nm)

L (r, 1;G)+ S(r, f) + S(r, g).

Proof. Using Lemmas 2.2, 2.4, 2.6 and 2.7 we obtain

(2.5)

N(r, 1;F ) +N(r, 1;G)

≤ 1

2[N(r, 1;F ) +N(r, 1;G)] +N(r, 1;F |= 1)

−(m

2− 1

2

)N f≥m+1(r, 1; f | g = 1) +N g≥m+1(r, 1; g | f = 1)

−(m− 1

2

)N

m)

L (r, 1; f) +Nm)

L (r, 1; g)

≤ n

2T (r, f) + T (r, g)+N(r, 0; f) +N(r, 0; g)

+N(r, α1; f |≥ 2) +N(r, α2; f |≥ 2)

+N(r, α1; g |≥ 2) +N(r, α2; g |≥ 2) +N(r, a;F |≥ 2)

+N(r, a;G |≥ 2) +N∗(r,∞; f, g)

−(m

2− 3

2

)NF≥m+1(r, 1;F | G = 1) +NG≥m+1(r, 1;G | F = 1)

−(m− 3

2

)N

m)

L (r, 1;F ) +Nm)

L (r, 1;G)

+N0(r, 0;F′) +N0(r, 0;G

′) + S(r, f) + S(r, g).

Using (2.5) in (2.3), the lemma follows in view of Definition 1.6.

Lemma 2.10 Let f , g be two non-constant meromorphic functions and supposeαi, i = 1, 2 has the same meaning as given in Lemma 2.8. Then

(n− 1)2(n− 2)2fn−2(f − α1)(f − α2)gn−2(g − α1)(g − α2) ≡ b,

where b is a non-zero constant and n ≥ 5 is an integer.

Proof. On the contrary, suppose that

(2.6) (n− 1)2(n− 2)2fn−2(f − α1)(f − α2)gn−2(g − α1)(g − α2) ≡ b.

Let z0 be a zero of f with multiplicity p. Then z0 is a pole of g with multiplicityq such that

(n− 2)p = (n− 2)q + 2q = nq.(2.7)

From (2.7) we see that 2q = (n− 2)(p− q) ≥ n− 2 and so p =n

n− 2q ≥ n

2.


Let z0 be a zero of f − αi i = 1, 2 with multiplicity p. Then z0 is a pole of gwith multiplicity q such that p = (n− 2)q + 2q = nq ≥ n.

Since the poles of f are the zeros of g and g − αi i = 1, 2, we get

N(r,∞; f) ≤ N(r, 0; g) +N(r, α1; g) +N(r, α2; g)

≤ 2

nN(r, 0; g) +

1

nN(r, α1; g) +

1

nN(r, α2; g)

≤ 4

nT (r, g).

By the second fundamental theorem we get

2T (r, f) ≤ N(r, 0; f) +N(r, α1; f) +N(r, α2; f) +N(r,∞; f) + S(r, f)

≤ 2

nN(r, 0; f) +

1

nN(r, α1; f) +

1

nN(r, α2; f) +

4

nT (r, g) + S(r, f)

≤ 4

nT (r, f) +

4

nT (r, g) + S(r, f).

i.e., (2− 4

n

)T (r, f) ≤ 4

nT (r, g) + S(r, f).(2.8)

Similarly, (2− 4

n

)T (r, g) ≤ 4

nT (r, f) + S(r, g)(2.9)

Adding (2.8) and (2.9) we get(2− 8

n

)T (r, f) + T (r, g) ≤ S(r, f) + S(r, g),

a contradiction for n ≥ 5. This proves the lemma.

Lemma 2.11 [5] Let f , g be two non-constant meromorphic functions and sup-pose n (≥ 6) is an integer. If

(2.10)

(n− 1)(n− 2)

2fn − n(n− 2)fn−1 +

n(n− 1)

2fn−2

≡ (n− 1)(n− 2)

2gn − n(n− 2)gn−1 +

n(n− 1)

2gn−2,

then f ≡ g.

Lemma 2.12 [22] If F , G share (∞, 0) and V ≡ 0 then F ≡ G.


Lemma 2.13 Let F , G be given by (2.1) and (2.2), where n ≥ 7 is an integerand V ≡ 0. If F , G share (1, 2), f , g share (∞, k), where 0 ≤ k < ∞, then thepoles of F and G are zeros of V and

(nk + n− 1) N(r,∞; f |≥ k + 1) = (nk + n− 1)N(r,∞; g |≥ k + 1)

≤ N(r, 0; f) +N(r, α1; f) +N(r, α2; f)

+N(r, 0; g) +N(r, α1; g) +N(r, α2; g)

+N∗(r, 1;F,G) + S(r, f) + S(r, g),

where αi i = 1, 2 has the same meaning as in Lemma 2.8.

Proof. Since f , g share (∞; k), it follows that F , G share (∞;nk) and so a poleof F with multiplicity p(≥ nk + 1) is a pole of G with multiplicity r(≥ nk + 1)and vice versa. We note that F and G have no pole of multiplicity q wherenk < q < nk + n. Now using the Milloux theorem [9, p. 55] and Lemma 2.7 weget from the definition of V

m(r, V ) = S(r, f) + S(r, g).

Hence

(nk + n− 1)N(r,∞; f |≥ k + 1) = (nk + n− 1)N(r,∞; g |≥ k + 1)

= (nk + n− 1)N(r,∞;F |≥ nk + n)

≤ N(r, 0;V )

≤ T (r, V ) +O(1)

≤ N(r,∞;V ) +m(r, V ) +O(1)

≤ N(r,∞;V ) + S(r, f) + S(r, g)

≤ N(r, 0;F ) +N(r, 0;G) +N∗(r, 1;F,G)

+S(r, f) + S(r, g)

≤ N(r, 0; f) +N(r, α1; f) +N(r, α2; f)

+N(r, 0; g) +N(r, α1; g) +N(r, α2; g)

+N∗(r, 1;F,G) + S(r, f) + S(r, g).

This proves the lemma.

Lemma 2.14 Let F , G be given by (2.1), where n ≥ 7 is an integer. Also let Sbe given as in Theorem 1.1. If Ef (S, 0) = Eg(S, 0), then S(r, f) = S(r, g).

Proof. Since Ef (S, 0) = Eg(S, 0), it follows that F and G share (1, 0). We firstnote that the polynomial

p(z) =(n− 1)(n− 2)

4zn − n(n− 2)

2zn−1 +

n(n− 1)

4zn−2 − 1

has only simple zeros. In fact,

p′(z) =

n(n− 1)(n− 2)

4zn−3(z − 1)2.


Also we note that p(0), p(1) = 0. Thus all the zeros of p(z) are simple andwe denote them by wj, j = 1, 2, ..., n. Since F , G share (1, 0) from the secondfundamental theorem we have

(n− 2)T (r, g) ≤n∑

j=1

N (r, wj; g) + S(r, g)

=n∑

j=1

N (r, wj; f) + S(r, g)

≤ nT (r, f) + S(r, g).

Similarly, we can deduce

(n− 2)T (r, f) ≤ nT (r, g) + S(r, f).

The last inequalities imply T (r, f) = O (T (r, g)) and T (r, g) = O (T (r, f)) and sowe have S(r, f) = S(r, g).

3. Proofs of the theorems

Proof of Theorem 1.1. Let F , G be given by (2.1) and (2.2). Since Ef (S, 2) =Eg(S, 2) and Ef (∞, k) = Eg(∞, k) it follows that F , G share (1, 2) and(∞, nk + n − 1). So N∗(r,∞; f, g) = N∗(r,∞;F,G) ≤ N(r,∞;F |≥ nk + n) =N(r,∞; f |≥ k + 1). By a simple computation it can be easily seen that 1 is a

root with multiplicity 3 of F − 1

2and hence F − 1

2= (f − 1)3 Qn−3(f), where

Qn−3(f) is a polynomial in f of degree n− 3 and thus

N2

(r,1

2;F

)≤ 2N(r, 1; f) +N (r, 0;Qn−3(f))

≤ 2N(r, 1; f) + (n− 3)T (r, f) + S(r, f).

Suppose that H ≡ 0. Then F ≡ G. So, it follows from Lemma 2.12 that V ≡ 0.

Hence, from Lemma 2.8 with a =1

2, m = 2, and Lemma 2.13, we obtain for

ε(> 0)(n2+ 1

)T (r, f) + T (r, g)

≤ 2N(r, 0; f) +N(r, 0; g) +N(r, 1; f) +N(r, 1; g)

+N(r,∞; f)

+N(r,∞; g) +N∗(r,∞; f, g)− 1

2N∗(r, 1;F,G) + S(r, f) + S(r, g)

≤ (5− 2Θ(0; f)− 2Θ(1; f)−Θ(∞; f) + ε)T (r, f)

+ (5− 2Θ(0; g)− 2Θ(1; g)−Θ(∞; g) + ε) T (r, g)

+1

nk + n− 1[3T (r, f) + 3T (r, g)] + S(r, f) + S(r, g).


That is

(3.1)

(n

2− 4 + Θf −

3

nk + n− 1− ε

)T (r, f)

+

(n

2− 4 + Θg −

3

nk + n− 1− ε

)T (r, g)

≤ S(r, f) + S(r, g).

Without loss of generality, we may suppose that there exists a set I with infinitelinear measure such that

T (r, g) ≤ T (r, f), r ∈ I.

From (3.1) and Lemma 2.14, we have[Θf +Θg − 8 + n− 6

nk + n− 1− 2ε

]T (r, g) ≤ S(r, g), r ∈ I\E,

which leads to a contradiction for sufficiently small ε > 0. Hence H ≡ 0. Then

(3.2) F ≡ aG+ b

cG+ d,

where a, b, c, d are constants such that ad− bc = 0. Also

(3.3) T (r, F ) = T (r,G) +O(1).

We now consider the following cases.

Case I. Let ac = 0. From (3.2) we get

(3.4) N(r,∞;G) = N(r,a

c;F

).

So, in view of (3.3), by the second fundamental theorem we get

T (r, F ) ≤ N(r, 0;F ) +N(r,∞;F ) +N(r,a

c;F

)+ S(r, F )

= N(r, 0; f) + 2T (r, f) +N(r,∞; f) +N(r,∞; g) + S(r, f)

≤ 5T (r, f) + S(r, f),

as r ∈ I and r −→ ∞. This, together with Lemma 2.7, gives n ≤ 5, whichcontradicts the condition n ≥ 7.

Case II. Let a = 0 and c = 0. Then F = αG+ β, where α =a

dand β =

b

d.

If F has no 1-point, by the second fundamental theorem we get

T (r, F ) ≤ N(r, 0;F ) +N(r,∞; f) + S(r, F )

≤ 3T (r, f) +N(r,∞; f) + S(r, f),



If F and G have a common 1-point, then α + β = 1, and so

(3.5) F ≡ αG+ 1− α.

Suppose α = 1. If 1−α = 12, then in view of (3.3) and the second fundamental

theorem we get

2T (r, F ) ≤ N(r, 0;F ) +N(r, 1− α;F ) +N

(r,1

2;F

)+N(r,∞;F ) + S(r, F )

≤ 3T (r, f) +N(r, 0;G) + (n− 2)T (r, f) +N(r,∞; f) + S(r, f)

≤ (n+ 5)T (r, f) + S(r, f),

as r ∈ I and r −→ ∞. This, together with Lemma 2.7, gives n ≤ 5, whichcontradicts the condition n ≥ 7. If α = 1

2, then we have from (3.5)

F ≡ 1

2(G+ 1).

So, by the second fundamental theorem we can obtain, using (3.3), that

2T (r,G) ≤ N(r, 0;G) +N

(r,1

2;G

)+N(r,−1;G) +N(r,∞;G) + S(r,G)

≤ 3T (r, g) + (n− 2)T (r, g) +N(r, 0;F ) +N(r,∞; g) + S(r, g)

≤ (n+ 5)T (r, g) + S(r, g),


So α = 1 and hence F ≡ G. So, by Lemma 2.11, we get f ≡ g.

Case III. Let a = 0 and c = 0. Then F ≡ 1

γG+ δ, where γ =

c

band δ =

d

b.

If F has no 1-points, then as in Case II we can deduce a contradiction. If F andG have a common 1-point, then γ + δ = 1 and so

(3.6) F ≡ 1

γG+ 1− γ.

Suppose γ = 1 If γ = −1, then by the second fundamental theorem we get

2 T (r, F ) ≤ N(r, 0;F ) +N(r,1

1− γ;F ) +N

(r,1

2;F

)+N(r,∞; f) + S(r, f)

≤ 3T (r, f) +N(r, 0;G) + (n− 2)T (r, f) +N(r,∞; f) + S(r, f)

≤ (n+ 5)T (r, f) + S(r, f),

as r ∈ I and r −→ ∞. This, together with Lemma 2.7, gives n ≤ 5, whichcontradicts the condition n ≥ 7. If γ = −1 from (3.6) we have

F ≡ 1

−G+ 2.


Now, the second fundamental theorem with the help of (3.3) yields

2T (r,G) ≤ N(r, 0;G) +N

(r,1

2;G

)+N(r, 2;G) +N(r,∞;G) + S(r,G)

≤ 3T (r, g) + (n− 2)T (r, g) +N(r,∞;F ) +N(r,∞;G) + S(r, g)

≤ (n+ 3)T (r, g) + S(r, g),

as r ∈ I and r −→ ∞. This, together with Lemma 2.7, gives n ≤ 3, whichcontradicts the condition n ≥ 7. So, we must have γ = 1 which implies FG ≡ 1,which is impossible by Lemma 2.10. This completes the proof of the theorem.

Proof of Corollary 1.1. Let F , G be given by (2.1) and (2.2). Since Ef (S, 2) =Eg(S, 2) and Ef (∞,∞) = Eg(∞,∞), it follows that f , g share (∞, k) for alllarge k. Also since minΘf ,Θg > 4 − n

2, for sufficiently large k we can have

minΘf ,Θg > 4− n2+ 3

nk+n−1and hence, by Theorem 1.1, we get the conclusion

of Corollary 1.1. So, Corollary 1.1 can be treated as a special case of Theorem 1.1.

Proof of Theorem 1.2. Let F , G be given by (2.1) and (2.2). Since Ef (S, 2) =Eg(S, 2) and Ef (∞, 2) = Eg(∞, 2) it follows that F , G share (1, 2) and(∞, 3n − 1). Suppose that H ≡ 0. Now proceeding in the same way as done in

the proof of Theorem 1.1, using Lemma 2.8 with a =1

2, m = 2 and Lemma 2.13

for k = 0 and k = 2 we obtain(n2+ 1

)T (r, f) + T (r, g)

≤ 2N(r, 0; f) +N(r, 0; g) +N(r, 1; f) +N(r, 1; g)

+ 2N(r,∞; f)

+N(r,∞; f |≥ 3)− 1

2N∗(r, 1;F,G) + S(r, f) + S(r, g)

≤ 4 T (r, f) + T (r, g)+ 6

n− 1T (r, f) + T (r, g)

+3

3n− 1T (r, f) + T (r, g)+ S(r, f) + S(r, g),

that is(n

2− 3− 6

n− 1− 3

3n− 1

)T (r, f) + T (r, g) ≤ S(r, f) + S(r, g).(3.7)

Clearly (3.7) implies a contradiction for n ≥ 8 and hence H ≡ 0 and the rest ofthe theorem can be proved in the line of proof of Theorem 1.1.

Proof of Theorem 1.3. Let F , G be given by (2.1) and (2.2). E3)(S, f) =E3)(S, g), Ef (∞,∞) = Eg(∞,∞) it follows that E3)(1, F ) = E3)(1;G), andF G share (∞,∞). We omit the detail proof since using Lemmas 2.9, 2.10 and2.11 the proof of the theorem can be carried out along the line of the proof ofTheorem 1.1.


Proof of Theorem 1.4. Let F , G be given by (2.1) and (2.2). Em)(S, f) =Em)(S, g), Ef (∞, k) = Eg(∞, k) it follows that Em)(1, F ) = Em)(1;G), andF G share (∞, nk + n− 1). We omit the detail proof since the same can be donein the line of proof of Theorem 1.1.

Acknowledgement. The authors wish to thank the referees for their valuablecomments and suggestions.

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Accepted: 1.08.2010


A NOTE ON P -REGULAR SEMIRINGS

M.K. Dubey

SAG, Metcalfe HouseDRDO ComplexNear Civil LinesDelhi-110054Indiae-mail: [email protected]

Abstract. The notion of P−regular semiring is introduced and characterization of the

same has been given. We also study the representation of the elements in terms of

quasi-ideals of weak P−regular semirings relative to the rightk−ideal P.

2000 Mathematical Subject Classification: 16Y30.


Throughout this paper S will denote a semiring. A semiring is a commutativemonoid (S,+, 0S) having additive identity zero 0S and a semigroup (S, ·) whichare connected by ring like distributivity and 0S · x = x · 0S = 0S for all x ∈ S.A left (right) ideal of a semiring S is a non-empty subset I of S such that a+b ∈ Iand ra(ar) ∈ I for all a, b ∈ I and r ∈ S. An ideal of a semiring S is a non-emptysubset I of S such that I is both left and right ideal of S. A left (right)ideal Iof S is called a left (right)k−ideal if x + y ∈ I, x ∈ S, y ∈ I imply that x ∈ I.An ideal I of a semiring S is called a k−ideal if it is both left k−ideal and rightk−ideal. If A is an ideal(resp. left, right) of a semiring S then

A = a ∈ S : a+ x ∈ A for some x ∈ A

is called k−closure of A. It can easily be verified that an ideal (resp. left, right) Aof S is a k−ideal if and only if A=A. An additive subsemigroup Q of a semiring Sis called a quasi-ideal of Sif QS∩SQ ⊆ Q. Clearly, every quasi-ideal of a semiringS is a subsemiring of S.

34 m.k. dubey

2. P−regular semirings

In this section, we study the concept of P−regular semiring and some propertiesrelated to the same.

Definition 2.1. Let S be a semiring and P be a right k− ideal of S. Thensemiring S is said to be a P−regular if for each a ∈ S, there exists x ∈ S suchthat a+ p1 = axa+ p2 for some p1, p2 ∈ P and axP ⊆ P and a semiring S is saidto be weak P−regular if for each a ∈ S, there exists x ∈ S such that a = axa+ pfor some p ∈ P and axP ⊆ P .

Definition 2.2. Let S be a semiring and P be a right k− ideal of S. An elemente ∈ S is called idempotent relative to P if e+ p′ = e2+ p′′ for some p′, p′′ ∈ P andeP ⊆ P.

Clearly, it is easy to see that if P = 0, then P−regular semiring is aregular and idempotent element is same as in semiring theory. From definitionit is also clear that ax = e is an idempotent element relative to P. Consider thesemiring Z4 = 0, 1, 2, 3 with respect to addition and multiplication modulo 4and I = 0, 2 is a right as well as k−ideal of Z4. Clearly Z4 is not regular because2 = 2⊙ x⊙ 2 for x ∈ Z4 but it is P−regular.

Theorem 2.3. The semiring S with unity is P−regular if and only if every rightideal of S be of the form aS + P where a ∈ S has the form aS + P = eS + P ,where e is an idempotent relative to P.

Proof. Suppose S is P−regular. Therefore for each a ∈ S, there exists x ∈ S suchthat a+p1 = axa+p2 for some p1, p2 ∈ P and axP ⊆ P. Now aS+P = axaS+P ⊆axS+P = eS+P. Therefore aS+P ⊆ eS+P. Again, eS+P = axS+P ⊆ aS+P.Form above conclusion, we have eS + P = aS + P. Conversely, suppose thataS + P = eS + P and e+ p′ = e2 + p′′ for some p′, p′′ ∈ P and eP ⊆ P. To showS is P−regular. From above, we can write a+ p1 = ey+ p2 and e+ p3 = ax+ p4.Now

ea+ p3a = axa+ p4a(1)

Again, ea+ ep1 = e2y+ ep2. Adding p′′y+ p2 on both sides, we get ea+ ep1+ p2+p′′y = e2y+ ep2+p2+p′′y = ey+ ep2+p′y+p2 = a+p1+p′y+ ep2, which implies

ea+ z1 = a+ z2(2)

where z1 = ep1+p2+p′′y, z2 = p′y+ep2+p1 ∈ P . Adding z1 in (1) and p3a in (2),we get a+z2+p3a = axa+p4a+z1. Thus a+p5 = axa+p6. Also, e+p3 = ax+p4implies ep + p3p = axp + p4p ∈ P (because ep ∈ eP ⊆ P and p3p ⊆ P ) impliesaxP ⊆ P (since P is a right k−ideal of S). Therefore S is a P−regular.

Proposition 2.4. Suppose P is a right k−ideal of S and I is an ideal of S suchthat P ⊆ I. If S is P−regular then I is P−regular.

a note on P-regular semirings 35

Proof. Suppose S is P−regular and I is an ideal of S such that P ⊆ I. Thenfor each a ∈ I there exists x ∈ S and p1, p2 ∈ P such that a + p1 = axa + p2and axP ⊆ P. Let y = xax ∈ I. Then axa + p1xa = axaxa + p2xa impliesp2+axa+ p1xa = aya+ p2xa+ p2 implies a+ p1+ p1xa = aya+ p2xa+ p2 impliesa+p′ = aya+p′′ for some p′, p′′ ∈ P. Also, ayP = a(xax)p = ax(axp) ∈ axP ⊆ P.Thus I is a P−regular.

Theorem 2.5. Let a semiring S be a P−regular. Then every right k− ideal Rand left k−ideal L of S has the form (P +R) ∩ (P + L) ⊆ P +RL.

Proof. Let S be a P− regular and let a ∈ (P +R)∩ (P + L). Then a ∈ S can bewritten as a = p+ r and a+p1+ l1 = p2+ l2 for some p, p1, p2 ∈ P, and r ∈ R andl1, l2 ∈ L. Since S is a P−regular therefore for each a ∈ S there exists x ∈ S suchthat a+ p3 = axa+ p4 for some p3, p4 ∈ P and axP ⊆ P. Now (a)x(a+ p1 + l1) =(p+ r)x(p2+ l2) which implies that axa+ axp1+ axl1 = pxp2+ pxl2+ rxp2+ rxl2implies axa + p4 + axp1 + axl1 = p4 + pxp2 + pxl2 + rxp2 + rxl2 which givesa+p3+axp1+axl1 = pxp2+pxl2+rxp2+rxl2+p4. Since axp2 = pxp2+rxp2 ∈ Pand P is a right k−ideal of S. Therefore rxp2 ∈ P and also axl1 = pxl1 + rxl1 ∈P +RL. Therefore a ∈ P +RL. Thus (P +R) ∩ (P + L) ⊆ (P +RL).

Note. The equality holds if P + R is a right k−ideal of S because supposea ∈ (P +RL). This implies a ∈ (P +R) and a ∈ (P + L). Therefore, a ∈ (P+R)∩(P + L) (as P +R is a right k−ideal of S). Hence (P +R)∩ (P + L) = P +RL.

Theorem 2.6. The semiring S is a weak P−regular if every right ideal R andleft ideal L of S has the form (P +R) ∩ (P + L) = P +RL.

Proof. The proof is same as in Theorem 2.6.

Proposition 2.7. Let S be a weakP−regular semiring and R be any right idealof S. Then the following holds:

(i) R + P = R2 + P

(ii) if R2 ⊆ P then R ⊆ P.

Proof. (i). Since S is a weakP−regular therefore for each a ∈ S there existsx ∈ S and some p ∈ P such that a = axa+p and axP ⊆ P. Suppose a ∈ R. Thena = (ax)a+ p ∈ R2+P . This implies R ⊆ R2+P . This implies R+P ⊆ R2+P.But R2 + P ⊆ R + P (always). Therefore R + P = R2 + P.

(ii). Since R2 ⊆ P therefore R2 + P ⊆ P . This gives R + P ⊆ P . ThereforeR ⊆ P (as P is a right a k−ideal of S).

Theorem 2.8. If a semiring S is weak P−regular, then every element of a quasi-ideal Q can be represented as the sum of two elements of P and Q.

Proof. Let S be a weak P−regular semiring and Q be a quasi-ideal of S. Thenany q ∈ Q can be written as q = qxq+ p and qxP ⊆ P for some p ∈ P and x ∈ S.Since Q is a quasi-ideal of S therefore qxq ∈ QSQ ⊆ QS ∩SQ ⊆ Q and thereforewe have q = p+ qxq ∈ P +Q.

36 m.k. dubey

Theorem 2.9. Let S be a weakP−regular semiring and Q1and Q2 are quasi-idealsof S. If q ∈ Q1 ∩ Q2, then this element q can be expressed as q = p + q1xq2 forsome p ∈ P, x ∈ S, q1 ∈ Q1 and q2 ∈ Q2 and also q1xq2xP ⊆ P.

Proof. Let q ∈ Q1∩Q2. Since S is a weak P−regular then there exists x ∈ S suchthat q = qxq + p for some p ∈ P and qxP ⊆ P. Since Q1 ∩Q2 is a quasi-ideal ofS therefore the element q ∈ Q1 ∩Q2 can be written as both q = p1 + q1 ∈ Q1 andq = p2+q2 ∈ Q2 for some p1, p2 ∈ P, q1 ∈ Q1, q2 ∈ Q2. Since S is a weak P−regulartherefore the element q ∈ S can be written as q = p3 + qxq for some p3 ∈ P andqxP ⊆ P . Now q = p3 + qxq = p3 + (p1 + q1)x(p2 + q2) = p3 + q1xp2 + q1xq2 +p1xp2+p1xq2 = p6+q1xq2 where p6 ∈ P because qxp2 = p1xp2+q1xp2 ∈ P and Pis a right k−ideal of S, therefore q1xp2 ∈ P . Again, qxP = p6xP + q1xq2xP ⊆ P .Since P is a right k−ideal of S therefore q1xq2xP ⊆ P.

Theorem 2.10. Let S be a weak P−regular. Then every quasi-ideal Q of S canbe written as P +Q = P +QSQ.

Proof. Let S be a weakP−regular and Q be a quasi-ideal of S. Then QSQ ⊆(QS) ∩ (SQ) ⊆ Q holds. Therefore P +QSQ ⊆ P +Q.

Conversely, let a ∈ P + Q. Then a = p + q for some p ∈ P and q ∈ Q.By weak P−regularity of S, we have a = axa + p1, for some p1 ∈ P, x ∈ S andaxP ⊆ P. This gives a = (p + q)x(p + q) + p1 = pxp + pxq + qxp + qxq + p1.Now axp = (p + q)xp = pxp+ qxp ∈ P and P is a right k−ideal of S. Thereforeqxp ∈ P . Thus we have a = (pxp + pxq + qxp + p1) + qxq ∈ P + QSQ. ThusP +Q ⊆ P +QSQ. Hence P +Q = P +QSQ.

Theorem 2.11. Let S be weak P−regular. If Q1and Q2are quasi-ideals of S, thenP + (Q1 ∩Q2) = P + (Q1SQ2) ∩ (Q2SQ1).

Proof. Let S be a weak P−regular and Q1 and Q2 be quasi-ideals of S. Ifq ∈ P+Q1∩Q2. Then q can be written as q = p+q′ for some p ∈ P and q′ ∈ Q1∩Q2.Also, by weak P−regularity of S, there exists x ∈ S such that q′ = p′ + q′xq′, forsome p′ ∈ P . Hence we have q = p+p′+q′xq′ ∈ P+(Q1SQ2)∩(Q2SQ1). Therefore,P +Q1 ∩Q2 ⊆ P +(Q1SQ2)∩ (Q2SQ1). Conversely, since (Q1SQ2)∩ (Q2SQ1) ⊆(Q1S∩SQ2)∩(Q2S∩SQ1) ⊆ Q1∩Q2, then P+(Q1SQ2∩Q2SQ1) ⊆ P+(Q1∩Q2).Hence P + (Q1SQ2 ∩Q2SQ1) = P + (Q1 ∩Q2).

Acknowledgment. The author wishes to express his thanks to the referee forgiving fruitful suggestion and valuable comments.

References

[1] Andrunakievich, A.V., Andrunakievich, V.A., Regularity of ring withrespect to right ideals, Dokl. Akad. Nauk, SSSR, 310, 2 (1990), 267–272.

[2] Choi, S.J., Quasi-ideals of a P−regular Near-Ring, Int. J. Algebra, 4 (11)(2010), 501–506.

[3] Steinfeld, O., Quasi-ideals in Rings and Semigroups, Akad. Kiado, Bu-dapest, 1978.



SOME CLASSES OF GENERALIZED MinMax POLYNOMIALS

H.M. Srivastava∗

Department of Mathematics and StatisticsUniversity of VictoriaVictoria, British Columbia V8W 3R4Canadae-mail: [email protected]

Gospava B. Djordjevic

Department of MathematicsFaculty of TechnologyUniversity of NisYU-16000 LeskovacRepublic of Serbiae-mail: [email protected]

Abstract. The so-called MinMax numbers Mn and their subsidiary numbers Nnfor the Pell numbers Pn were studied by (for example) A.F. Horadam [3]. TheseMinMax numbers Mn are positive integers which are the minimal and maximalrepresentations by means of the Pell numbers. Analogous results for the MinMax num-bers Dn and their subsidiary numbers Rn, and for the modified Pell numbers qn,are obtained in [3], Qn := 2qn being the Pell-Lucas numbers. A.F. Horadam [4], onthe other hand, expanded this MinMax number system to the algebraic polynomialsMn(x), Nn(x), Dn(x) and Rn(x). Our aim in this paper is to investigate re-sults, which are similar to those in [4], but which hold true instead for the followinggeneralized sequences of polynomials:

Pn,m(x) and Qn,m(x) (m ∈ N; n ∈ N ∪ 0),

N being the set of positive integers.

Keywords and phrases: Pell numbers; Modified Pell numbers; MinMax numbers;

Pell-Lucas numbers; Recurrence relations; Explicit representations; Algebraic polyno-

mials; MinMax polynomials; Fibonacci polynomials; Lucas polynomials.

2010 Mathematics Subject Classification: Primary 11B83; Secondary 11B37,

11B39.

∗ Corresponding author.

38 h.m. srivastava, g.b. djordjevic

1. Introduction

Throughout this paper, we denote by N the set of positive integers. We also write

N0 := N ∪ 0 = 0, 1, 2, ....

The generalized Pell polynomials Pn,m(x), and the generalized Pell-Lucas poly-nomials Qn,m(x) and q∗n,m(x), are defined by (see [1])

(1.1) Pn,m(x) = 2xPn−1,m(x) + Pn−m,m(x) (n = m; m,n ∈ N0)

with P0,m(x) = 0 and Pn,m(x) = (2x)n−1 (n = 1, ...,m− 1);

(1.2) Qn,m(x) = 2xQn−1,m(x) +Qn−m,m(x) (n = m; m,n ∈ N0)

with Q0,m(x) = 2 and Qn,m(x) = (2x)n (n = 1, ...,m− 1; m ∈ N\1; n ∈ N) and

(1.3) q∗n,m(x) = 2xq∗n−1,m(x) + q∗n−2m,m(x) (n = 2m; m,n ∈ N0)

with q∗0,m(x) = 1 and q∗n,m(x) = (2x)n−1 (n = 1, ..., 2m− 1; m,n ∈ N).In their special case when m = 2, the polynomials

Pn,2

(x2

)and Qn,2

(x2

)are the same as the familiar Fibonacci and Lucas polynomials, respectively (see[2]; see also [6]).

By using the recurrence relations (1.1), (1.2) and (1.3), and their correspon-ding initial values, we can compute the first few members of the polynomials

Pn,2(x) ≡ Pn(x), Qn,2(x) ≡ Qn(x) and q∗n,1(x) ≡ q∗n(x),

which are given in Table 1, Table 2 and Table 3 below.

Table 1

P0(x) = 0

P1(x) = 1

P2(x) = 2x

P3(x) = 4x2 + 1

P4(x) = 8x3 + 4x

P5(x) = 16x4 + 12x2 + 1

P6(x) = 32x5 + 32x3 + 6x

P7(x) = 64x6 + 80x4 + 24x2 + 1

P8(x) = 126x7 + 192x5 + 80x3 + 8x

P9(x) = 256x8 + 448x6 + 240x4 + 40x2 + 1

P10(x) = 512x9 + 1024x7 + 672x5 + 160x3 + 10x.

some classes of generalized MinMax polynomials 39

Table 2

Q0(x) = 2

Q1(x) = 2x

Q2(x) = 4x2 + 2

Q3(x) = 8x3 + 6x

Q4(x) = 16x4 + 16x2 + 2

Q5(x) = 32x5 + 40x3 + 10x

Q6(x) = 64x6 + 96x4 + 36x2 + 2

Q7(x) = 128x7 + 224x5 + 112x3 + 14x

Q8(x) = 256x8 + 512x6 + 320x4 + 64x2 + 2

Q9(x) = 512x9 + 1152x7 + 864x5 + 240x3 + 18x

Q10(x) = 1024x10 + 2560x8 + 2240x6 + 800x4 + 100x2 + 2.

Table 3

q∗0(x) = 1

q∗1(x) = 1

q∗2(x) = 2x+ 1

q∗3(x) = 4x2 + 2x+ 1

q∗4(x) = 8x3 + 4x2 + 4x+ 1

q∗5(x) = 16x4 + 8x3 + 12x2 + 4x+ 1

q∗6(x) = 32x5 + 16x4 + 32x3 + 12x2 + 6x+ 1

q∗7(x) = 64x6 + 32x5 + 80x4 + 32x3 + 24x2 + 6x+ 1

q∗8(x) = 128x7 + 64x6 + 192x5 + 80x4 + 80x3 + 24x2 + 8x+ 1

q∗9(x) = 256x8 + 128x7 + 448x6 + 192x5 + 240x4 + 80x3 + 40x2 + 8x+ 1.

Our present investigation is motivated essentially by the earlier works of Ho-radam (see, for details, [3]) dealing with the so-called MinMax numbers Mn andtheir subsidiary numbers Nn for Pell numbers Pn, and also with the MinMaxnumbers Dn and their subsidiary numbers Rn and the modified Pell numbersqn, Qn := 2qn being the Pell-Lucas numbers. These MinMax numbers Mn arepositive integers which are the minimal and maximal representations by meansof the Pell numbers. Analogous results for the MinMax numbers Dn and theirsubsidiary numbers Rn, and also for the modified Pell numbers qn. Sub-sequently, Horadam [4] expanded this MinMax number system to the algebraicpolynomials Mn(x), Nn(x), Dn(x) and Rn(x). The main object of thispaper is to present results, which are similar to those in [4], but which hold trueinstead for the following generalized sequences of polynomials:

Pn,m(x) and Qn,m(x) (m ∈ N; n ∈ N ∪ 0),


which are defined here by (1.1) and (1.2), respectively.

2. The MinMax polynomials Mn,m(x)

The polynomials Mn,m(x) (m ∈ N; n ∈ N0) are defined by means of thefollowing recurrence relation:

(2.1) Mn,m(x) = 2xMn−1,m(x) +Mn−2m,m(x) + l (n = 2m; m,n ∈ N)

with M0,m(x) = 0 and Mn,m(x) = (2x)n−1 (n = 1, ...,m) and Mm+1,m(x) =(2x)m + 1 and Mn,m(x) = 2xMn−1,m(x) (n = m+ 2, ..., 2m− 1), where

l =

0 (n = mk + 1; k ∈ N0)

1 (n = mk + 1).

One of our main results is asserted by Theorem 1 below.

Theorem 1. For m ∈ N and n = m, each of the following equalities holds true:

(2.2) Mn,m(x) =

[n/m]∑i=0

Pn−mi,2m(x)

and

(2.3) Mn,m(x) =Pn+1,2m(x) + Pn+1−m,2m(x)− l

2x,

where

l =

0 (n = mk; k ∈ N0)

1 (n = mk).

Proof. We make use of the principle of mathematical induction on n in order toprove equalities (2.2) and (2.3). First of all, it is easy to prove equality (2.2) for

n = 0, 1, ..., 2m− 1.

Suppose now that (2.2) holds true for n = mn (n ∈ N). Then, for n = mn + 1,


we apply the recurrence relation (2.1) to get

Mmn+1,m(x) = 2xMmn,m(x) +Mmn+1−2m,m(x) + 1

= 2xn∑

i=0

Pmn−mi,2m(x) +n−2∑i=0

Pmn+1−2m−mi,2m(x) + 1

=n−2∑i=0

(2xPmn−mi,2m + Pmn+1−2m−mi,2m)

+ 2x(Pm,2m(x) + P0,2m(x)) + 1

=n−2∑i=0

Pmn+1−mi,2m(x) + Pm+1,2m(x) + P1,2m(x)

=n∑

i=0

Pmn+1−mi,2m(x),

which proves the the first assertion (2.2) of Theorem 1.Next, clearly, equality (2.3) holds true for n = 0, 1, · · · , 2m− 1, by means of

the recurrence relations (1.1) and (2.1). Suppose that (2.3) holds true for n = mn.Then, for n = mn+ 1, we find that

Mmn+1,m(x) = 2xMmn,m(x) +Mmn+1−2m,m(x) + 1

= 2xPmn+1,2m(x) + Pmn+1−m,2m(x)− 1

2x

+Pmn+2−2m,2m(x) + Pmn+2−3m,2m(x)

2x+ 1

=Pmn+2,2m(x) + Pmn+2−m,2m(x)

2x,

which, by the principle of mathematical induction on n, proves the second asser-tion (2.3) of Theorem 1. Our proof of Theorem 1 is thus completed.

3. The polynomials Nn,m(x)

The polynomials Nn,m(x) (m ∈ N; n ∈ N0) are defined by

(3.1) Nn,m(x) = Mn+1,m(x) +Mn+1−2m,m(x) (n = 2m),

which, in conjunction with (2.1), shows that the polynomials Nn,m(x) satisfythe following recurrence relation:

(3.2) Nn,m(x) = 2xNn−1,m(x) +Nn−2m,m(x) + l

with N0,m(x) = 1 and Nn,m(x) = (2x)n (n = 1, ...,m−1) and Nm,m(x) = (2x)m+1and Nn,m(x) = 2xNn−1,m(x) (n = m+ 1, ..., 2m− 1), where

l =

0 (n = mk; k ∈ N0)

2 (n = mk).


Theorem 2. The polynomials Nn,m(x) satisfy the following relationship:

(3.3) Nn,m(x) =Qn+1,2m +Qn+1−m,2m(x)− l

2x,

where

l =

0 (n = mk +m− 1; k ∈ N0)

2 (n = mk +m− 1).

Proof. By using (1.2) and (3.2), we readily obtain

Nm−1,m(x) =Qm,2m(x) +Q0,2m(x)− 2

2x,

which shows that relationship (3.3) holds true for n = m− 1. If we assume that(3.3) is holds true for n = mk, then, for n = mk, we find by using the recurrencerelation (3.2) that

Nmk,m(x) = 2xNmk−1,m(x) +Nmk−2m,m(x) + 2

= 2x

(Qmk,2m(x) +Qmk−m,2m(x)− 2

2x

)+

Qmk+1−2m,2m(x) +Qmk+1−3m,2m(x)

2x+ 2

=Qmk+1,2m(x) +Qmk+1−m,2m(x)

2x,

which evidently completes our proof of Theorem 2.

By some suitable algebraic manipulations based upon (3.3), we get the fol-lowing result.

Theorem 3. For m ∈ N and n = m, the following relations hold true:

(3.4) Nn,m(x)−Nn−m,m(x) = Qn,2m(x)

and

(3.5) Nn,m(x)−Nn−2m,m(x) = Qn,2m(x) +Qn−m,2m(x).

Proof. Upon setting n = m in (3.4), we get

Nm,m(x)−N0,m(x) = Qm,2m(x),

which, by virtue of (3.2) and Table 2, coincides with the identity:

(2x)m + 1− 1 = (2x)m.


If (3.4) holds true for some fixed positive integer n (n = m), then

Nn+1,m(x)−Nn+1−m,m(x)

= 2xNn,m(x) +Nn+1−2m,m(x)

− 2xNn−m,m(x)−Nn+1−3m,m(x)

= 2x [Nn,m(x)−Nn−m,m(x)]

+ [Nn+1−2m,m(x)−Nn+1−3m,m(x)]

= 2xQn,2m(x) +Qn+1−2m,2m(x)

= Qn+1,2m(x),(3.6)

which implies that relation (3.4) is holds true when n is replaced by n+ 1.The second assertion (3.5) of Theorem 3 is a direct consequence of the first

assertion (3.4).

4. The subsidiary MinMax polynomials Dn,m(x)

Instead of the MinMax polynomials for the generalized Pell polynomials Pn,m(x)defined by (1.1), we now consider the analogous polynomials for the polynomi-als Qn,m(x) and q∗n,m(x), which are defined by means of the recurrence re-lations (1.2) and (1.3), respectively. We thus define the MinMax polynomialsDn,m(x)n∈N0 by means of the following recurrence relation:

(4.1) Dn,m(x) = 2xDn−1,m(x) +Dn−2m,m(x) + l (n = 2m; m,n ∈ N)

with D0,m(x) = 0 and Dn,m(x) = (2x)n−1 (n = 1, · · · ,m) and Dm+1,m(x) =(2x)m + 2 and Dn,m(x) = 2xDn−1,m(x) (n = m+ 2, · · · , 2m− 1), where

l =

0 (n = mk + 1; k ∈ N0)

2 (n = mk + 1).

Theorem 4 below provides the connection between the classes ofpolynomials Mn,m(x) and Dn,m(x).

Theorem 4. For m ∈ N and n = m the following relationships hold true:

(4.2) Mn,m(x) +Mn−m,m(x) = Dn,m(x) (n > m);

(4.3) Dn,m(x) =q∗n+1,m(x) + q∗n+1−m,m(x)− l

2x(n = m),

where

l =

0 (n = mk; k ∈ N0)

2 (n = mk);


(4.4) Dn,m(x)−Dn−m,m(x) = q∗n,m(x) (n = m)

and

(4.5) Dn,m(x)−Dn−2m,m(x) = q∗n,m(x) + q∗n−m,m(x) (n = 2m).

Proof. We first assume that (4.2) holds true for some positive integer n. Then,for n 7→ n+ 1, we get

Mn+1,m(x) +Mn+1−m,m(x) = 2xMn,m(x) +Mn+1−2m,m(x) + l

+ 2xMn−m,m(x) +Mn+1−3m,m(x) + l

= 2x [Mn,m(x) +Mn−m,m(x)] +Mn+1−2m,m(x)

+Mn+1−3m,m(x) + 2l

= 2xDn,m(x) +Dn+1−2m,m(x) + 2l

= Dn+1,m(x) (2l = 2 or 2l = 0).

Next, we suppose that the relationship (4.3) holds true for n = mk. Then,for n = mk + 1, we get

Dmk+1,m(x) = 2xDmk,m(x) +Dmk+1−2m,m(x) + 2

= 2xq∗mk+1,m(x) + q∗mk+1−m,m(x)− 2

2x

+q∗mk+1−2m,m(x) + q∗mk+1−3m,m(x)

2x+ 2

=1

2x

[2xq∗mk+1,m(x) + q∗mk+1−2m,m(x)

+ 2xq∗mk+1−m,m(x) + q∗mk+1−3m,m(x)]

=q∗mk+2,m(x) + q∗mk+2−m,m(x)

2x.

For n = m in (4.4), we are led at once to the following obvious identity:

Dm,m(x)−D0,m(x) = q∗m,m(x),

so (4.4) holds true for n = m. Suppose that (4.4) holds true for some positiveinteger n (n = m). Then, for n 7→ n+ 1, we get

Dn+1,m(x)−Dn+1−m,m(x) = 2xDn,m(x) +Dn+1−2m,m(x) + l

− 2xDn−m,m(x)−Dn+1−3m,m(x)− l

= 2x(Dn,m,(x)−Dn−m,m(x)) +Dn+1−2m,m(x)

−Dn+1−3m,m(x)

= 2xq∗n,m(x) + q∗n+1−2m,m(x)

= q∗n+1,m(x).


The last assertion (4.5) of Theorem 4 is an immediate consequence of assertion(4.4). The proof of Theorem 4 is thus completed.

We now present another interesting result.

Theorem 5. The polynomials Dn,m(x) possess the following representation:

(4.6) Dn,m(x) =

[(n−1)/m]∑i=0

q∗n−mi,m(x) (m,n ∈ N).

Proof. For n = 1 in (4.6), we have the following obvious special case:

D1,m(x) =0∑

i=0

q∗1−mi,m(x),

which shows that that formula (4.6) holds true for n = 1.

Suppose now that (4.6) holds true for n = mk. Then, for n = mk+1, we get

Dmk+1,m(x) = 2xDmk,m(x) +Dmk+1−2m,m(x) + 2

= 2x

[(mk−1)/m]∑i=0

q∗mk−mi,m(x) +

[(mk−2m)/m]∑i=0

q∗mk+1−2m,m(x) + 2

= 2xk−1∑i=0

q∗mk−mi,m(x) +k−2∑i=0

q∗mk+1−2m,m(x) + 2

=k−2∑i=0

(2xq∗mk−mi,m(x) + q∗mk+1−2m−mi,m(x))

+ 2xq∗mk−m(k−1),m(x) + 2

=k−2∑i=0

q∗mk+1−mi,m(x) + 2xq∗m,m(x) + 1 + 1

=k−2∑i=0

q∗mk+1−mi,m(x) + q∗m+1,m(x) + 1

=k∑

i=0

q∗mk+1−mi,m(x)(k =

[ nm

]).


For n = mk + 2, we find from the recurrence relation (4.1) that

Dmk+2,m(x) = 2xDmk+1,m(x) +Dmk+2−2m,m(x)

= 2xk∑

i=0

q∗mk+1−mi,m(x) +k−2∑i=0

q∗mk+2−2m−mi,m(x)

=k−2∑i=0

q∗mk+2−mi,m(x) + 2xq∗m+1,m(x) + 2xq∗1,m(x)

=k−2∑i=0

q∗mk+2−mi,m(x) + q∗m+2,m(x) + q∗2,m(x)

=k∑

i=0

q∗mk+2−mi,m(x).

This, evidently, completes our proof of Theorem 5.

5. The subsidiary MinMax polynomials Rn,m(x)

In this section, we introduce and investigate the sequence of polynomialsRn,m(x)m∈N, which are the subsidiary MinMax polynomials of Dn,m(x) forqn,m(x), where

2qn,2(x) = Qn,2(x).

Thus, by definition, we have

(5.1) Rn,m(x) = 2xRn−1,m(x) +Rn−2m,m(x) + l (n = 2m; m ∈ N)

with R0,m(x) = 0 and Rn,m(x) = (2x)n (n = 1, ...,m−1) and Rm,m(x) = (2x)m+2and Rn,m(x) = 2xRn−1,m(x) (n = m+ 1, ..., 2m− 1), where

l =

0 (n = mk; k ∈ N0)

2 (n = mk).

The connection between the polynomials Rn,m(x), Nn,m(x) and Dn,m(x)is given by Theorem 6 below.

Theorem 6. Each of the following relationships holds true:

(5.2) Rn,m(x) = Dn+1,m(x) +Dn+1−2m,m(x) (n = 2m),

(5.3) Rn,m(x) = Nn,m(x) +Nn−m,m(x) (n = m),

(5.4) Rn,m(x)−Rn−m,m(x) = q∗n+1,m(x) + q∗n+1−2m,m(x) (n = 2m)

and

(5.5) Rn,m(x)−Rn−m,m(x) = Nn,m(x)−Nn−2m,m(x) (n = 2m).


Proof. Suppose that (5.2) is correct for n = mk. Then, for n = mk + 1, therecurrence relation (5.1) yields

Rmk+1,m(x) = 2xRmk,m(x) +Rmk+1−2m,m(x)

= 2x(Dmk+1,m(x) +Dmk+1−2m,m(x)) +Dmk+2−2m,m(x) +Dmk+2−4m,m(x)

= Dmk+2,m(x) +Dmk+2−2m,m(x),

which proves relationship (5.2).

By using the initial values for Rn,m(x) and Nn,m(x), we can easily proverelationship (5.3). Let relationship (5.3) hold true for n = mk. Then

Rmk,m(x) = 2xRmk−1,m(x) +Rmk−2m,m(x) + 4

= 2x(Nmk−1,m(x) +Nmk−1−m,m(x)) +Nmk−2m,m(x) +Nmk−3m,m(x) + 4

= 2xNmk−1,m(x) +Nmk−2m,m(x) + 2 + 2xNmk−1−m,m(x) +Nmk−3m,m(x) + 2

= Nmk,m(x) +Nmk−m,m(x),


Suppose now that the relationship (5.4) holds true for some positive integern. Then, for n 7→ n+ 1, we find from the recurrence relation (5.1) in conjunctionwith (3.1) that

(5.6)

Rn+1,m(x)−Rn+1−m,m(x)

= 2xRn,m(x) +Rn+1−2m,m(x) + l − 2xRn−m,m(x)−Rn+1−3m,m(x)− l

= 2x [Rn,m(x)−Rn−m,m(x)] +Rn+1−2m,m(x)−Rn+1−3m,m(x)

= 2xq∗n+1,m(x) + 2xq∗n+1−2m,m(x) + q∗n+2−2m,m(x) + qn+2−4m,m(x)

= q∗n+2,m(x) + q∗n+2−2m,m(x),


It is easily observed that the last assertion (5.5) of Theorem 6 is a directconsequence of relationship (5.3).

6. A set of sequences of numbers

In their special cases when x = 0, the above-investigated MInMax polynomialsMn,m(x), Nn,m(x), Dn,m(x) and Rn,m(x) would lead us to some interes-ting sequences of MinMax numbers, which we denote by Mn,m, Nn,m, Dn,mand Rn,m, respectively. Some of these sequences of MinMax numbers are givenbelow:


Mn,2 : 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 3, 0, 4, 0, 4, ...;Mn,3 : 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 3, ...;Mm,m : 0, 1, 0, ..., 0︸︷︷︸

m−1

, 1, 0, ..., 0︸︷︷︸m−1

, 2, 0, ..., 0︸︷︷︸m−1

, ....

Nn,2 : 1, 0, 1, 0, 3, 0, 3, 0, 5, 0, 5, ...;Nn,3 : 1, 0, 0, 1, 0, 0, 3, 0, 0, 3, 0, 0, 5, 0, 0, 5, ...;Nn,m : 1, 0, ..., 0︸︷︷︸

m−1

, 1, 0, ..., 0︸︷︷︸m−1

, 3, 0, ..., 0︸︷︷︸m−1

, 3, 0, ..., 0︸︷︷︸m−1

, 5, ....

Dn,2 : 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...;Dn,3 : 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 4, 0, 0, 5, ...;Dn,m : 0, 1, 0, ..., 0︸︷︷︸

m−1

, 2, 0, ..., 0︸︷︷︸m−1

, ....

Rn,2 : 0, 0, 2, 0, 4, 0, 6, 0, 8, 0, 10, ...;Rn,3 : 0, 0, 0, 2, 0, 0, 4, 0, 0, 6, 0, 0, 8, 0, 0, 10, ...;Rn,m : 0, ..., 0︸︷︷︸

m−1

, 2, 0, ..., 0︸︷︷︸m−1

, ....

Acknowledgements. The second-named author was supported, in part, by theMinistry of Science of the Republic of Serbia under Grant 144003.

References

[1] Djordjevic, G.B., On the kth order derivative sequences of generalizedFibonacci and Lucas polynomials, Fibonacci Quart., 43 (2005), 290–298.

[2] Djordjevic, G.B., Generalizations of the Fibonacci and Lucas poly-nomials, FILOMAT, 23 (2009), 287–297.

[3] Horadam, A.F., MinMax sequences for Pell numbers, in Applicationsof Fibonacci Numbers, vol. 6 (Pullman, Washington, 1994), 231–249,Kluwer Academic Publishers, Dordrecht, Boston and London, 1996.

[4] Horadam, A.F., MinMax polynomials, Fibonacci Quart., 34 (1996),7–17.

[5] Horadam, A.F., Mahon, J.M., Pell and Pell-Lucas polynomials, Fi-bonacci Quart., 23 (1985), 7–20.

[6] Raina, R.K., Srivastava, H.M., A class of numbers associated withthe Lucas numbers, Math. Comput. Modelling, 25 (7) (1997), 15–22.



ON γ-s-URYSOHN CLOSED AND γ-s-REGULAR CLOSED SPACES

Sabir Hussain

Department of MathematicsCollege of ScienceQassim UniversityP.O. Box 6644, Buraydah 51452Saudi Arabiae-mail: [email protected]

Ennis Rosas

Departamento de MatematicasUniversidad de OrienteNucleo de SucreVenezuelae-mail: [email protected]

Abstract. In this paper, we introduced and studied γ-s-Urysohn spaces and γ-s-regular

closed spaces. Several characterizations and properties of these classes of spaces have

been obtained.

Keywords. γ-closed (open), γ-interior (closure), γ-semi-open(closed),γ-s-Urysohn,γ-s-

regular, γ-s-adherent, γ-irresolute.

AMS Subject Classification: 54A05, 54A10, 54D10, 54D99.

1. Introduction

S. Kasahara [14] introduced and discussed an operation γ of a topology τ into thepower set P (X) of a space X. H. Ogata [16] introduced the concept of γ-opensets and investigated the related topological properties of the associated topologyτγ and τ by using operation γ.

S. Hussain and B. Ahmad [1]-[6] and [10]-[13] continued studying the pro-perties of γ-operations on topological spaces and investigated many interestingresults. Recently, B. Ahmad and S. Hussain [2], [13] defined and discussed γ-semi-open sets in topological spaces. They explored many interesting propertiesof γ-semi-open sets. It is interesting to mention that γ-semi-open sets generalizedγ-open sets introduced by H. Ogata [17].

50 sabir hussain, ennis rosas

In 1982, S.P. Arya and M.P. Bhamini introduced the concept of s-Urysohn.A space X is said to be s-Urysohn, if for any two distinct points x and y of X,there exist semi-open sets U and V containing x and y respectively such thatcl(U) ∩ cl(V ) = ∅. The concept of s-regular space was introduced by Mahaesh-wari and Prasad in 1975. In 1984, Arya and Bhamini introduced and discusseds-regular-closed sets [8].

In 2009, S. Hussain and B. Ahmad defined a new axiom called γ-s-regularity.It is interesting to mention that this axiom is a generalization of the axiom ofs-regularity [15] as well as semi-regularity [9].

In this paper, we introduced and studied γ-s-Urysohn spaces and γ-s-regularclosed spaces. A γ-s-Urysohn (resp., γ-s-regular [12]) space X is said to be γ-s-Urysohn-closed (resp., γ-s-regular-closed), if it is γ-closed in every γ-s-Urysohn(respt. γ-s-regular [12]) space in which it can be embedded. Several characteriza-tions and properties of these classes of spaces have been obtained.

First, we recall some definitions and results used in this paper. Here after,we shall write a space in place of a topological space.

Preliminaries

Definition 2.1. [14] Let X be a space. An operation γ : τ → P (X) is a functionfrom τ to the power set of X such that V ⊆ V γ, for each V ∈ τ , where V γ

denotes the value of γ at V . The operations defined by γ(G) = G, γ(G) = cl(G)and γ(G) = intcl(G) are examples of operation γ.

Definition 2.2. [14] Let A ⊆ X. A point x ∈ A is said to be γ-interior point ofA if there exists an open nbd N of x such that Nγ ⊆ A and we denote the set ofall such points by intγ(A). Thus

intγ(A) = x ∈ A : x ∈ N ∈ τ and Nγ ⊆ A ⊆ A.

Note that A is γ-open [14] iff A = intγ(A). A set A is called γ-closed [1] iffX − A is γ-open.

Definition 2.3. [16] A point x ∈ X is called a γ-closure point of A ⊆ X, ifUγ ∩ A = ϕ, for each open nbd U of x. The set of all γ-closure points of A iscalled γ-closure of A and is denoted by clγ(A). A subset A of X is called γ-closed,if clγ(A) ⊆ A. Note that clγ(A) is contained in every γ-closed superset of A.

Definition 2.4. [16] An operation γ on τ is said to be regular, if for any opennbds U, V of x ∈ X, there exists an open nbd W of x such that Uγ ∩ V γ ⊇ W γ.

Definition 2.5. [16] An operation γ on τ is said to be open, if for any open nbdU of each x ∈ X, there exists γ-open set B such that x ∈ B and Uγ ⊇ B.

Definition 2.6. [13] A subset A of a space X is said to be a γ-semi-open set, ifthere exists a γ-open set O such that O ⊆ A ⊆ clγ(O). The set of all γ-semi-open

on γ-s-urysohn closed and γ-s-regular closed spaces 51

sets is denoted by SOγ(X). A is γ-semi-closed if and only if X−A is γ-semi-openin X. Note that A is γ-semi-closed if and only if intγclγ(A) ⊆ A [2].

Definition 2.7. [2] Let A be a subset of a space X. The intersection of allγ-semi-closed sets containing A is called γ-semi-closure of A and is denoted bysclγ(A). Note that A is γ-semi-closed if and only if sclγ(A) = A. The set of allγ-semi-closed subsets of A is denoted by SCγ(A).

Definition 2.8. [2] Let A be a subset of a space X. The union of γ-semi-opensubsets contained in A is called γ-semi-interior of A and is denoted by sintγ(A).

Definition 2.9. [13] An operation γ on τ is said be semi-regular, if for any semi-open sets U, V of x ∈ X, there exists a semi-open W of x such that Uγ∩V γ ⊇ W γ.

Note that, if γ is semi-regular operation, then intersection of two γ-semi-opensets is γ-semi-open [13].

Definition 2.10. [2] A subset A of a space X is said to be γ-semi-regular, if itis both γ-semi-open and γ-semi-closed. The class of all γ-semi-regular sets of Xis denoted by SRγ(A). Note that, if γ is a regular operation, then the union ofγ-semi-regular sets is γ-semi-regular.

Definition 2.11. [16] A spaceX is said to be γ-T2 space, if for each distinct pointsx, y ∈ X there exists open sets U, V such that x ∈ U , y ∈ V and Uγ ∩ V γ = ϕ.

Definition 2.12. [3] Let X be a space and A ⊆ X. A point x ∈ X is said to bea γ-adherent point of A, if Uγ ∩ A = ϕ, for every γ-open set U such that x ∈ U .A point x is a γ-adherent point for A if and only if x ∈ clγ(A).

3. γ-S-Urysohn closed spaces

Definition 3.1. A space X is said to be γ-s-Urysohn, if for any two distinctpoints x and y there exist γ-semi-open sets U and V such that x ∈ U , y ∈ V andclγ(U) ∩ clγ(V ) = ϕ.

Definition 3.2. A γ-s-Urysohn space X is said to be γ-s-Urysohn-closed, if it isγ-closed in every γ-s-Urysohn space in which it can be embedded.

Definition 3.3. A filter base ξ is said to be a γ-s-Urysohn filter base, if wheneverx is not γ-adherent point of ξ, there exists a γ-semi-open set U containing x suchthat clγ(U) ∩ clγ(F ) = ϕ, for some F ∈ ξ. A γ-open cover δ of X is said to be aγ-s-Urysohn cover, if there exists a γ-semi-open cover ϱ of X such that for eachV ∈ ϱ, there is a U ∈ δ such that clγ(V ) ⊆ U .

We are interesting in characterize the γ-s-Urysohn-closed space when it is aγ-T2 and γ-s-Urysohn space.

Theorem 3.4. Let X be a γ-s-Urysohn and γ-T2 space and γ be a regular andopen operation. Then the following are equivalent:


(1) X is γ-s-Urysohn-closed.

(2) Every γ-s-Urysohn cover δ of X has a finite subfamily δ∗ such that the γ-closures of members of δ∗ cover X.

(3) Every γ-open γ-s-Urysohn filter base has nonempty γ-adherence.

Proof. (1) ⇒ (2). Let X be γ-s-Urysohn-closed and let δ be a γ-s-Urysohn coverof X. Suppose on the contrary that for no finite subfamily of δ, the γ-closuresof the members of the subfamily cover X. Let p /∈ X and let Y = p ∪X. Letℑ = τ ∪ p ∪ X − ∪n

i=1clγ(Ui), Ui ∈ δ. Then ℑ is a topology on Y . Definean operation γℑ on ℑ as follows:

γℑ(U) = γ(U) if U ∈ τ and

γℑ(U) = U otherwise.

We shall prove that (Y,ℑ) is γℑ-s-Urysohn. Let x and y be two distinct pointsin Y . If x = p = y, there exist disjoint γ-semi-open sets H and K containing xand y respectively such that clγℑ(H) ∩ clγℑ(K) = ∅. Suppose now, that one of xand y, say is p. Since δ is a γ-s-Urysohn cover of X, there exists a γ-semi-opencover ϱ of X such that for each V ∈ ϱ, there exists a UV ∈ δ where clγ(V ) ⊆ UV .Let x ∈ V ∈ ϱ. Now, p ∪ (X − clγ(UV )) is γℑ-open(and hence γℑ-semi-open)set containing y = p. Since clγ(V ) ⊆ UV . clγℑ(V ) ∩ (p ∪ (X − UV )) = ∅. Thatis, clγℑ(V ) ∩ clγℑ(p ∪ (X − clγ(UV ))) = ∅, since γℑ is regular. Hence (Y,ℑ) isγℑ-s-Urysohn. But X is not a γ-closed subset of (Y,ℑ) since p ∈ clγℑ(X). Thisis a contradiction to the fact that X is γ-s-Urysohn-closed. Hence (2) is true.

(2) ⇒ (3). We suppose contrarily that ξ be a γ-s-Urysohn filter base withoutany γ-adherent point. Then δ = X− clγ(F ) : F ∈ ξ is a γ-open cover of X. Weshall prove that δ is a γ-s-Urysohn cover. Let x ∈ X. Since x is not a γ-adherentpoint of ξ, there exists a γ-semi-open set Vx such that clγ(Vx)∩clγ(Fx) = ∅ for someFx ∈ ξ. Therefore, there exist X − clγ(Fx) ∈ δ such that clγ(Vx) ⊆ X − clγ(Fx).Hence, ϱ = Vx : x ∈ X and clγ(Vx) ⊆ X − clγ(Fx) is a γ-semi-open over

of X. Thus δ is a γ-s-Urysohn cover. Therefore, X =n∪

i=1

clγ(X − clγ(Fxi) ⊆

n∪i=1

clγ(X−Fxi) =

n∪i=1

(X− (Fxi), since each Fxi

is γ-open. Thus X = X−n∩

i=1

(Fxi)

which means that∩n

i=1(Fxi) = ϕ, a contradiction. Thus (3) is proved.

(3) ⇒ (1). Contrarily, suppose that X be a γ-s-Urysohn space which is notγ-s-Urysohn-closed. Let Y be a γ-s-Urysohn space in which X is embedded. Ifpossible, suppose that X is not a γ-closed subset of Y . Let p ∈ clγ(X) − Xand δ = U ∩ X where U is a γ-open subset of Y containing p. Then δ is aγ-s-Urysohn filter base in X. Since X is a γ-s-Urysohn and γ − T2, it can beeasily verified that δ has no γ-adherent point in X. This is a contradiction. Thiscompletes the proof.


Theorem 3.5. Every γ-clopen subset of a γ-s-Urysohn-closed space is γ-s-Urysohn-closed.

Proof. Let X be a γ-s-Urysohn-closed and let Y ⊆ X be γ-open. Let ξ be a γ-s-Urysohn filter base in Y with empty γ-adherence. Y being γ-open is γ-s-Urysohn.Also ξ is a γ-open filter base in X. We shall claim that ξ is a γ-s-Urysohn filterbase in X. If every point of x is a γ-adherent point of ξ in X, then ξ is of courseγ-s-Urysohn in X. Let x be a point in X which is not a γ-adherent point of ξ in X.Since every γ-semi-open subset V of X containing x have nonempty intersectionwith every F ∈ ξ. Then V ∩ Y is a γ-semi-open subset of X [2] and hence ofY . Since Y is a γ-semi-closed subset of X, x ∈ Y . Thus V ∩ Y is a γ-semi-opensubset of Y containing x having nonempty intersection with every member of ξ.Also, Y being γ-open, every γ-semi-open subset of Y is of the form V ∩ Y , whereV is a γ-semi-open subset of X. Hence every γ-semi-open subset of Y containingx intersects every member of ξ, which is a contradiction to the fact that ξ is aγ-s-Urysohn filter base in Y . Y being γ-closed. ξ can not have a γ-adherent pointin X, since it has empty γ-adherent in Y . This is a contradiction. Hence theproof.

Definition 3.6. A point x is said to be a γ-s-adherent point of a filter base ξ, ifx ∈ sclγ(F ), for every F ∈ ξ.

Definition 3.7. [17] Let (X, τ) and (Y, δ) be spaces. Let γ :→ P (X) andβ : δ → P (Y ) be operations. Let (X × Y, τ × δ) be the product space and letρ : τ × δ → P (X×Y ) be an operations on τ × δ. Then ρ is called associative with(γ, β), if (U × V )ρ = Uγ × V β holds for each nonvoid U ∈ τ and nonvoid V ∈ δ.

It is known in [17] that, if A ⊆ X and B ⊆ Y . Then clρ(A× B) = clγ(A)×clβ(B).

Lemma 3.8. Let X be a space and γ be a regular operation such that every γ-openfilter base in X has nonempty γ-s-adherent and let Y be an arbitrary space. If ξis a ρ-open ρ-s-Urysohn filter base in X × Y , then Py(ξ) is a ρ-open ρ-s-Urysohnfilter base in Y, where Py is the projection function from X × Y onto Y. Whereγ, β and ρ are operations on X, Y and X × Y respectively.

Proof. ξ is a ρ-open filter base in X × Y . Then Px(ξ) is a γ-open filter basein X and hence has a γ-s-adherent point, say x ∈ X. We shall prove that Px(ξ)is a β-s-Urysohn filter base in Y . Py(ξ) is a β-open filter base in Y . Supposethat y is not a β-adherent point of Py(ξ). Then (x, y) is not a ρ-adherent pointof ξ in X × Y . It is given that ξ is a ρ-s-Urysohn filter base in X × Y . Hencethere exists a ρ-semi-open subset U × V of X × Y containing (x, y) such thatclρ(U × V ) ∩ clρ(F ) = ∅, for some F ∈ ξ. Since U × V is a ρ-semi-open subset ofX×Y containing (x, y), U is a γ-semi-open subset of X containing x and V is a β-semi-open subset of Y containing y [2]. Also, we have clρ(U×V )∩clρ(F ) = ∅. Thatis, (clγ(U)× clβ(V )) ∩ clρ(F ) = ∅ for some F ∈ ξ. Hence clβ(V ) ∩ clβ(Py(F )) = ∅for some F ∈ ξ, since x is a γ-s-adherent point of Px(ξ).


4. γ-s-Regular spaces

Definition 4.1. A γ-s-regular space is said to be γ-s-regular-closed, if it is γ-closed in every γ-s-regular space in which it can be embedded.

Definition 4.2. A cover δ is said to be a γ-s-regular cover, if there exists aγ-semi-open cover ϱ such that the γ-semi-closures of members of ϱ refine δ.

Definition 4.3. A filter base is said to be a γ-s-regular filter base, if it is equivalentto a γ-semi-closed filter base.

Theorem 4.4. Let X be a γ-T2 space and γ be a regular operation. Then thefollowing are equivalent for a γ-s-regular space X:

(1) X is γ-s-regular-closed.(2) Every γ-open γ-s-regular cover has a finite subcover.(3) Every γ-s-regular filter base has nonempty γ-adherence.

Proof. (1) ⇒ (2). Let δ be a γ-open γ-s-regular cover of X and suppose thatδ does not have a finite subcover and p /∈ X. Let Y = p ∪ X. Let ℑ =τ ∪ p ∪ X −

∪ni=1 Ui : Ui ∈ δ. Then ℑ is a topology on Y . Define an

operation γℑ on ℑ as follows: γℑ(U) = γ(U) if U ∈ τ and γℑ(U) = U otherwise.We shall prove that (Y,ℑ) is γℑ-s-regular.

Case (1). Let x ∈ Y , where x = p and B be a ℑ-closed set containing neither xnor p. Then Y −B is a ℑ-open set containing both x and p. Since δ is a γ-s-regularcover, there exists a γ-semi-open cover ϱ of X such that sclγ(V ) : V ∈ ϱ is arefinement of δ. Since x ∈ X, x ∈ V for some V ∈ ϱ. Now, x ∈ Y − B where(Y −B) is a γℑ-open set containing p. Hence Y −B = p∪X−

∪ni=1(Ui) : Ui ∈

δ. Therefore, x ∈ V ∩ (p ∪ (X −n∪

i=1

(Ui))). The set V ∩ (p ∪ (X −n∪

i=1

(Ui)))

is a ℑ-open set, being an intersection of ℑ-open(and hence ℑ-semi-open) set.

Thus V ∩ (p ∪ (X −n∪

i=1

(Ui))) is a ℑ-semi-open set containing x. Also Y −B =

p∪(X−n∪

i=1

(Ui)) = Y −n∪

i=1

(Ui). Hence B =n∪

i=1

(Ui). Thus V ∩(p∪(X−n∪

i=1

(Ui)))

andn∪

i=1

(Ui) are disjoint ℑ-semi-open sets containing x and B respectively.

Case 2. Now, let us suppose that x = p and B is ℑ-closed set not containing p.

Then Y − B = p ∪ X −n∪

i=1

(Ui) : Ui ∈ δ. Therefore, p ∪

(X −

n∪i=1

(Ui)

)is

ℑ-open set and hence a γℑ-semi-open set containing p andn∪

i=1

Ui is a ℑ-open set

containing B.


Case 3. Suppose that x = p and let B be a γℑ-closed set containing p. ThenY − B is a ℑ-open set containing x and not containing p. Hence Y − B is aτ -open set containing x. Therefore there exists a γ-semi-open subset V of X suchthat x ∈ V ⊆ sclγ(V ) ⊆ Y − B. Thus (Y,ℑ) is γℑ-s-regular. But X is not aγ-closed subset of Y since every γℑ-open set containing p intersect X. This is acontradiction. Hence δ has a finite subcover.

(2) ⇒ (3). Let ξ be a γ-s-regular filter base without a γ-adherent point inX. Then X − clγ(F ) : F ∈ ξ is a γ-open cover of X. We shall prove that thisis a γ-s-regular cover. Let x ∈ X. Then x is not a γ-adherent point of ξ, so thatx /∈ clγ(Fx), for some Fx ∈ ξ. Since X is γ-s-regular, there exists a γ-semi-openset Vx containing x such that x ∈ Vx ⊆ sclγ(Vx) ⊆ X − clγ(Fx). Thus Vx : x ∈ Xand sclγ(Vx) ⊆ X− clγ(Fx) = δ is a γ-semi-open cover of X such that sclγ(Vx) :Vx ∈ δ is a refinement of X − clγ(Fx) : Fx ∈ δ. Hence X − clγ(F ) : F ∈ δ isa γ-s-regular cover of X. Therefore there exist finite many members F1, F2, ..., Fn

of δ such that X =∪n

i=1(X − clγ(Fi)) ⊆∪n

i=1(X − Fi) = X −n∩

i=1

Fi. That is

n∩i=1

Fi = ϕ, a contradiction to the fact that δ is a filter base.

(3) ⇒ (1). If possible, suppose that X is not γ-s-regular-closed. Let X beembedded in a γ-s-regular space Y and let clγy(X)−X = ϕ. Let p ∈ clγy(X)−X.Let δ = X∩U : U is a nbd of p in Y and let ϱ = X∩V : V is the γ-semi-closureof a γ-semi-open subset of Y containing p . Since Y is γ-s-regular, δ and ϱ areequivalent and they are filter bases in X. It can be verified that each V ∈ ϱ isa γ-semi-closed subset of X. Hence δ is a γ-s-regular filter base in X. δ does nothave a γ-adherent point in X, since X is γ-T2. This is a contradiction. Hence theproof.

Definition 4.5. [6] A function f : X → Y is said to be γ-irresolute, if the inverseimage of every γ-semi-open set is γ-semi-open.

Theorem 4.6. If X = Πi∈IXi is γ-s-regular-closed, then each Xi is γ-s-regular-closed provided each Xi is γ-s-regular.

Proof. Let ξi be a γ-s-regular filter base in Xi. Let δi be the γ-semi-closedfilter base in Xi which is equivalent to ξi. Then P−1

i (Fi) : Fi ∈ ξi is a filterbase in X and P−1

i (Ui) : Ui ∈ ξi is a γ-semi-closed filter base in X, since theprojection mapping is γ-irresolute[6]. Also P−1

i (Ui) : Ui ∈ ξi is equivalent toP−1

i (Fi) : Fi ∈ ξi. Thus P−1i (Fi) : Fi ∈ ξi is a γ-s-regular filter base in X.

If x = (xi) is a γ-adherent of P−1i (Fi) : Fi ∈ ξi in X. Then xi is a γ-adherent

point of ξi. Thus Xi is γ-s-regular-closed.


References

[1] Ahmad, B., Hussain, S., Properties of γ-Operations on Topological Spaces,Aligarh Bull. Math., 22 (1) (2003), 45-51.

[2] Ahmad, B., Hussain, S., γ-Semi-Open Sets in Topological Spaces. II,Southeast Asian Bull. Maths., 34 (2010), 997-1008.

[3] Ahmad, B., Hussain, S., γ-Convergence in Topological Space, SoutheastAsian Bull. Maths., 29 (2005), 832-842.

[4] Ahmad, B., Hussain, S., γ∗-Regular and γ-Normal Space, Math. Today.,22 (1) (2006), 37-44.

[5] Ahmad, B., Hussain, S., On γ-s-Closed Subspaces, Far East Jr. Math. Sci.,31 (2)(2008), 279-291.

[6] Ahmad, B., Hussain, S., Noiri, T., On Some Mappings in TopologicalSpace, Eur. J. Pure Appl. Math., 1 (2008), 22-29.

[7] Arya, S.P., Bhamini, M.P., Some Generalizations of Urysohn Spaces,(1982) (preprint).

[8] Arya, S.P., Bhamini, M.P., P-Closed spaces, Indian J. Pure Appl. Math.,15 (1) (1984), 89-98.

[9] Dotsett, C., On Semi-Regular Spaces, Soochow Jr. Math., 10 (1975), 347-350.

[10] Hussain, S., Ahmad, B., On Minimal γ-Open sets, Eur. J. Pure Appl.Maths., 2(3) (2009), 338-351.

[11] Hussain, S., Ahmad, B., On γ-s-Closed Spaces, Sci. Magna Jr., 3 (4)(2007), 89-93.

[12] Hussain, S., Ahmad, B., On γ-s-Regular Spaces and Almost γ-s-Con-tinuous Functions, Lobackevskii. J. Math., 30 (4) (2009), 263-268. DOI:10.1134/ 51995080209040039.

[13] Hussain, S., Ahmad, B., Noiri, T., γ-Semi-Open Sets in Topologi-cal Spaces, Asian Eur. J. Math., 3 (3) (2010), 427-433. DOI: 10.1142/S1793557110000337.

[14] Kasahara, S., Operation-compact spaces, Math. Japon., 24 (1979), 97-105.

[15] Maheshwari, S.N., Prasad, R., On s-Regular Spaces, Glasnik Math., 10(1975), 347-350.

[16] Ogata, H., Operations on topological spaces and associated topogy, Math.Japon., 36 (1) (1991), 175-184.

[17] Rehman, F.U., Ahmad, B., Operations on topological spaces. II, Math.Today, 11 (1993), 13-20.



(0, 2; 0)-INTERPOLATION ON THE UNIT CIRCLE

Swarnima Bahadur

Department of Mathematics & AstronomyUniversity of LucknowLucknow 226007Indiae-mail: [email protected]

Abstract. In this paper, we study the explicit representation and convergence of Pal-

type weighted (0, 2; 0)-interpolation on two pairwise disjoint sets of nodes on the unit

circle, which are obtained by projecting vertically the zeros of(1− x2

)Pn (x) and P ′

n (x)

respectively on the unit circle, where Pn (x) stands for nth Legendre polynomial.

Keywords: Legendre polynomial, weighted (0, 2)-interpolation, Pal-type interpolation,

regularity, explicit forms, convergence.

2000 Mathematics Subject Classification: 41A05, 30E10.

1. Introduction

P. Turan initiated the study of (0, 2)-interpolation in order to get approximatesolution of the differential equation y” + f · y = 0. The first result was publishedby J. Suranyi and P. Turan [19] in 1955. J. Balazs [2] introduced a generalizationof this problem and considered the weighted (0, 2)-interpolation on the zeros of

ultraspherical polynomial P(λ)n (x) , λ > −1. Then several mathematicians [5], [8],

[9], [13], [15], [16], [22], [23], [24] considered the weighted (0, 2)-interpolation onvarious set of nodes. O. Kis [11] was first to consider the interpolation problem onthe unit circle and then a lot of mathematicians [17], [18], [20] studied differentinterpolation problems on the unit circle. In paper [1], author proved the con-vergence theorem of weighted (0, 2)∗-interpolation on the projected nodes on theunit circle.

In case of Pal-type interpolation, P. Mathur and S. Datta [14] considered theweighted Pal-type (0, 2; 0)−interpolation on (−∞,∞) . Recently M. Lenard [12]has considered the (0, 2; 0) and (0; 0, 2)-type interpolation problem on the zeros ofLegendre polynomial Pn (x) and give the explicit formulae. H.P. Dikshit [7] con-sidered the existence of Pal-type interpolation on the non-uniformly distributednodes on the unit circle. Later on, in [2], [3], author considered the convergenceof (0, 1; 0) and (0; 0, 1)-interpolation on the sets obtained by projecting vertically

58 swarnima bahadur

the zeros of (1− x2)Pn (x) and P ′n (x) respectively on the unit circle, where Pn (x)

stands for nth Legendre polynomial. In paper [4], author considered the (0; 0, 2)-interpolation on the projected nodes of (1− x2)Pn (x) and P ′

n (x) respectively onthe unit circle and proved a convergence theorem. The aim of this paper is toconsider explicit representation and convergence of weighted Pal-type (0, 2; 0)-interpolation on the unit circle.

Let Zn = zk = k = 0, 1, ..., 2n+ 1 satisfying

(1.1)

z0 = 1, z2n+1 = −1

zk = cos θk + i sin θk,

zn+k = −zk, k = 1 (1)n

and Tn = ωk = k = 1, ..., 2n− 2 such that

(1.2)

ωk = cosϕk + i sinϕk,

ωn+k = −ωk, k = 1 (1)n− 1

be two set of nodes such that the weighted (0, 2)-interpolation is prescribed onthe one set of nodes, whereas Lagrange interpolation on the points of other one

In Section 2 we give some preliminaries, in Section 3 we describe the pro-blem, in Section 4 we represent the explicit forms of interpolatory polynomials,and in Sections 5 and 6 we give the estimates and convergence of interpolatorypolynomials.

2. Preliminaries

In this section, we shall give some well known results, which we shall use in ourpresent paper.

The differential equation satisfied by Pn (x) is

(2.1) (1− x2) P”n (x) − 2x P ’

n (x) + n (n+ 1) Pn (x) = 0

(2.2) W (z) =2n∏k=1

(z − zk) = KnPn

(1 + z2

2z

)zn

(2.3) H(z) =2n−2∏k=1

(z − ωk) = K∗nP

′n

(1 + z2

2z

)zn−1

We shall require the fundamental polynomials of Lagrange interpolation basedon Zn and Tn

(2.4) Lk (z) =W (z)

W ′ (zk) (z − zk)

(0,2;0)-interpolation on the unit circle 59

(2.5) lk (z) =H(z)

H ′ (ωk) (z − ωk)

(2.6) Jk (z) =

∫ z

0

z2n+2 Lk (z) dz

(2.7) J1j (z) =

∫ z

0

z2n+j W (z) dz, j = 0, 1,

which satisfies

(2.8) J1j (−1) = (−1)j+1 J1j (1) .

For −1 ≤ x ≤ 1,

(2.9) |Pn (x)| ≤ 1

(2.10) (1− x2)14 |Pn (x)| ≤

(2

nπ

) 12

(2.11) (1− x2)34 |P ′

n (x)| ≤√2n

(2.12) |P ′n (x)| ≤

n (n+ 1)

2.

If uk be the zeros of P ′n (x) , then by [6]

(2.13) Pn (uk) >1√8πk

.

Let xk = cos θk, (k = 1, 2, .., n) be the zeros of the nth Legendre polynomialPn (x), with 1 > x1 > x2 > ... > −1, then

(2.14)

(1− x2

k)2 ≥ k2n−2, k = 1, ..,

[n2

](1− x2

k)2 ≥ (n− k + 1)2 n−2, k =

[n2

]+ 1, .., n

(2.15)

|P ′

n (xk)| ≥ k− 32n2, k = 1, ..,

[n2

]|P ′

n (xk)| ≥ (n− k + 1)−32 n2, k =

[n2

]+ 1, .., n

For more details, see [ 20], [21].

60 swarnima bahadur

3. The Problem

Let zk2n+1k=0 and ωk2n−2

k=1 be two disjoint set of nodes obtained by projectingvertically the zeros of (1− x2)Pn (x) and P ′

n (x) respectively on the unit circle,where Pn (x) stands for n

th Legendre polynomial and w (z) ∈ C be a given function,called weight function. Here we are interested to determine the convergence ofinterpolatory polynomials satisfying the conditions:

(3.1)

Rn (zk) = αk, k = 0, 1, 2, .., 2n+ 1

w (z)Rn (z) ”z=zk = βk, k = 1, 2, .., 2n

Rn (ωk) = γk, k = 1, 2, .., 2n− 2

where w (z) = (1− z2)52 and αk, βk and γk are complex constants.

4. Explicit representation of interpolatory polynomials

We shall write Rn (z) satisfying (3.1) as

(4.1) Rn (z) =2n+1∑k=0

αkAk (z) +2n∑k=1

βkBk (z) +2n−2∑k=1

γkCk (z) ,

where Ak (z) , Bk (z) and Ck (z) are unique polynomials each of degree ≤ 6n − 1determined by the following conditions:

For k = 0 (1) 2n+ 1

(4.2)

Ak (zj) = δkj, j = 0 (1) 2n+ 1(1− z2)

52 Ak (z)

”

z=zj= 0, j = 1 (1) 2n

Ak (ωj) = 0, j = 1 (1) 2n− 2

For k = 1 (1) 2n

(4.3)

Bk (zj) = 0, j = 0 (1) 2n+ 1(1− z2)

52 Bk (z)

”

z=zj= δkj, j = 1 (1) 2n

Bk (ωj) = 0, j = 1 (1) 2n− 2

For k = 1 (1) 2n− 2

(4.4)

Ck (zj) = 0, j = 0 (1) 2n+ 1(1− z2)

52 Ck (z)

”

z=zj= 0, j = 1 (1) 2n

Ck (ωj) = δkj, j = 1 (1) 2n− 2


Theorem 4.1. For k = 1 (1) 2n− 2

(4.5)

Ck (z) =z−2n (z2 − 1)W 2(z)Lk (z)

ω−2nk (ω2

k − 1)W 2(ωk)Lk (ωk)lk (z)

+z−2n−1W (z)H (z)

ω−2nk (ω2

k − 1)W 2(ωk)H ′ (ωk)Lk (ωk)Gk (z) + c1J10 (z) + c2J11 (z)

where

(4.6) Gk (z) =

z∫0

(1− z2

) zW ′ (z) + ckW (z)

z − ωk

Lk (z) dz

with

(4.7)

ck = −ωkW ′ (ωk)

W (ωk)

c1 = −Gk (1)−Gk (−1)

2J10 (1)and

c2 = −Gk (1) +Gk (−1)

2J11 (1).

Proof. Consider (4.5).

Clearly, Ck (zj) = 0 for j = 1 (1) 2n and Ck (ωj) = δkj for j = 1 (1) 2n− 2.

Also, for z = ±1, we get (4.7) .

Further, from(1− z2)

52 Ck (z)

”

z=zj= 0 for j = 1 (1) 2n, we get G′

k (z) ,

which on integration gives (4.6) .

Theorem 4.2. For k = 1 (1) 2n

(4.8) Bk (z) = z−2n−1W (z)H (z) bkJk (z) + b1J10 (z) + b2J11 (z)

where

(4.9) bk =1

2zk (z2k − 1)52 W ′(zk)H (zk)

(4.10) b1 = −bkJk (1)− Jk (−1)

2J10 (1)and b2 = −bk

Jk (1) + Jk (−1)

2J11 (1).

Proof. Consider (4.8).

62 swarnima bahadur

Clearly, Bk (zj) = 0 for j = 1 (1) 2n and Bk (ωj) = δkj for j = 1 (1) 2n− 2.

Again, for z = ±1, we get (4.10) .

Further, from(1− z2)

52 Bk (z)

”

z=zj= δkj for j = 1 (1) 2n, we get (4.9) for

j = k, owing to (2.7) and (2.14)-(2.15).

For j = k, one can verify the result.

Theorem 4.3. For k = 1 (1) 2n,

(4.11)

Ak (z) =(z2 − 1)H(z)L2

k (z)

(z2k − 1)H(zk)

+z−2n−1W (z)H (z)

(z2k − 1)W ′(ωk)H(zk)Tk (z) + a1J10 (z) + a2J11 (z)+ akBk (z)

where

(4.12)

ak = −(z2k − 1

) 52

H”(zk)

H ′(zk)+

2zk(z2k − 1)

H ′(zk)

H(zk)+

35zk

(z2k − 1)2

+7

(z2k − 1)+

28zk(z2k − 1)

L′k (zk) + 4

H ′(zk)

H(zk)L′k (zk)

(4.13) Tk (z) =

z∫0

z2n+1(1− z2

) Lk (z)− L′k (zk)Lk (z)

z − zkdz

(4.14) a1 = −Tk (1)− Tk (−1)

2J10 (1)and a2 = −Tk (1) + Tk (−1)

2J11 (1).

Further, for k = 0, 2n+ 1

(4.15) Ak (z) = z−2n−1W (z)H (z) a1J10 (z) + a2J11 (z)

where

(4.16) a1 =1

2W (1)H (1) J10 (1)

(4.17) a2 =1

2W (1)H(1)J11(z).

Proof. One can check that in this theorem (4.11) and (4.15) are polynomials ofrequired degree satisfying (4.2) , so we omit the details of proof.


5. Estimation of interpolatory polynomials

For convergence of interpolatory polynomial, we need the estimates of fundamentalpolynomials:

Lemma 1. Let Ak (z) be defined in (4.11), then for |z| ≤ 1, we have

(5.1)2n∑k=1

|Ak (z)| ≤ cn32 log n

(5.2) |A0 (z)| ≤ c, |A2n+1 (z)| ≤ c,

where c is a constant independent of n and z.

Lemma 2. Let Bk (z) be defined in (4.8), then for |z| ≤ 1, we have

(5.3)2n−2∑k=1

|Bk (z)| ≤ clog n

n12


Lemma 3. Let Ck (z) be defined in (4.5), then for |z| ≤ 1, we have

(5.4)2n−2∑k=1

|Ck (z)| ≤ cn32 log n


Proof of Lemma 3. Consider (4.5) and using (2.10)-(2.15), we get (5.4) , owingto results from [20], [21]. So we can omit the details of the proof.

Proof of Lemma 2. Consider (4.8), we have

(5.5) |Bk (z)| ≤ |Pn (x)P′n (x)| |bk| |Jk (z)|+ |b1| |J10 (z)|+ |b2| |J11 (z)| ,

where

(5.6) |bk| ≤1

(1− x2k)

74 |P ′

n (xk)|2

(5.7) |Jk (z)| ≤ max|z|≤1

|Lk (z)|z∫

0

t2n+2dt,

by substituting z = eiθ (0 ≤ θ < 2π) .

Combining (5.5)-(5.8) and using (2.10)-(2.15), we get (5.3) .Similarly one can prove Lemma 1.

64 swarnima bahadur

6. Convergence

In this section we prove the following:

Theorem 6.1. Let f (z) be continuous in |z| ≤ 1 and analytic in |z| < 1. Letthe arbitrary numbers βk’s be such that

(6.1) |βk| = o(n

32ω2(f, n

−1)), k = 1 (1) 2n.

Then the sequence Rn be defined by

(6.2) Rn (z) =2n+1∑k=0

f(zk)Ak (z) +2n∑k=1

βkBk (z) +2n−2∑k=1

γkCk (z)

satisfies the relation

(6.3) | Rn (z)− f(z)| = o(n

32 log n.ω2(f, n

−1))

where ω2 (f, n−1) is the modulus of smoothness of f (z) .

Remark. Let f(z) be continuous in |z| ≤ 1 and f ′ ∈ Lip α. α > 12, then the

sequence Rn converges uniformly to f (z) in |z| ≤ 1, follows from (6.3) provided

(6.4) ω2(f, n−1) = o

(n−1−α

).

To prove Theorem 6.1, we shall need the following:

Let f(z) be continuous in |z| ≤ 1 and analytic in |z| < 1. Then there existsa polynomial Fn (z) of degree 2n− 2 satisfying Jackson’s inequality

(6.5) |f(z)− Fn (z)| ≤ cω2

(f, n−1

), z = eiθ (0 ≤ θ < 2π)

and also an inequality due to O. Kis [11] viz. :

(6.6) F (m)n (z) ≤ cnmω2

(f,

1

n

),

for positive integer m.

Proof. Let z = eiθ (0 ≤ θ < 2π), using (6.1), (6.2) , (6.4)–(6.6) and Lemmas 1,2 and 3, we get the result.

References

[1] Bahadur, S., Mathur, K.K., Weighted (0, 2)∗− interpolation on theunit circle, Int. J. Appl. Math. Stat., 20 (M11) (2011), 73-78.


[2] Swarnima Bahadur, Pal-type (0, 1; 0)-interpolation on the unit circle,Advances in Theo. and Appl. Math., 6 (1) 2011, 35-39.

[3] Swarnima Bahadur, Pal-type (0; 0, 1)−interpolation on the unit circle,Int. Journal of Math. Analysis, vol. 5 (2011), no. 29, 1429-1434

[4] Swarnima Bahadur, Pal-type (0; 0, 2)−interpolation on the unit circle,Int. J. Math. Sci., vol. 11, no. 1-2 (2012), 103-111.

[5] Eneduanya, S.A.N, The weighted (0, 2) interpolation I, DemonstracioMathematica, 18 (1) (1985), 9-13.

[6] Balazs , J., Turan , P., Notes on Interpolation III, Acta Math. Sci.Hungar., 9 (1958), 195-214.

[7] Balaz s, J., Sulyozott (0, 2)-interpolacio ultraszferikus polinomok gyokein,MTA III. oszt. Kozl., 11 (1961), 305-338.

[8] Dikshit, H.P., Pal-type interpolation on non-uniformly distributed nodeson the unit circle, Journal of computational and Applied Mathematics, 155(2) (15) (2003), 253-261.

[9] Joo, I., On weighted (0, 2) interpolation, Annales Univ. Sci. Budapest.,Sect. Math., 38 (1995), 185-222.

[10] Joo, I., Szili, L., On weighted (0, 2)-interpolation on the roots of Jacobipolynomials, Acta Math. Hung., 66 (1-2) (1995), 25-50.

[11] Kis, O., Remarks on interpolation (Russian), Acta Math. Sci. Hungar., 11(1961), 49-64.

[12] Lenard, Margit, On weighted (0.2)- type interpolation, Elect. Trans.Num. Anal., 25 (2006), 206-223.

[13] Lenard, Margit, Weighted (0, 2)-Interpolation with interpolatory boun-dary conditions, Annales Univ. Sci. Budapest., Sect. Comp. 24 (2004),253-273.

[14] Mathur, P., Datta, S., On pal type weighted lacunary (0, 2; 0)-inter-polation on the infinite interval (−∞,∞), Approx. Theo. & its Appl., 17(4) (2001), 1-10.

[15] Prasad, J., Balazs -type interpolation on Laguerre abscissas, Math. Japo-nica, 13 (1967), 47-53.

[16] Prasad, J., On the weighted (0, 2)-interpolation, SIAM J. Numer. Anal.,7 (1970), 428-446.

[17] Sharma, A., Lacunary interpolation in the roots of unity, Z. Angew. Math.,46 (1964), 41-49.

66 swarnima bahadur

[18] Sharma, A., Riemanschneider, S.D., Birkhoff interpolation at the nth

roots of unity, Convergence, Cand. J. Math., 33 (2) (1981), 362-371.

[19] Su ranyi, J., Turan, P., Notes on interpolation. I. On some interpola-torical properties of the ultraspherical polynomials, Acta Math. Acad. Sci.Hung., 6 (1955), 66-79.

[20] Xie, S., (0, 1, 3)∗-interpolation on the unit circle, Acta Math. Sinica, 39 (5)(1996), 690-700.

[21] Szego, G., Orthogonal Polynomials, Amer. Math. Soc. Coll. Publ., NewYork, 1959.

[22] Szili, L., Weighted (0, 2)-interpolation on the roots of Hermite polynomials,Annales Univ. Sci. Budapest., Sect. Math., 27 (1985), 153-166.

[23] Szili, L., Weighted (0, 2) interpolation on the roots of the classical ortho-gonal polynomials, Bull. Allahabad Math. Soc., 8-9 (1993-94), 111-120.

[24] Szili, L., A survey on (0, 2) interpolation, Annales Univ. Sci. Budapest.,Sect. Comp., 16 (1996), 377-390.



ON LAPLACE TRANSFORMS OF GENERALIZED WHITTAKERFUNCTION OF MULTI-VARIABLES Mλ,µ1···µk

(x1, ..., xk)

M. Kamarujjama

Waseem A. Khan

Department of Applied MathematicsFaculty of Engineering and TechnologyAligarh Muslim UniversityAligarh – 202002Indiae-mails: [email protected]

waseem08 [email protected]

Abstract. In this paper, a Laplace transform of generalized Whittaker function is

derived which is used to obtain further some partly bilateral and partly unilateral gene-

rating function and series expansion. Some special cases are also discussed.

Keywords: Generalized Whittaker function, Lauricella’s function, Appell’s function

and Laplace transform.

2000 AMS Subject Classification: 33C15, 33C65, 33C70.

1. Introduction and definition

A Whittaker function Mλ,µ was introduced by Whittaker [2] (see also Whittakerand Watson [3]) in terms of the confluent hypergeometric function 1F1

(or Kummer’s function)

(1.1) Mλ,µ(x) = xµ+1/2e(−1/2)x1F1

(1

2+ µ− λ, 2µ+ 1; x

).

Further generalization of the Whittaker functionMλ,µ was introduced by Humbert[6; p.63 (15)] in the following form

(1.2)Mλ,µ1···µk

(x1, ..., xk) = xµ+1/21 · · · xµk+1/2

k exp

[−1

2(x1 + · · ·+ xk)

]·Ψ(k)

2 [µ1 + · · ·+ µk − λ+ k/2; 2µ1 + 1, ..., 2µk + 1; x1, ..., xk],

where Ψ(k)2 denotes Humbert’s confluent hypergeometric function of n-variables

[6; p. 62(11)]

68 m. kamarujjama, w.a. khan

(1.3)

Ψ(k)2 [a; c1, ..., ck;x1, ..., xk]

=∞∑

m1···mk=0

(a)m1+···+mk

(c1)m1 · · · (ck)mk

xm11 · · · xmk

k

m1! · · ·mk!(max|x1|, |x2|, ..., |xk| < ∞).

A Lauricella function F(k)C [6; p. 60] generalized the Appell function F4 to a

function of k variables which is defined as

(1.4)

F(k)C [a, b; c1, ..., ck;x1, ..., xk] = F 2:0;...;0

0:1;...;1

a, b : ; ...; ;x1, ..., xk

: c1; ...; ck ;

=

∞∑m1,m2,...,mk=0

(a)m1+···+mk(b1)m1+···+mk

(c1)m1 · · · (ck)mk

xm11 · · · xmk

k

m1! · · ·mk!

(√

|x1|+ · · ·+√|xk| < 1).

2. Integral transforms

We first establish the following integral

(2.1)

I =

∫ ∞

0

uν−1e−puMλ,µ1···µk(x1u, ..., xku)du

=xµ1+1/21 · · · xµk+1/2

k Γ(b)

(p+X)bF

(k)C

[a, b; 2µ1+1, ..., 2µk+1;

x1

p+X, ...,

xk

p+X

],

where a = µ1 + · · · + µk +k

2− λ, b = µ1 + · · · + µk +

k

2+ ν, X =

x1 + · · ·+ xk

2and Re(p+X) > 0.

For k = 2, equation (2.1) reduces to

(2.2)

I =

∫ ∞

0

uν−1e−puMλ,µ1,µ2(x1u, x2u)du

=xµ1+1/21 x

µ2+1/22 Γ(µ1 + µ2 + 1 + ν)

(p+X)µ1+µ2+1+ν

·F4

[µ1+µ2+1−λ, µ1+µ2+1+ν; 2µ1+1, 2µ2+1;

x1

p+X,

x2

p+X

](Re(p+X) > 0)

where as for k = 1, equation (2.1) reduces to a known result [1; p. 215(11)].

Proof of result (2.1)

By using definitions (1.2) and (1.3), the L.H.S. of integral (2.1) is given by

on laplace transforms of generalized whittaker function ... 69

I =

∫ ∞

0

uν−1e−pu(x1u)µ1+1/2 · · · (xku)

µk+1/2 exp

[−1

2(x1u+ · · ·+ xku)

]·Ψ(k)

2 [µ1 + · · ·+ µk − λ+ k/2; 2µ1 + 1, ..., 2µk + 1; x1u, ..., xku]du

= xµ1+1/21 · · · xµk+1/2

k

∞∑m1···mk=0

(a)m1+···+mk

(2µ1 + 1)m1 · · · (2µk + 1)mk

(x1)m1 · · · (xk)

mk

m1! · · ·mk!

·∫ ∞

0

uµ1+···+µk+k/2+m1+···+mk+ν−1e−(p+x1+···+xk

2)udu,

and then, using the integral transform [1, p.137(1)], we get the main result (2.1).In view of integral (2.1), we can also establish the following integral

(2.3)

I =

∫ ∞

0

uν−1e−pu−zu2

Mλ,µ1···µk(x1u, ..., xku)du

= xµ1+1/21 · · · xµk+1/2

k

∞∑q=0

(−1)qzqΓ(b+ 2q)

q!(p+X)b+2q

·F (k)C

[a, b+ 2q; 2µ1 + 1, ..., 2µk + 1;

x1

p+X, ...,

xk

p+X

].

where a = µ1 + · · · + µk +k2− λ, b = µ1 + · · · + µk +

k

2+ ν, X =

x1 + · · ·+ xk

2and Re(p+X) > 0.

Clearly, if z = 0 in (2.3), the above result reduces to the result (2.1).

3. Generating relations

A well known modified generating relation of Exton is given by [5; p. 147(3)]

(3.1) exp

(s+ t− xt

s

)=

∞∑m=−∞

∞∑n=m∗

sm tn

m! n!1F1 (−n; m+ 1 ; x)

where 1F1 (−n; m+ 1 ; x) /m!n! = L(m)n (x)/(m + n)!, 1F1 and L

(m)n are confluent

hypergeometric function and Laguerre polynomials, respectively (see [4; p. 200(1)]).Pathan and Yasmeen [7] modified the above result (3.1) of Exton, by definingm∗ = max[0,−m] and

L(n)n (x)

(m+ n)!=

1

n!

n∑r=m∗

(−n)rxr

(m+ r)!r!if n ≥ m∗

= 0 if 0 ≤ n < m∗.

A set of expansions

xr = 2r∞∑

m=−∞

∞∑n=m∗

(−n)r(x/2)m+n

m!n!1F1 (−n; m+ 1 ; x) (3.2)


for r = 0, 1, 2, ..., has been obtained by Exton [5, p. 148(8)] from (3.1) by takingsuccessive partial derivatives with respect to t and letting s = t = x/2.

On replacing s, t and x by su, tu and xu, respectively, in (3.1), multiplyingboth sides by

uν−1e−puMλ,µ1···µk(x1u, ..., xku)

integrating the multiple series with respect to u between the limits zero and infi-nity. Using integral (2.1) and definition (1.2) and adjusting the parameters, we get

(3.3)

F(k)C [a, b; 2µ1+1, ..., 2µk+1;

x1

p−s−t+ xts+X

, ...,xk

p−s−t+xts+X

]

=(p−s−t+xt

s+X)b

(p+X)b

∞∑m=−∞

∞∑n=m∗

sm tn(b)m+n

m! n!(p+X)m+n

n∑l=0

(−n)l (b+m+n)l(m+1)l l!

(x

p+X

)l

F(k)C

[a, b+m+n+ l; 2µ1+1, ..., 2µk+1;

x1

p+X, ...,

xk

p+X

]

where a = µ1 + · · · + µk +k

2− λ, b = µ1 + · · · + µk +

k

2+ ν, X =

x1 + · · ·+ xk

2and Re(p) > 0, Re(p+X) > 0.

Similarly, using the result (3.2) in place of (3.1), we get

(3.4)

F(k)C [a, b+ r; 2µ1 + 1, ..., 2µk + 1;

x1

p+X, ...,

xk

p+X]

=2r

(b)rxr

∞∑m=−∞

∞∑n=m∗

(−n)r(b)m+n(x/2)m+n

m!n!(p+X)m+n−r

·n∑

l=0

(−n)l (b+m+ n)l(m+ 1)l l!

(x

p+X

)l

·F (k)C

[a, b+m+ n+ l ; 2µ1 + 1, ..., 2µk + 1;

x1

p+X, ...,

xk

p+X

]

where a = µ1 + · · · + µk +k

2− λ, b = µ1 + · · · + µk +

k

2+ ν, X =

x1 + · · ·+ xk

2and Re(p) > 0, Re(p+X) > 0.

By setting s = t = x/2 in (3.3), the result reduces to

(3.5)

F(k)C [a, b; 2µ1 + 1, ..., 2µk + 1; x1

p+X, ..., xk

p+X]

=∞∑

m=−∞

∞∑n=m∗

(x/2)m+n(b)m+n

m! n!(p+X)m+n

n∑l=0

(−n)l (b+m+ n)l(m+ 1)l l!

(x

p+X

)l

·F (k)C

[a, b+m+ n+ l; 2µ1 + 1, ..., 2µk + 1;

x1

p+X, ...,

xk

p+X

]

on laplace transforms of generalized whittaker function ... 71

where a = µ1 + · · · + µk +k

2− λ, b = µ1 + · · · + µk +

k

2+ ν, X =

x1 + · · ·+ xk

2and Re(p) > 0, Re(p+X) > 0. which can also be obtained from (3.4) by takingr = 0.

4. Special cases

On setting x = 0 in (3.3), the result reduces to

(4.1)

F(k)C [a, b; 2µ1 + 1, ..., 2µk + 1;

x1

p− s− t+X, ..., ...,

xk

p− s− t+X]

=(p− s− t+X)b

(p+X)b

∞∑m=−∞

∞∑n=m∗

sm tn(b)m+n

m! n! (p+X)m+n

·F (k)C

[a, b+m+ n; 2µ1 + 1, ..., 2µk + 1;

x1

p+X, ..., ...,

xk

p+X

]

where a = µ1 + · · · + µk +k

2− λ, b = µ1 + · · · + µk +

k

2+ ν, X =

x1 + · · ·+ xk

2and Re(p) > 0, Re(p+X) > 0.


(4.2)

F4

[a, b; 2µ1 + 1, 2µ2 + 1;

x1

p− s− t+X,

x2

p− s− t+X

](p− s− t+X)b

(p+X)b

∞∑m=−∞

∞∑n=m∗

sm tn(b)m+n

m! n! (p+X)m+n

·F4

[a, b+m+ n ; 2µ1 + 1, 2µ2 + 1;

x1

p+X,

x2

p+X

]

where a = µ1 + µ2 + 1 − λ, b = µ1 + µ2 + 1 + ν, X =x1 + x2

2and Re(p) > 0,

Re(p+X) > 0.


(4.3)

F4

[a, b; 2µ1 + 1, 2µ2 + 1;

x1

p+X,

x2

p+X

]=

∞∑m=−∞

∞∑n=m∗

(x/2)m+n(b)m+n

m! n!(p+X)m+n

n∑l=0

(−n)l (b+m+ n)l(m+ 1)l l!

(x

p+X

)l

·F4

[a, b+m+ n+ l ; 2µ1 + 1, 2µ2 + 1;

x1

p+X,

x2

p+X

]

where a = µ1 + µ2 + 1 − λ, b = µ1 + µ2 + 1 + ν, X =x1 + x2

2and Re(p) > 0,

Re(p+X) > 0.



(4.4)

2F1[a, b; 2µ1 + 1;x1

p+X]

=∞∑

m=−∞

∞∑n=m∗

(x/2)m+n(b)m+n

m! n!(p+X)m+n

n∑l=0

(−n)l (b+m+ n)l(m+ 1)l l!

(x

p+X

)l

·2F1

[a, b+m+ n+ l; 2µ1 + 1;

x1

p+X

]where a = µ1 +

1

2− λ, b = µ1 +

1

2+ ν, X =

x1

2and Re(p) > 0, Re(p+X) > 0.

By putting x1 = 0 and x2 = 0 in (4.3), the above result reduces to a knownresult of Pathan and Yasmeen [7; p. 242(2.3)]

(4.5) 1 =∞∑

m=−∞

∞∑n=m∗

(x/2p)m+n (c)m+n

(m+ n)!P (m,c−1)n

(p− 2x

p

)

where the Jacobi polynomials P(α,β)n (x) is defined by [4; p. 254(1)]

P (α,β)n (x) =

(1 + α)nn!

2F1

[−n, 1 + α + β + n ;

1− x

21 + α ;

].

References

[1] Erdelyi, A. et al., Table of Integral Transfoms, vol. 1, McGraw-Hill NewYork, 1954.

[2] Whittaker, E.T., An expression of certain known function as generalizedhypergeometric functions, Bull. Amer. Math. Soc., 10 (1903), 125-134.

[3] Whittaker, E.T., Watson, G.N., A Course in Modern Analysis, 4thed. Cambridge University Press. Cambridge, England, 1990.

[4] Rainville, E.D., Special functions, The Macmillan Co., New York, 1960.

[5] Exton, H., A new generating functions for associated Laguerre polynomialsand resulting expansions, Jnanabha, 13 (1983), 147-149.

[6] Srivastava, H.M., Manocha, H.L., A treatise on generating functions,Halsted Press (Ellis Horwood Ltd., Chichester), John Wiley and Sons, NewYork, 1984.

[7] Pathan, M.A., Yasmeen, On partly bilateral and partly unilateral gene-rating functions, J. Austr. Math. Soc., Ser. B., 28 (1986), 240-245.



FEKETE-SZEGO PROBLEM FOR CONCAVE UNIVALENTFUNCTIONS DEFINED BY SALAGEAN OPERATOR

Alawiah Ibrahim

School of Engineering and ScienceVictoria UniversityP.O. Box 14428, Melbourne City, MC 8001AustraliaSchool of Mathematical SciencesFaculty of Science and TechnologyUniversiti Kebangsaan Malaysia43600 Bangi, SelangorMalaysiae-mail: [email protected]

Maslina Darus1

School of Mathematical SciencesFaculty of Science and TechnologyUniversiti Kebangsaan Malaysia43600 Bangi, SelangorMalaysiae-mail: [email protected]

Sever S. Dragomir

School of Engineering and ScienceVictoria UniversityP.O. Box 14428, Melbourne City, MC 8001Australiae-mail: [email protected] of Computational and Applied MathematicsUniversity of WitwatersrandPrivate Bag-3, Wits-2050, JohannesburgSouth Africae-mail: [email protected]: http://www.staff.vu.edu.au/rgmia/dragomir

Abstract. Let C0 (α) denote the class of concave univalent functions defined in the

open unit disk U. In this paper, we investigate the sharp upper bounds of Fekete-Szego

functional with real and complex parameter λ for the class of concave univalent functions

defined by Salagean differential operator.

Keywords: Concave, Univalent function, Salagean differential operator.

2000 Mathematics Subject Classification: 30C45, 30C10.

1Corresponding author.

74 a. ibrahim, m. darus, s.s. dragomir

1. Introduction

Let S denote the class of all analytic and univalent functions

f(z) = z +∞∑n=2

anzn(1.1)

defined on the open unit disk U = z ∈ C : |z| < 1.Denote by S∗ (β), C (β) and K (α, β), the classes of starlike functions of order

β, convex functions of order β and close-to-convex functions of order α type βrespectively, which are analytically defined as follows:

(i) S∗ (β) =

f ∈ A : Re

(zf ′ (z)

f (z)

)> β, z ∈ U, 0 ≤ β < 1

,

(ii) C(β) =

f ∈ A : Re

(1 +

zf ′′ (z)

f ′ (z)

)> β, z ∈ U , 0 ≤ β < 1

,

(iii) K (α, β) =

f ∈ A : Re

(f ′ (z)

g′ (z)

)> α, g (z) ∈ C (β) ,

z ∈ U, 0 ≤ α < 1, 0 ≤ β < 1

.

In 1933, Fekete and Szego [19] obtained the maximum value of |a3 − λa22| asa function of the real parameter λ, namely∣∣a3 − λa22

∣∣ ≤ 1 + 2 exp

(−2λ

1− λ

),

for the class S of analytic and univalent functions given by (1.1). This inequalityis sharp for each λ ∈ [0, 1]. In the literature, there exists a large number of resultsof the Fekete-Szego functional |a3 − λa22| for various subclasses of S, such as theclass of S∗ (β) , C (β) and K (α, β). For instance, Keogh and Merkers [10], Kaplan[26], Koepf [27] solved the Fekete-Szego problem for close-to-convex functions.Nasr and Gawad [20], Gawad and Thomas [12], Darus and Thomas [18], Ibrahimand Darus [4] and others generalized this result for the class of functions that areclose-to-convex functions of order α and type β. Later, Avkhadiev et al. [8], [9]and Bhowmik et al. [5], [6], they gave another treatment of Fekete-Szego problemby considering the class of concave univalent functions given by (1.1).

Also, there are several authors that proved this type of result for the Fekete-Szego functional for the class of function defined by differential operator, see [16],[3], for example, by using the Salagean differential operator Dk [11], for f ∈ Swhich is defined by

(i) D0f(z) = f(z),

(ii) D1f(z) = Df(z) = z +∞∑n=2

nanzn,

fekete-szego problem for concave univalent functions ... 75

(iii) Dkf(z) = D(Dk−1f (z)

)= z +

∞∑n=2

nkanzn; k = 1, 2, ... .

Denote by S∗k , the class of k-starlike functions which is analytically defined as

follows:

S∗k =

f(z) ∈ S : Re

(Dk+1f(z)

Dkf(z)

)> 0, k = 0, 1, 2, ... , z ∈ U

.(1.2)

In this paper, we investigated the sharp upper bounds of Fekete-Szego func-tional |a3 − λa22| for the class of concave univalent functions with real and complexparameter λ, where the function of f is defined by Salagean differential opera-tor (1.2).

2. Preliminary results

A function f : U → C is said to belong to the family C0 (α) if f satisfies thefollowing conditions:

(a) f is analytic in U with the standard normalization f (0) = f ′ (0) − 1 = 0.In addition it satisfies f (1) = ∞.

(b) f maps conformally onto a set whose complement with respect to C is con-vex.

(c) The opening angle of f (U) at ∞ is less than or equal to πα, α ∈ (1, 2].

The class C0 (α) is referred to concave univalent functions and for a detaileddiscussion about concave functions we refer to [8], [9], [17] and the referencestherein. Recently, the class C0 (α) of concave function was considered by Bhowmiket al. [5], [6].

We recall the analytic characterization for the functions in C0 (α), α ∈ (1, 2]:f ∈ C0 (α) if and only if RePf (z) > 0, z ∈ U, where

Pf (z) =2

α− 1

[(α + 1)

2

1 + z

1− z− 1− z

f ′′ (z)

f ′ (z)

].

In [5], [6] they used this characterization and proved the following theorem.

Theorem 1 Let α ∈ (1, 2]. A function f ∈ C0 (α) if, and only if, there exist astarlike function ϕ ∈ S∗ such that f (z) = Λϕ (z) where

Λϕ (z) =

z∫0

1

(1− t)α+1

(t

ϕ (t)

)(α−1)/2

dt

and S∗ denote the family of starlike functions g defined by g ∈ S∗ if and only if

Re

(zg′ (z)

g (z)

)> 0.


The objective of the present paper is to give some generalizations of the resultof Fekete-Szego problem given by Bhowmik et al. [5] for the starlike functiondefined by Salagean differential operator Dkf, k = 0, 1, 2, ..., which is f ∈ S∗

k ischaracterized by the condition (1.2).

In order to prove our main results, we need to recall the following lemma.

Lemma 1 [27] Let g(z) = z+∞∑n=2

bnzn ∈ S∗. Then |b3 − λb22| ≤ max 1, |3− 4λ|,

which is sharp for the Koebe function k if |λ− 3/4| ≥ 1/4 and for (k (z))1/2 =z

1− z2if |λ− 3/4| ≤ 1/4.

3. Main result and its proof

We consider the Fekete-Szego functional |a3 − λa22| for real and complex parameterλ. Our results are contained in the following theorems.

Theorem 1 Let f ∈ C0 (α) have the expansion given by (1.1), α ∈ (1, 2],k = 0, 1, 2, ... . If λ is real, then we have

12 |a3 − λa22|

≤

(3 + 22k

)(2− 3λ)α2

+3(1− 22k

)(1− 2α)λ

+6(1− 3k

)α + 2

(3k+1 − 22k

),

if λ ≤ λ0;

4 [(2− 3λ)α2 + 1] , if λ0 ≤ λ ≤ 2 (α− 1)

3α;

4 [(10− 9λ)α + (2− 3λ)]

3 (2− λ)− (2− 3λ)α, if

2 (α− 1)

3α≤ λ ≤ 2

3;

12 (1− λ)α√

12(1−λ)

(4−3λ)2−(3λ−2)2α2 , if 23≤ λ ≤ λ2;

4 [(3λ− 2)α2 − 1] , if λ2 ≤ λ ≤ 2 (α + 2)

3 (α + 1);

(3 + 22k

)(3λ− 2)α2

+3(22k − 1

)(1− 2α)λ

+6(3k − 1

)α + 2

(22k − 3k+1

),

if λ ≥ 2 (α + 2)

3 (α + 1);

where

λ0 =22k−2 (α + 1)− 3k

3 (22k−3) (α− 1)and λ2 =

2

3+

1

6α2

(√8α2 + 1− 1

).

The inequalities are sharp.


Theorem 2 Let f ∈ C0 (α) have the expansion given by (1.1), α ∈ (1, 2],k = 0, 1, 2, ... . If λ are complex numbers, then we have∣∣a3 − λa22

∣∣ ≤ max

1,

1

12(α + 1) ν (α, λ)

,

whereν (α, λ) = |(2− 3λ) (α + 1) + 2|+ 2 (α− 1) |3λ− 2|

+

(α− 1

α + 1

)|6 + [2− 3 (α− 1)λ]| .

Proof. We recall from Theorem 1 that for f ∈ C0 (α) if and only if there exist a

function ϕ (z) = z +∞∑n=2

ϕnzn ∈ S∗

k , k = 0, 1, 2, ... such that

f ′ (z) =1

(1− z)α+1

(z

Dnϕ (z)

)(α−1)/2

,(3.1)

where f has the form given by (1.1) and Dn is the Salagean operator. Comparingthe coefficients of z and z2 on the both sides of the series expansion (3.1), weobtain that

a2 =(α + 1)

2− 2k−2 (α− 1)ϕ2

and

a3 =1

6(α+ 1) (α + 2)− 2k−1

3

(α2 − 1

)ϕ2

−3k−1

2(α− 1)ϕ3 +

22k−3

3(α− 1)ϕ2

2,

respectively.A computation yields that

a3 − λa22 =(α + 1)2

4

[2 (α+ 2)

3 (α+ 1)− λ

]+2k−2

(α2 − 1

)(λ− 2

3

)ϕ2 −

3k−1

2(α− 1)

×

[ϕ3 −

(22k−2 (α + 1)− 3λ

(22k−3

)(α− 1)

3k

)ϕ22

](3.2)

Now, we need to investigate the maximum values of the function |a3 − λa22| byconsidering several cases of λ.

Case 1: Consider the first case for all λ ≤ 22k−2 (α + 1)− 3k

3 (22k−3) (α− 1).

We observe that the assumption on λ is seen to be equivalent to

1

3k[22k−2 (α + 1)− 3λ

(22k−3

)(α− 1)

]≥ 1


and the first term in equation (3.2) is nonnegative. Hence, using the Lemma 1 forthe last term in (3.2), we have∣∣∣∣∣ϕ3 −

(22k−2 (α + 1)− 3λ

(22k−3

)(α− 1)

3k

)ϕ22

∣∣∣∣∣≤

22k (α + 1)− 3λ(22k−1

)(α− 1)

3k− 3

and noticing that for ϕ ∈ S∗k , |ϕn| ≤ n1−k, k = 2, 3, ... , we have from the equality

(3.2) that

∣∣a3 − λa22∣∣ ≤ (α + 1)2

4

[2 (α + 2)

3 (α + 1)− λ

]+ 2k−2

(α2 − 1

)(2

3− λ

)|ϕ2|

+3k−1

2(α− 1)

∣∣∣∣∣ϕ3 −

(22k−2 (α+ 1)− 3λ

(22k−3

)(α− 1)

3k

)ϕ22

∣∣∣∣∣=

(α + 1) (α + 2)

6− λ

4(α + 1)2 +

(α2 − 1)

2

(2

3− λ

)+3k−1

2(α− 1)

(22k (α + 1)− 3λ

(22k−1

)(α− 1)

3k− 3

).

It can be simplified to

|a3 − λa22| ≤ 1

12

[(3 + 22k

)(2− 3λ)α2 + 3

(1− 22k

)(1− 2α)λ

+6(1− 3k

)α + 2

(3k+1 − 22k

)],

for λ ∈(∞,

22k−2 (α + 1)− 3k

3 (22k−3) (α− 1)

).

Case 2: Let λ ≥ 2 (α + 2)

3 (α + 1).

For this case, the first term in (3.2) is nonnegative. The condition on λ in

particular gives λ ≥ 2

3and therefore our assumption on λ implies that

22k−2 (α + 1)− 3λ(22k−3

)(α− 1)

3k≤ 22k

3k

(1

2

).

Again, it follows from Lemma 1, that∣∣∣∣∣ϕ3 −22k−2 (α + 1)− 3λ

(22k−3

)(α− 1)

3k

∣∣∣∣∣ϕ22 ≤ 3−

22k (α + 1)− 3λ(22k−1

)(α− 1)

3k.


In view of these observation and an use of the inequality that |ϕ2| ≤ 21−k, equality(3.2) gives∣∣a3 − λa22

∣∣ ≤ (α + 1)2

4

[λ− 2 (α + 2)

3 (α + 1)

]+ 2k−2

(α2 − 1

)(λ− 2

3

)(21−k

)+3k−1

2(α− 1)

(3−

22k (α+ 1)− 3λ(22k−1

)(α− 1)

3k

).

(3.3)

Thus, simplifying the right hand side expression (3.3), we obtain that

|a3 − λa22| ≤ 1

12

[(3 + 22k

)(3λ− 2)α2 + 3

(22k − 1

)(1− 2α)λ

+6(3k − 1

)α− 2

(3k+1 − 22k

)],

for λ ∈[2 (α + 2)

3 (α + 1),∞).

Case 3: Consider λ, where

λ ∈(22k−2 (α+ 1)− 3k

3 (22k−3) (α− 1),2 (α + 2)

3 (α + 1)

).

Now we deal with the case by using the formulas (3.1) and (3.2) together withthe representation formula for ϕ (z) ∈ S∗

k . Let us define w (z) by

Dk+1ϕ (z)

Dkϕ (z)=

1 + zw (z)

1− zw (z); (w(z) = 1)(3.4)

where w : U → U is a function analytic in U with the Taylor series

w (z) =∞∑n=0

cnzn.

Comparing the coefficients of z and z2 in (3.4), we get that

ϕ2 = 21−kc0 and ϕ3 =1

3k(c1 + 3c20

).(3.5)

Inserting these resulting formulas (3.5) into (3.2) yields

a3 − λa22 ≤ (α + 1)2

4

[2 (α + 2)

3 (α + 1)− λ

]+2k−2

(α2 − 1

)(λ− 2

3

)(21−kc0

)+3k−1

2(α− 1)

[1

3k(c1 + 3c20

)−

(22k−2 (α + 1)− 3λ

(22k−3

)(α− 1)

3k

)(22−2k

)c20

]= A+Bc0 + Cc20 +Dc1,

(3.6)


where

A =1

6(α + 2) (α + 1)− λ

4(α + 1)2 ,

B =1

6

(α2 − 1

)(3λ− 2) ,

C = − 1

12(α− 1) [4− 2α+ 3λ (α− 1)] ,

D = −1

6(α− 1) .

Hence, by using the well known inequalities that |c0| ≤ 1 and |c1| ≤ 1−|c0|2, from(3.6) we obtain that∣∣a3 − λa22

∣∣ ≤ ∣∣A+Bc0 + Cc20∣∣+ 1

6(α− 1)

(1− |c0|2

).(3.7)

Now, in order to determine the maximum value of (3.7), let c0 = reiθ, then weconsider the quadratic expression

f (r, θ) = |A+Bc0 + Cc20|2

= (A− Cr2)2+B2r2 + 2Br (A+ Cr2) cos θ + 4ACr2 cos2 θ,

(3.8)

where cos θ ∈ [−1, 1] , r ∈ (0, 1]. For getting the upper bounds of |a3 − λa22|, wehave to find the biggest value of (3.8) for r in the interval (0, 1]. So, let x = cos θ,then from (3.8) we have

h(x) =(A− Cr2

)2+B2r2 + 2Br

(A+ Cr2

)x+ 4ACr2x2.(3.9)

We have to determine the maximum value of (3.9) for x ∈ [−1, 1]. So, for this,we need to consider the several subclasses of λ, where

λ ∈(22k−2 (α+ 1)− 3k

3 (22k−3) (α− 1),2 (α + 2)

3 (α + 1)

).

Case 3A: First, consider

λ ∈(22k−2 (α + 1)− 3k

3 (22k−3) (α− 1),2 (α− 2)

3 (α− 1)

).

We observe that for λ in this interval, we have A > 0, B < 0, C > 0 andA + Cr2 > 0 for r ∈ (0, 1], and (3.9) attains its maximum value at x = −1.Therefore, it gives that∣∣a3 − λa22

∣∣ ≤ g(r) = A−Br + Cr2 +1

3(α− 1)

(1− r2

).(3.10)

By a simple calculation, we show that the maximum value of (3.10) attains at theboundary of r, i.e. r = 1. Therefore

g(r) ≤ g(1) = A−B + C =1

3

[(2− 3λ)α2 + 1

].


Case 3B: Let λ =2 (α− 2)

3 (α− 1).

In this case, we get C = 0, therefore h(x) becomes a linear function,

h(x) = A2 +B2r2 + 2BrAx.(3.11)

It is easy to show that the maximum value of (3.11) occurs at x = −1 and r = 1.Again we get the maximum value of |a3 − λa22| as the previous case.

Case 3C: Let λ ∈(2 (α− 2)

3 (α− 1),2 (α− 1)

3α

).

In this interval, the quadratic function (3.9) has maximum value at

x (r) = −B

4

(1

Cr+

r

A

),

where x (r) is monotonic increasing in r ∈ (0, 1] and x (1) < −1. Hence we getthe upper bound as in Cases 3A and 3B. As conclusion, from the Cases 3A, 3Band 3C give us that ∣∣a3 − λa22

∣∣ ≤ 1

3

[(2− 3λ)α2 + 1

]for

λ ∈(22k−2 (α + 1)− 3k

3 (22k−3) (α− 1),2 (α− 1)

3α

).

Case 3D: Let λ ∈[2 (α− 1)

3α,2

3

).

From the Case 3C, the inequality x (1) < −1 gives that

2 (3λ+ 4α2 − 12α2λ+ 9α2λ2 − 4)

[(3λ− 4) + α (3λ− 2)] [α (3λ− 2)− (3λ− 4)]< 0,

hence it shows that

p(λ) = 9α2λ2 +(3− 12α2)λ+ 4(α2 − 1

)< 0

where λ < 23. Factorizing p (λ), we have

λ1 =2

3− 1

6α2

(1 +

√8α2 + 1

)(3.12)

and

λ2 =2

3− 1

6α2

(1−

√8α2 + 1

).(3.13)

It is clear that λ1 < λ2. Therefore, for λ ∈[2 (α− 1)

3α, λ1

), functions (3.9) and

(3.10) have their maximum value at

x = −1 and rm =−3B

−6C + α− 1∈ (0, 1]


respectively. Hence the upper bound of Fekete-Szego functional is given by

|a3 − λa22| ≤ g(rm) = A−Brm + Cr2m +1

3(α− 1)

(1− r2m

)=

4 [(10− 9λ)α + (2− 3λ)]

3 (2− λ)− (2− 3λ)α.

(3.14)

Next, we consider for λ ∈[λ1,

2

3

). In this interval, the quadratic equation (3.9)

attains its maximum value at

x (r) =−B (A+ Cr2)

4ACr

with

h (x (r)) = − 1

4AC

(B2 − 4AC

)(A− Cr)2 .

Hence, the Fekete-Szego functional satisfies the following inequality

|a3 − λa22| ≤√

h (x (r)) +(α− 1)

6

(1− r2

)= (A− Cr)

√1− B2

4AC+

(α− 1)

6

(1− r2

)= k (r) .

(3.15)

The maximum value of g(r),

g(r) = A−Br + Cr2 +(α− 1)

6

(1− r2

)and (3.15) occurs at

rm =−B

−2C + (α−1)3

and r0 =B

2C +√

1− B2

4AC

respectively. It is easy to show that (3.15) is monotonic decreasing for r ≥ r0.Hence, the maximum value of |a3 − λa22| is given by (3.14).

For λ =2

3, we get B = 0 and C = 1

6(1− α) . Thus, the maximum value

∣∣a3 − λa22∣∣ = α

3,(3.16)

occurs at x = cos θ = 0 and r ∈ (0, 1].From (3.14), (3.15) and (3.16) we concluded that∣∣a3 − λa22

∣∣ ≤ 4 [(10− 9λ)α + (2− 3λ)]

3 (2− λ)− (2− 3λ)α

for λ ∈[2 (α− 1)

3α,2

3

].


Case 3E: Let λ ∈(2

3, λ2

], where λ2 is given by (3.13).

In this interval, we have B > 0. So that (3.9) attains its maximum value atx = 1. Then, we consider the function

l (r) = h (1) = A+Br + Cr2 +(α− 1)

6

(1− r2

).

Again, by a simple calculation shows that the maximum value of l (r) to beoccured at

rn =B

−2C + (α−1)3

,

hence the maximum of the function (3.15) to be attained at

r1 =B

−2C

(1 +

√1− B2

4AC

) ∈ (0, 1] .

It is easily to prove that r1 < rn ≤ 1. Since k(r) is monotonic increasing function,then

k (r) ≤ k (1) = (A− C)

√1− B2

4AC,

which gives that

∣∣a3 − λa22∣∣ ≤ k(1) = (1− λ)α

√12 (1− λ)

(4− 3λ)2 − (3λ− 2)2 α2

for λ ∈(2

3, λ2

].

Case 3F: Finally, we consider the case for λ ∈(λ2,

2 (α+ 2)

3 (α+ 1)

).

For these λ, we see that A < 0, B > 0, C < 0, A+Cr2 < 0 and the maximumvalue of function (3.7) is attained for x = −1, i.e.

η (x) = −A+Br − Cr2 +(α− 1)

6

(1− r2

).

We get η (r) ≤ η (1) for all λ in these interval and hence∣∣a3 − λa22∣∣ ≤ −A+B − C =

1

3

[(3λ− 2)α2 − 1

].

Thus, the proof of Theorem 1 is complete.Further, substitute (3.5) into (3.2) yields

12 (a3 − λa22) = (α + 1) [(2− 3λ) (α + 1) + 2] + 2(α2 − 1

)(3λ− 2) c0

+(α− 1) (6 + [2− 3 (α− 1)λ]) c20 + 2 (1− α) c1.


Hence for λ complex numbers, we have

12 |a3 − λa22| ≤ (α + 1) |(2− 3λ) (α + 1) + 2|+2 (1− α) |c1|+ 2

(α2 − 1

)|3λ− 2| |c0|

+(α− 1) |6 + [2− 3 (α− 1)λ]| |c0|2 .(3.17)

Using the well known inequality that |c0| ≤ 1 and |c1| ≤ 1−|c0|2, then from (3.17)we get

12∣∣a3 − λa22

∣∣ ≤ 1

12(α + 1) ν (α, λ)

for Re ν (α, λ) > 0, where

ν (α, λ) = |(2− 3λ) (α+ 1) + 2|+ 2 (1− α) |3λ− 2|

+(α− 1)

α + 1|6 + [2− 3 (α− 1)λ]| .

Thus, the proof of Theorem 2 is complete.

Remark 1 Taking k = 0 and λ real numbers, we deduce a result of Bhowmiket al. [5].

Other problems related to Fekete-Szego functional for further reading can be foundin ([1], [2], [7], [13], [14], [15], [21], [22], [23], [24], [25]).

Acknowledgements: The second author is fully supported by LRGS/TD/2011/UKM/ICT/03/02.

References

[1] Mohammed, A., Darus, M., On Fekete-Szego problem for certain subclassof analytic functions, Int. J. Open Problems Compt. Math., 3 (4) (2010),510-520.

[2] Khaled, A.A., Darus, M., On the Fekete-Szego theorem for the generalizedOwa-Srivastava operator, Proceedings of the Romanian Academy-series A:Mathematics, 12 (3)(2011), 179-188.

[3] Catas, A., Oros, G.I., Oros, G., Differential subordinations associatedwith multiplier transformation, Abstract Appl. Anal., ID 845724 (2008), 1-11.

[4] Ibrahim, A., Darus, M., The Fekete-Szego theorem: to determine thesharp upper bounds for close-to-convex functions, Jour. of Inst. Math. &Comp. Sci (Math. Ser.), 14 (3) (2001), 207-216.


[5] Bhowmik, B., Ponnusamy, S., Wirths, K.-J., On the Fekete-Szego pro-blem for concave univalent functions, Jour. of Math. Analysis & Appl., 373(2011), 432-438.

[6] Bhowmik, B., Ponnusamy, S., Wirths, K.-J., Characterization the pre-Schwarzian norm estimate for concave univalent functions, Monatsh Math.,161 (2010), 59-75.

[7] Eljamal, E.A., Darus, M., On Fekete-Szego problems for certain subclassof analytic functions, Inter. Jour. Pure and Appl. Math., 71 (4) (2011), 571-580.

[8] Avkhadiev, F.G., Pommerenke, S., Wirths, K.-J., Sharp inequalitiesfor coefficient of concave schlicht function, Comment. Math. Helv., 81 (2006),801-807.

[9] Avkhadiev, F.G., K-J. Wirths, K.-J., Concave schlicht functions withbounded opening angle at infinity, Lobachevskii J., 17 (2005), 3-10.

[10] Keogh, F.R., Merkers, E.P., A coefficient inequality for certain class ofanalytic functions, Proc. Amer. Math. Soc., 20 (1969), 8-12.

[11] Salagean, G.S., Subclasses of Univalent Function, Lecture Notes in Math.1013, Springer-Verlag, Berlin (1983), 362-372.

[12] El-Gawad, H.R., Thomas, D.K., The Fekete-Szego problem for stronglyclose-to-convex function, Math. Japonica, 44 (3) (1996), 8-12.

[13] Faisal, I., Darus, M., On Fekete-Szego theorem for certain analytic func-tions, Journal of Ouality Measurement and Analysis, 5 (1) (2009), 85-91.

[14] Al-Shaqsi, K., Darus, M., On Fekete-Szego problems for certain subclassof analytic functions, Appl. Math. Sci., 2 (2008), 431-441.

[15] Al-Shaqsi, K., Darus, M., Fekete-Szego problem for univalent functionswith respect to k-symmetric points, The Australian J. Math. Anal. Appl., 5(2008), 1-12.

[16] Babalola, K.O., Bounds on the coefficients of certain analytic and univa-lent functions, Mathematica, Tome 50 (73) (2008), 139-148.

[17] Cruz, L., Pommerenke, C., On concave univalent function, Complex Var.Elliptic Equ., 52 (2007), 153-159.

[18] Darus, M., Thomas, D.K., On the Fekete-Szego theorem for close-to-convex functions, Math. Japonica, 47 (1) (1998), 125-132.

[19] Fekete, M., Szego, G., Eine bemerkung uber ungerade Schlichte funktio-nen, London Math. Soc., 8 (1993), 85-89.


[20] Nasr, M.A., El-Gawad, H.R., On the Fekete-Szego problem for close-to-convex functions of order ρ, New Trend in Geometric Function Theory andApplications (Madras, 1990), World Sci. Publishing, River Edge, NJ (1991),66-74.

[21] M.H. Al-Abbadi and M. Darus, The Fekete-Szego theorem for a certain classof analytic functions, Sains Malaysiana, 40(4) (2011), 385-389.

[22] Al-Abbadi, M.H., Darus, M., The Fekete-Szego theorem for certain classof Analytic Functions, General Mathematics, 19 (3) (2011), 4151.

[23] Tuneski, N., Darus, M., Fekete-Szego functional for non-Bazilevich func-tions, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, 18 (2)(2002), 63-65.

[24] Ramadan, S.F., Darus, M., A multiplier transformation defined by con-volution involving a differential operator, Int. J. Open Problems ComplexAnal., 2 (2) (2010), 2074-2827.

[25] Ramadan, S.F., Darus, M., On the Fekete-Szeg ”o inequality for a classof analytic functions defined by using generalized differential operator, ActaUniversitatis Apulensis, 26 (2011), 167-178.

[26] Kaplan, W., Close-to-convex schlicht functions, Michigan Math. J., 1(1952/53), 169-185

[27] Koepf, W., On the Fekete-Szego problem for close-to-convex functions,Proc. Amer. Math. Soc., 101 (1987), 420-433.



r-WEAK cb SPACES

D. Bhattacharya

Department of MathematicsNIT AgartalaWest Tripura, Pin: 799055Indiae-mail: bhattacharyad [email protected]

L. Dey

Bhavan’s Tripura College of Teacher EducationNarsingarhWest Tripura, Pin: 799015Indiae-mail: [email protected]

Abstract. The purpose of the paper is to introduce the notion of r-weak cb spaces as a

generalization of weak cb spaces. The r-weak cb property of a space is defined with the

help of a stronger form of normal upper semi continuous functions viz. strongly normal

upper semi continuous (s-nusc) function. A stronger form of regular closed subsets

called strongly regular (s-regular) closed subsets turns out to be the natural tool for

defining the new function. The r-weak cb spaces are characterized by a increasing cover

of s-regular open (i.e. complement of s-regular closed) subsets and a decreasing sequence

of s-regular closed subsets. Some of the properties of the newly introduced spaces and

its interrelationship with other spaces are investigated.

Keywords and phrases: locally finite, σ-locally finite, cozero refinement, s-regular

closed, s-nusc.


1. Introduction

Throughout by a space we mean a completely regular (Hausdorff) space. For basicdefinitions of zero-sets, cozero-sets, Fσ-sets, Gδ-sets, regular open and regularclosed sets, refinement, cover etc. we refer [9] and [13].

The concept of regular Gδ-subsets was introduced by J. Mack [14]. A subsetH of a topological space X is called a regular Gδ-subset if H is an intersection ofa sequence of closed sets whose interiors contain H.

88 d. bhattacharya, l. dey

Equivalently, if H =∞∩n=1

Gn =∞∩n=1

ClXGn, where each Gn is open subset

of X, then H is a regular Gδ-subset. The complement of a regular Gδ-subsetis called regular Fσ, i.e., a subset V of a space X is said to be a regular Fσ ifV =

∪∞n=1 Fn =

∪∞n=1 IntXFn, where each Fn is closed subset of X. Properties

of regular Gδ and regular Fσ-subsets have been studied in [3]. The intersection(resp. union) of two regular Fσ (resp. regular Gδ)-subsets is a regular Fσ (resp.regular Gδ)-subset. The countable union (resp. intersection) of regular Fσ (resp.regular Gδ)-subsets is a regular Fσ (resp. regular Gδ)-subset. Also every zero-setis regular Gδ and every cozero-set is regular Fσ.

A real valued function on a topological space X is said to be locally boundedif each point has a neighbourhood (nbd) on which the function is bounded. LetC(X) denotes the set of all real valued continuous functions on X. A collectionof subsets Aα of a topological space is said to be locally finite (resp. σ-locallyfinite) if each point x ∈ X has a nbd which has non-empty intersection only with afinite (resp. countable) number of members of the collection. Also for two familiesof sets U and V , V is said to be a refinement of U , if for every V ∈ V there existsU ∈ U such that V ⊆ U . A family of continuous functions F is locally finite(resp. subordinate to a cover U) if the collection of cozero-sets associated with Fis locally finite collection of sets (resp. is a refinement of U). A family F ⊆ C(X)is a partition of unity of a space X if each function in F maps X into [0, 1] and∑

f(x) = 1 at each point x ∈ X, where the summation is taken over all f ∈ F .The constant function, on any set, whose constant value is the real number r isdenoted by r, e.g. 1 is used to represent the constant function whose constantvalue is 1, i.e., 1(x) = 1 for all x. For all real valued function f on a set, thesymbols f+ and f− are respectively defined as follows:

f+(x) =

f(x) for f(x) > 00 for f(x) ≤ 0

and

f−(x) =

f(x) for f(x) < 00 for f(x) ≥ 0

Thus with these notations the function hn = [(n + 1)f − 1]+ × [(n − 1)f − 1]−

is explained as hn(x) = [(n + 1)f(x) − 1]+ × [(n − 1)f(x) − 1]−. For real valuedfunctions f and g, the functions f ∨ g and f ∧ g are defined by the formulae(f ∨ g)(x) = maxf(x), g(x) and (f ∧ g)(x) = minf(x), g(x) respectively.If f, g ∈ C(X) then the functions f ∨ g, f ∧ g, f+ and f− are also the membersof C(X).

Real valued semi continuous functions play an important role in topology.Lower (resp. upper) semi continuous functions abbreviated as lsc (resp. usc)functions and their stronger forms are found in the literature. A function f :X → R is lsc (resp. usc) at a point x0 ∈ X, if for each ϵ > 0, there existsan nbd of x0, say N(x0), such that for all x ∈ N(x0), f(x) > f(x0) − ϵ (resp.

r-weak cb spaces 89

f(x) < f(x0)− ϵ). Equivalently, a function f : X → R is lsc (resp. usc) iff for allr ∈ R, the set x : f(x) > r (resp.x : f(x) < r) is an open set [13]. A lsc (resp.usc) function f : X → R is normal lower (resp. upper) semi continuous or nlsc(resp. nusc) at a point x0 ∈ X, if f(x0) < r (resp. f(x0) > r) and any open setU containing x0, there exists a non-empty open set V , such that ClXV ⊆ U andf(y) < r (resp. f(y) > r) for all y ∈ V . Equivalently, a lsc (resp. usc) functionf : X → R is nlsc (resp. nusc) iff for all r ∈ R, the set x : f(x) < r (resp.x : f(x) > r) is a union of regular closed sets [8].

A Tychonoff space X is Oz iff every regular closed set of X is a zero-set[5]. Equivalently, a Tychonoff space X is Oz iff every regular closed set is theintersection of a countable collection of regular closed nbds [6]. A topologicalspace X is said to be a cb space if for each locally bounded function h, there existsf ∈ C(X) such that f ≥ |h| [12]. The cb property is stronger than countableparacompactness, but equivalent to it for normal spaces. A space X is said to beweak cb if each locally bounded lsc function is bounded above by a continuousfunction. The weak cb and countably paracompactness are independent propertiesin Tychonoff spaces [15]. Two important characterizations of cb (resp. weak cb)spaces are: a spaceX is cb (resp. weak cb) iff (i) given an usc (resp. nusc) functionh on X there exists f ∈ C(X) such that f ≥ h, (ii) for each decreasing sequenceFn of closed (resp. regular closed) subsets of X with empty intersection, thereexists a sequence of zero-sets Zn with Zn ⊇ Fn and

∩n

Zn = ϕ.

It would be interesting to study whether a further generalization of weak cbspaces could be introduced characterized by equivalent conditions like (i) and (ii)of weak cb spaces, where the nusc function in (i) and the regular closed subsetsin (ii) are respectively replaced by their stronger forms. Further it is known thatthe characteristic function of regular closed (resp. regular open) subset of a spaceis nusc (resp. nlsc). This motivates us to search for a class of subsets which isstronger than that of regular closed subsets and whose characteristic function isat the same time stronger than the nusc function. In this direction it is foundthat the desired characteristics are present in the class of s-regular closed subsetswhich is defined below.

2. s-Regular closed subset and s-nusc function

Definition 2.1 A subset F (resp. G) of X is said to be strongly regular closedor s-regular closed (resp. strongly regular open or s-regular open) if F = ClXB(resp. G = IntXA), where B (resp. A) is a regular Fσ (resp. regular Gδ)-subsetof X.

Theorem 2.2 The complement of an s-regular open subset is s-regular closed andvice versa.

Proof. Let G be an s-regular open subset of X. Then G = IntXA, where Ais a regular Gδ-subset. Now X − G = X − IntXA = ClX(X − A) = ClXB,


where B = X −A is a regular Fσ-subset of X. Conversely, the complement of ans-regular closed set is s-regular open.

From the definition it follows that every s-regular open (resp. s-regular closed)subset is regular open (resp. regular closed). Now we cite an example to showthat a regular open (resp. regular closed) subset may not be s-regular open (resp.s-regular closed).

Example 2.3 Let us consider the spaces of ordinals, W=[0, ω1) and W∗=[0, ω1],the one point compactification of W, where ω1 denotes the first uncountableordinal. Again let us consider a subset A of W which is cofinal in W, but not atail. Then A is open in W and hence in W∗ and B = ClW∗A is a regular closedsubset of W∗, which contains ω1 but not a tail of W∗. Next we show that Bis not a regular Gδ. Now any open set in W∗ containing ω1 must contain a tailW−W (α) = σ : σ > α, α is an ordinal. If possible, let B is a regular Gδ-subsetofW∗. Then B =

∩n

Gn =∩n

ClW∗Gn, where each Gn is open inW∗ and contains

ω1. Then for each n ∈ N , there exists an ordinal αn such that σ : σ > αn ⊆ Gn.Let α∗ > supn αn, then α∗ ∈ W and σ : σ > α∗ ⊆

∩n

Gn = B, i.e., B contains

a tail W−W (α∗), which is a contradiction. Thus B is not even a Gδ and hencenot a regular Gδ. Therefore any regular Gδ-subset containing ω1 must contain atail σ : σ > α, α is an ordinal. Let C be a regular Gδ-subset of W

∗ containingω1. Then B = C and also IntW∗B = A and IntC contains a tail, which in turnimplies that IntB = IntC and hence the regular open subset IntB is not equalto the s-regular open subset IntC.

This example also shows that the regular closed subset X−A = X−IntXB =ClX(X −B), where X = W∗, is not s-regular closed.

With the help of s-regular closed subsets next we define functions to be calledstrongly nlsc (in short s-nlsc) and strongly nusc (in short s-nusc) functions.

Definition 2.4 A lsc (resp. usc) function f : X → R is said to be s-nlsc (resp.s-nusc) if for each real number r ∈ R, the set x : f(x) < r (resp. x : f(x) > r)is a countable union of s-regular closed subsets of X.

The rationale of taking countable unions of the above definition stems fromthe fact that closures of countable unions of s-regular closed subsets are s-regularclosed (Lemma 3.4), unlike regular closed subsets where closures of arbitraryunions of regular closed subsets are again regular closed. The following theoremprovides us with examples of such functions.

Theorem 2.5 The characteristic function of an s-regular open subset is s-nlsc.

Proof. Let G be an s-regular open subset in a space X. The characteristicfunction f of G is defined by

f(x) = 1, for all x ∈ G= 0, for all x ∈ X −G

r-weak cb spaces 91

It can be easily seen that f is lsc.We prove that f is s-nlsc. For this we show that x : f(x) < r is a countable

union of s-regular closed subsets of X. We consider the following cases.

Case (i). r < 0, x : f(x) < r = ϕ = ClXϕ, and ϕ is regular Fσ.

Case (ii). 0 ≤ r < 1, x : f(x) < r = X −G, which is s-regular closed and canbe thought of as a countable union of s-regular closed subsets.

Case (iii). r ≥ 1, x : f(x) > r = X, which is s-regular closed.

Hence f is s-nlsc.

Dually we can prove that the characteristic function of an s-regular closedsubset is s-nusc.

Remark 2.6 From Hardy and Woods [11] we quote here a similar result for nusc(resp. nlsc) function. The characteristic function of a regular closed (resp. open)subset is nusc (resp. nlsc).

Next, we find the relationship between nlsc and s-nlsc functions. That everys-nlsc function is nlsc follows directly from Definition 2.4 and the characterizationof nlsc functions [8]. But the converse is not true which follows from the followingexample.

Example 2.7 In Example 2.3 we have seen that the subset A of X = W∗ isregular open, but not s-regular open and X − A = F is regular closed, but nots-regular closed. Thus, in view of Remark 2.6, it follows that the characteristicfunction χ

A of A is nlsc. We want to show that χA is not s-nlsc. Let us suppose thecontrary. Then, by definition, x : χA(x) < r is a countable union of s-regular

closed subsets of X. Then, for r =1

2, x : χ

A(x) < r = X − A and thus

X − A is a countable union of s-regular closed subsets. Hence, X − A is anFσ-subset, since each s-regular closed subset is closed. In Example 2.3, it hasbeen shown that ClXA is regular closed, but not a regular Gδ (not even a Gδ).Thus X − ClXA = IntX(X − A) = X − A (since X − A is open in X) is notregular Fσ (not even a Fσ) and hence cannot be expressed as a countable unionof closed subsets. Thus, we arrive at a contradiction. Therefore, χA is not s-nlsc.

Similarly, the characteristic function χF of F is nusc, but not s-nusc.

3. r-Weak cb spaces and its properties

In this section, we introduce r-weak cb spaces as a generalization of weak cb spacesand study its properties.

Definition 3.1 A space X is called r-weak cb space if for each locally boundeds-nusc function h in X, there exists f ∈ C(X) such that |h| ≤ f .


As remarked earlier, our aim is to characterize the r-weak cb space in terms ofan increasing cover of s-regular open and a decreasing sequence of s-regular closedsubsets. To achieve our goal, we first prove some results in the form of lemmaswhich will be utilized in the desired characterization.

Lemma 3.2 If f is strictly positive s-nlsc function on X, then1

fis s-nusc.

Proof. Let f be an s-nlsc function on X which is strictly positive. Now, we show

that g =1

fis s-nusc. Clearly, x : g(x) > 0 = X is a countable union of s-regular

closed subsets. Let r > 0. Now,

x : g(x) > r =

x :

1

f(x)> r

=

x : f(x) <

1

r

,

which is a countable union of s-regular closed subsets, since f is s-nlsc. Thus, foreach r ∈ R, the set x : g(x) > r is a countable union of s-regular closed subsets.

Hence g =1

fis s-nusc.

Lemma 3.3 Let Bn be a countable increasing cover of X consisting of s-regular

open subsets and g be a function defined by g(x) = 1 if x ∈ B1 and g(x) =1

nfor

x ∈ Bn −Bn−1, n ≥ 2, then g is s-nlsc function.

Proof. Let us consider a countable increasing s-regular open cover Bn of

X. Now, let us define a function g by g(x) = 1 if x ∈ B1 and g(x) =1

nfor

x ∈ Bn − Bn−1, n ≥ 2. Clearly, g is lsc. Now, we see that the function g isactually s-nlsc. For this, we need to consider the following cases only as g > 0 forall x.

Case (i). Let 0 < r ≤ 1, we can always find n ∈ N such that1

n+ 1≤ r <

1

n.

Now, x ∈ X : g(x) < r =

x ∈ X : g(x) <

1

n

= X − Bn, which is

s-regular closed and can be thought of a countable union of s-regular closedsubsets.

Case (ii). Let r > 1, then x : g(x) < r = X, which is a countable union ofs-regular closed subsets. Thus, for each real number r, the set x : g(x) < ris a countable union of s-regular closed subsets of X. Hence g is an s-nlscfunction.

Lemma 3.4 If A =∪n

ClXGn, where each Gn is regular Fσ in X, n ∈ N , then

ClXA is s-regular closed.

r-weak cb spaces 93

Proof. Let V =∪n

Gn, which is regular Fσ. Now, we show that ClXA = ClXV .

Clearly, V ⊆ A and so ClXV ⊆ ClXA.

Again, ClXV = ClX

(∪n

Gn

)⊇

∪n

ClXGn = A, which implies ClXClXV ⊇

ClXA, i.e., ClXV ⊇ ClXA. Therefore, ClXA = ClXV , which is an s-regularclosed.

We are now in a position to characterize the r-weak cb spaces in terms of anincreasing cover of s-regular open subsets and a decreasing sequence of s-regularclosed subsets having empty intersection.

Theorem 3.5 Let X be any topological space. Then the following statements areequivalent.

a) X is r-weak cb.

b) For a given strictly positive s-nlsc function g on X, there exists f ∈ C(X)such that 0 < f(x) ≤ g(x) for all x ∈ X.

c) For each countable increasing s-regular open cover of X, there exists a locallyfinite partition of unity subordinate that cover.

d) Each countable increasing s-regular open cover of X has a locally finite cozerorefinement.

e) Each countable increasing s-regular open cover of X has a σ-locally finitecozero refinement.

f) Each countable increasing s-regular open cover of X has a countable cozerorefinement.

g) Given a decreasing sequence Pn : n ∈ N of s-regular closed subsets ofX with

∩n∈N

Pn = ϕ, there exists a decreasing sequence Zn : n ∈ N of

zero-sets with∩

n∈NZn = ϕ, such that Pn ⊆ Zn for every n.

Proof. (a) ⇒ (b). Let g be a strictly positive s-nlsc function on X. Then1

g

is s-nusc (Lemma 3.2). Again,1

gbeing strictly positive and usc, it is easy to

see that1

gis locally bounded. Now, by (a), there exists f1 ∈ C(X) such that

0 <1

g(x)≤ f1(x) for all x ∈ X. Then f(x) =

1

f1(x)is such that f ∈ C(X) and

0 < f(x) ≤ g(x) for all x ∈ X.

(b) ⇒ (c). Let us consider a countable increasing s-regular open cover Bnof X. Now, let us define a function g by g(x) = 1 if x ∈ B1 and g(x) =

1

n


for x ∈ Bn −Bn−1, n ≥ 2. Then, g is an s-nlsc function (Lemma 3.3), which isalso strictly positive. So, there exists f ∈ C(X) such that 0 < f(x) ≤ g(x) forall x ∈ X. Now, let us define a sequence of continuous functions hn on X ashn = [(n+1)f − 1]+[(n− 1)f − 1]−, n ∈ N , where 1(x) = 1 for all x ∈ X. We see

that hn(x) = 0 for f(x) ≤ 1

n+ 1or f(x) ≥ 1

n− 1for n > 1. Thus

Z(hn) =

x : f(x) ≤ 1

n+ 1

∪x : f(x) ≥ 1

n− 1

,

i.e.,

X − Z(hn) =

x :

1

n+ 1< f(x) <

1

n− 1

, for all n > 1, and

X − Z(h1) =

x :

1

2< f(x)

, for n = 1.

Now, we show that the sequence of cozero-sets X − Z(hn) is locally finite.Let us consider a point y ∈ X such that f(y) = r, then 0 < r ≤ 1. Sincef is continuous, the set x : r − δ < f(x) < r + δ, δ > 0 is open and clearlyy ∈ x : r−δ < f(x) < r+δ. Now, by the Archimedean property of real number,

we can choose m ∈ N such that r − δ >1

m− 1. Then x : r − δ < f(x) < r + δ

is a nbd of y which has empty intersection with X − Z(hn) for n ≥ m. So thecozero-sets X − Z(hn) forms a locally finite family, i.e., hn is locally finite.Now, hn being a locally finite family of continuous functions, h(x) =

∑hn(x)

is continuous. Next, we show that h is non-vanishing. For this let p ∈ X and

f(p) = a, 0 < a ≤ 1. If a = 1, then p ∈x :

1

2< f(x)

= X − Z(h1), so

h1(p) = 0. On the other hand, if 0 < a < 1, we can find n ∈ N such that

p ∈x :

1

n+ 1< f(x) <

1

n− 1

= X − Z(hn). In this case, hn(p) = 0. Thus

h(p) =∑

hn(p) = 0. Since h is non-vanishing,1

his defined and belongs to C(X)

and

hn

h

is a locally finite partition of unity. Since

X − Z

(hn

h

)=

x :

1

n+ 1< f(x) <

1

n− 1

, for all n > 1 and

X − Z

(hn

h

)=

x :

1

2< f(x)

, for n = 1,

we have

X − Z

(hn

h

)⊆

x :

1

n+ 1< f(x)

= Bn and

∪n

X − Z

(hn

h

)= X.

r-weak cb spaces 95

(c) ⇒ (d). Let us consider a countable increasing s-regular open cover Bnof X, then there exists a locally finite partition of unity Fα, say α ∈ Λ, an indexset, subordinate to the cover Bn. So by definition X −Z(Fα) forms a locallyfinite cozero refinement of the cover Bn.

(d) ⇒ (e). Since every locally finite collection is σ-locally finite, the result isobvious.

(e) ⇒ (f). Let A be a σ-locally finite cozero refinement of the countableincreasing s-regular open cover Bn of X. Then A = ∪Am, where each Am islocally finite collection of cozero-sets. Let Am,n be the union of the sets Am,αin Am such that Am,α ⊆ Bn. Since Am,n is the union of a locally finite family ofcozero-sets, it is a cozero-set [1]. Thus Am,n is a countable cozero refinementof Bn.

(f) ⇒ (g). Let us consider a decreasing sequence Fn : n ∈ N of s-regularclosed subsets of X with

∩n

Fn = ϕ. Then X−Fn is an increasing s-regular open

cover of X. Let M be a countable subset of C(X) such that X−Z(f), f ∈ M,is a refinement of the cover X−Fn. Let us denote by Zn, the intersection of allthose Z(f), f ∈ M for which Z(f) ⊇ Fn. Each Zn, being a countable intersectionof zero sets, is a zero set [9]. Now

∩n

Zn ⊆∩

f∈MZ(f) = ϕ, since X − Z(f) is a

cover of X. Thus Zn is the desired sequence of zero-sets.

(g) ⇒ (a). Let h be a locally bounded s-nusc function on X and definePn = x : h(x) > n, n ∈ N . The set Pn need not be countable but each Pn

can be expressed as a countable union of s-regular closed subsets of X (Definition2.4). Hence Fn = ClXPn (⊆ x : h(x) ≥ n) is an s-regular closed subset (Lemma3.4). Then, clearly, Fn is a decreasing sequence of s-regular closed subsets ofX with

∩n

Fn = ϕ. So, by the given condition, there exists a decreasing sequence

Zn = Z(gn) : n ∈ N of zero sets with∩

n∈NZ(gn) = ϕ such that Fn ⊆ Z(gn)

for every n, gn ∈ C(X). Let us define fn = 1 −(∨

i≤n

n|gi|)∧ 1, n ∈ N . Since∩

n∈NZ(gn) = ϕ for each x ∈ X, there exists i ∈ N such that gi(x) = 0. So,

we can find j ∈ N such that |gi(y)| >1

jfor all y in an nbd N(x) of x by the

continuity of the function gi and the Archimedean property of real numbers. Thusj|gi(y)| > 1 for all y ∈ N(x). If n ≥ i and n ≥ j, then n|gi(y)| > j|gi(y)| > 1 forall y ∈ N(x). Choosing n ≥ maxi, j, n|gi(y)| > 1 and thus (∨i≤nn|gi|)(y) > 1

and, consequently,

(∨i≤n

n|gi|)∧1 = 1. Thus, by the definition of fn, fn(y) = 0 for

all y ∈ N(x), provided n ≥ maxi, j, i.e., y ∈ X−Z(fn), y ∈ N(x) only for finitenumber of values of n. These show that fn is locally finite. Therefore, by thelocally finite property of fn,

∑n

fn ∈ C(X), it follows that f = 1+∑

fn ∈ C(X).


On Fn, we have gi(x) = 0 for i ≤ n and hence fi(x) = 1 for i ≤ n. Thus,f(x) ≥ 1+n > h(x) on Fn −Fn+1, for each n ∈ N . Also, for X −F1, h(x) < 1,but f(x) ≥ 1. So, f(x) ≥ h(x), for all x ∈ X. This proves the theorem.

Next, we study some properties of r-weak cb spaces. In the following theoremsit will be shown that, like weak cb spaces, this generalized space also possessesthe properties that a cozero subspace of an r-weak cb space and its product witha locally compact paracompact space are again r-weak cb spaces. To prove thefirst result we need the following lemma.

Lemma 3.6 If h is s-nusc on X, then h+ is also s-nusc.

Proof. Let us consider the set x : h+(x) < λ, λ ∈ R. Since h+ ≥ 0, we needonly to consider the case λ ≥ 0. Now, x : h+(x) < λ = x : h(x) < λ, whichis open. Thus h+ is usc.

Now, consider the set x : h+(x) > λ, λ ∈ R.

Case (i). λ < 0, x : h+(x) > λ = X, which is s-regular closed and can bethought of a countable union of s-regular closed subsets.

Case (ii). λ ≥ 0, x : h+(x) > λ = x : h(x) > λ, which a countable unionof s-regular closed subsets, h being s-nusc.

Hence h+ is also s-nusc.

Theorem 3.7 Each cozero subspace of an r-weak cb space is r-weak cb.

Proof. Let X be an r-weak cb space and Y be a cozero subspace of X. Then

Y = X − Z(g) for some g ∈ C(X), 0 ≤ g ≤ 1. Let Fn =

x ∈ X :

1

n≤ g(x)

.

For a given locally bounded s-nusc function h on Y , let us define a sequence offunctions hn, n ∈ N on X as follows:

hn(x) = h+(x) on Fn ⊆ Y and hn(x) = 0 on X − Fn.

It can be easily seen that hn is a locally bounded s-nusc on X for each n ∈ N .Since X is r-weak cb, there exists fn ∈ C(X) such that hn ≤ fn for each n.

Next, we define the continuous functions gn : X → [0, 1] for n > 2, asgn = 1− [(n− 1)(n− 2)g− (n− 2)]+ ∧ 1, where 1(x) = 1 etc. for all x ∈ X. Now

Fn−2 =

x ∈ X :

1

n− 2≤ g(x)

= x ∈ X : (n− 1)(n− 2)g(x)− (n− 1) ≥ 0= x ∈ X : (n− 1)(n− 2)g(x)− (n− 2) ≥ 1.

So, gn(x) = 1−1 = 0 for all x ∈ Fn−2, which implies that Z(gn) ⊇ Fn−2. Similarly,it can be checked that gn(x) = 1 for all x ∈ X − Fn−1. Assuming g1 = g2 = 1, weset f = ∨n(fngn)|Y . Next, we show that gn|Y is a locally finite family in C(Y ).

r-weak cb spaces 97

If y ∈ Y , then g(y) = 0. So, we can find an integer i ∈ N such that g(y) >1

i, i.e.,

ig(y) − 1 > 0. As g ∈ C(X), the set x : ig(x) − 1 > 0 = M , say, is open andy ∈ M . Then, M is a nbd of y which does not meet X − Z(gn+2) for n ≥ i, sinceX − Z(gn+2) ⊆ X − Fn = x : ng(x) − 1 < 0. Thus, gn|Y is a locally finitefamily in C(Y ), which implies that (fngn)|Y is also locally finite. So f ∈ C(Y ).

Again, since gn(x) = 1 for x ∈ X − Fn−1, Fn−1 ⊆ Fn and hence gn(x) = 1 forx ∈ Fn − Fn−1. Therefore, on Fn − Fn−1 ⊆ Fn ⊆ Y , f = ∨nfngn = ∨nfn ≥ fnfor each n ∈ N . Finally, on Fn − Fn−1, h

+ = hn ≤ fn ≤ f for all n ∈ N , whichimplies that h(x) ≤ f(x) on Y . Hence Y is r-weak cb.

Since every clopen subset is cozero, we have the following:

Corollary 3.8 Every clopen subspace of r-weak cb space is r-weak cb.

To prove that the product of an r-weak cb space with a locally compactparacompact space is again r-weak cb we first prove the result considering a specialcase when the other space is compact. To that end, the following lemma is needed.

Lemma 3.9 Let p : X × Y → X be the projection map and Y be compact(T1-space). Then for each s-regular closed subset A of X×Y , p(A) is an s-regularclosed subset of X.

Proof. Let A = ClX×YG, where G is a regular Fσ-subset of X × Y , be ans-regular closed subset of X × Y . We are to show that p(A) is an s-regularclosed subset of X. It is known that the projection map is open, continuous andsurjective. Further since Y is compact, the projection of X×Y onto X is a closedmapping [14]. Therefore, p(A) = p(ClX×YG) = ClX(p(G)). It remains to showthat p(G) is a regular Fσ-subset of X. Let G =

∪n

Fn =∪

n IntX×Y Fn, where

each Fn is closed in X × Y . Let Kn = p(Fn), then each Kn is closed in X. Also

p(G) = p

(∪n

Fn

)=

∪n

p(Fn) =∪n

Kn. Finally we show that∪n

Kn =∪n

IntXKn.

It suffices to show that∪n

Kn ⊆∪n

IntXKn. Now, p being an open mapping,

p(IntX×Y Fn) ⊆ IntXp(Fn) = IntXKn, for each n ∈ N . Hence∪n

p(IntX×Y Fn) ⊆∪n

IntXKn, i.e., p

(∪n

IntX×Y Fn

)⊆

∪n

IntXKn, i.e., p

(∪n

Fn

)⊆

∪n

IntXKn,

i.e.,∪n

p(Fn) ⊆∪n

IntXKn, i.e.,∪n

Kn ⊆∪n

IntXKn. Therefore, p(G) =∪n

Kn =∪n

IntXKn is a regular Fσ-subset of X.

Theorem 3.10 The product of an r-weak cb space and a compact (Hausdorff)space is r-weak cb.

Proof. Let X be an r-weak cb space and Y be a compact (Hausdorff) space.Let Kn, n ∈ N be a decreasing sequence of s-regular closed subsets of X × Ywith

∩n

Kn = ϕ. Let p : X × Y → X be the projection map. By Lemma 3.9,


p(Kn) = An, say, is an s-regular closed subset ofX for each n ∈ N and An is alsodecreasing. Next, we prove that

∩n

An = ϕ. For any x ∈ X, x × Y is compact.

Thus, the family Kn∩(x×Y ) of closed sets does not have the finite intersectionproperty, in other words x×Y fails to meet Kn for some n. Therefore An hasempty intersection. Now, X being r-weak cb, there exists a decreasing sequenceZn of zero-sets having empty intersection such that Zn ⊇ An. Lastly, Zn×Y is a sequence of zero-sets in X×Y with empty intersection such that Zn×Y ⊇ Kn

for each n ∈ N . Hence, in view of Theorem 3.5 (g), X × Y is r-weak cb.

The conditions of the above theorem could be weakened by assuming locallycompact paracompact (Hausdorff) property of the space Y . This will be shownin Theorem 3.17. For this purpose, we require some basic results regarding em-bedding properties of s-regular closed subsets, which are discussed first. In thisdirection, we consider an example to illustrate the difficulty in embedding ans-regular closed subset of a space to a subspace of it.

Example 3.11 Let X=[0, 1] ∪ [2, 3] and Y=0 ∪ [2, 3] ⊆ X. Now G=

(0,

1

2

)∪

(2, 2.5) is regular Fσ in X and hence F = ClXG =

[0,

1

2

]∪ [2, 2.5] is s-regular

closed in X. Then, V = G∩Y = (2, 2.5) is also a regular Fσ in Y . But 0 /∈ ClY V ,whereas 0 ∈ ClXG. Thus F ∩Y = 0∪ [2, 2.5] and ClY V = [2, 2.5], which showsthat F ∩ Y = ClY V .

Thus, we see that if F = ClXG, where G is regular Fσ in X, be ans-regular closed subset of a space X and Y ⊆ X, then for V = G ∩ Y theequality F ∩ Y = ClY V may not be true always.

Now, we investigate the condition under which F ∩ Y = ClY V holds.

Theorem 3.12 If Y be a subset of X with the property that ClXG∩ Y ⊆ ClY V ,where G is regular Fσ in X and V = G∩Y , then, for each s-regular closed subsetF = ClXG of X, F ∩ Y = ClY V is s-regular closed in Y .

Proof. Let F = ClXG, where G is regular Fσ-subset of X. Then, G ∩ Y is aregular Fσ-subset of Y [4]. Let V = G∩ Y . Then ClY V is s-regular closed subsetof Y . We are to show that F ∩ Y = ClY V . Now, ClY V ⊆ ClXV ⊆ ClXG = F ,i.e., ClY V ∩ Y ⊆ F ∩ Y , i.e., ClY V ⊆ F ∩ Y as ClY V ⊆ Y .

Conversely, let p ∈ F ∩Y = ClXG∩Y . Then, by our assumption, p ∈ ClY V .Hence, F ∩ Y ⊆ ClY V . Therefore, F ∩ Y=ClY V is s-regular closed subset of Y .

The assumption made in the above theorem is fulfilled by wide category ofsubspaces, e.g. open subspaces, dense subspaces [9, Art. 0.12] etc.

Next, we investigate the relationship between the properties of preservationof trace of an s-regular closed subset and the restriction of an s-nlsc (or s-nusc)function of a space to a subspace of it. In this direction first we see that thepreservation of trace of an s-regular closed subset implies that the restriction ofan s-nlsc (resp. s-nusc) function to a subspace is s-nlsc (resp. s-nusc).

r-weak cb spaces 99

Theorem 3.13 The restriction of an s-nusc (resp. s-nlsc) function of a space Xto a subset Y of X is also s-nusc (resp. s-nlsc), provided that the trace of eachs-regular closed subset of X to Y is an s-regular closed subset of Y .

Proof. First we prove the theorem for the s-nusc function. The proof for s-nlsccase is similar. Let f be an s-nusc function on X and Y is a subset of X. Letf |Y = g. For each real number r, Sr(f) = x ∈ X : f(x) > r is a countableunion of s-regular closed subsets of X, since f is s-nusc on X. Now, Sr(f) ∩ Yis a countable union of s-regular closed subsets of Y , by our assumption. Hence,Sr(f) ∩ Y = y ∈ Y : g(y) > r. Hence, g is s-nusc function on Y , sincey ∈ Y : g(y) > r = Sr(f) ∩ Y .

But it is interesting to see that the restriction of an s-nlsc (resp. s-nusc)function of a space to a closed subspace is again s-nlsc (resp. s-nusc).

Theorem 3.14 The restriction of an s-nlsc (resp. s-nusc) function of a space Xto a closed subset C having Lindeloff frontier is also s-nlsc (resp. s-nusc).

Proof. Let us prove the theorem for the s-nlsc function. The s-nusc case maybe dealt dually. Let f be an s-nlsc function on X and let C be a closed subsetof X. Clearly, g = f |C is lsc. Now consider the set Fr = x : f(x) < r,r ∈ R. By definition, Fr is a countable union of s-regular closed subsets of X,i.e. Fr =

∪n

ClXUn, where Un is a regular Fσ-subset of X for each n ∈ N . Now,

t ∈ C : g(t) < r = Fr ∩C =∪n

(ClXUn ∩C). We propose to show that the trace

ClXUn ∩ C of s-regular closed subsets ClXUn generated by the s-nlsc functionf on X is again s-regular closed in C. Let Vn = Un ∩ C, then Vn is a regularFσ-subset of C [4] and there exists a regular Fσ-subset Wn, n ∈ N , Wn ⊇ Vn suchthat

∪n

ClXUn ∩C =∪n

ClCWn. Since ClCVn ⊆ ClXUn ∩C for each n, so we have∪n

ClCVn ⊆∪n

(ClXUn ∩ C). Now let p ∈∪n

(ClXUn ∩ C). Then, p ∈ ClXUn ∩ C

for some n, and every nbd of p in X meets Un. Now, we claim that either everynbd of p in C meets Vn, i.e., p ∈ ClCVn or else p is an isolated point in C.Clearly, f(p) = g(p) < r. Now, f being lsc, the sets x ∈ X : f(x) > f(p) − ϵ,ϵ > 0 form basic nbds of p in X. Then, each such nbd meets Vn = Un ∩ C andhence p ∈ ClXVn = ClCVn provided x ∈ C : r > f(x) > f(p) − ϵ − p = ϕ.Otherwise, p would be an isolated point of C. As C is closed, all such points psatisfying p ∈ ClXUn∩C, but p /∈ ClXVn for some n, must be a boundary point ofC. Let G be the collection of all such p’s. Now, we see that G is indeed a regularFσ-subset of C. By our assumption, FrXC is Lindeloff and G ⊆ FrXC consists ofisolated points. Hence G must be a countable set, which in turn implies that G is

a regular Fσ subset. Now, it is plain to see that

(∪n

ClCVn

)∪G =

∪n

(ClXUn∩C)

and t ∈ C : g(t) < r = Fr ∩ C =

(∪n

ClCVn

)∪ G =

∪n

(ClXUn ∩ C) is an

s-regular closed of C and hence g is s-nlsc.


Corollary 3.15 Let X × C ⊆ X × Y , where C is compact in Y . Then for eachs-nlsc (resp. s-nusc) function f of X × Y , f |(X × C) is s-nlsc (resp. s-nusc).

Proof. Since compact subsets are closed in Hausdorff spaces and FrX×Y (X ×C)is compact and hence Lindeloff, the corollary at once follows from Theorem 3.14.

Finally, before going to the proof of the main theorem we quote a resultfrom [2].

Result 3.16 Let S ⊆ X be open in X and f and g are respectively members ofC(S) and C(X). Then, the function h on X defined by h = f +g on S and h = gon X − S is continuous.

Theorem 3.17 The product of a r-weak cb space and a locally compact paracom-pact (Hausdorff) space is r-weak cb.

Proof. Let X be a r-weak cb space and Y be a locally compact paracompact(Hausdorff) space. Since Y is locally compact, each point of Y has a compactnbd. Consider a compact nbd Ky containing y ∈ Y . Then there exists an open setOy in Y such that y ∈ Oy ⊆ Ky. Again Y being paracompact, the cover of Y bythe open sets Oy : y ∈ Y has a locally finite open refinement U = Uα : α ∈ Λ,say. Then, clearly, the members of the family U = ClUα : α ∈ Λ are compactsets. Since Y is normal (as it is paracompact Hausdorff) and the open cover U ofY is point finite (as it is locally finite), there is an open cover V = Vα : α ∈ Λsuch that ClVα ⊆ Uα for each α ∈ Λ [7].

Now, let h be a locally bounded s-nusc function on X×Y . By Theorem 3.10,X × ClUα is r-weak cb. The function h|X × ClUα being s-nusc on X × ClUα

(Corollary 3.15) and X ×ClUα is r-weak cb, the function h|X ×ClUα is boundedabove by a continuous function f in X×ClUα. The disjoint sets ClVα and Y −Uα

are such that ClVα is compact and Y −Uα is a closed set of the completely regularspace Y . Hence ClVα and Y − Uα are completely separated [9] and this impliesthat X × Vα ⊆ X × ClVα and X × (Y − Uα) are completely separated. Thus,there exists a function eα in C(X × Y ) such that eα = 1 on X × Vα and eα = 0on X × (Y − Uα). Finally, define a function gα on X × Y as gα = eα + f onX×Vα and gα = eα elsewhere. Since X×Vα is open in X×Y , it can be seen thatgα ∈ C(X × Y ) (Result 3.16) and gα(X × Y ) ≥ |h(x, y)| on X × Vα and gα = 0on X × (Y − Uα). Again, since Uα : α ∈ Λ is locally finite, g = ∨α∈Λgα existsin C(X × Y ) and clearly g ≥ h. Hence the theorem.

Next we study the relationship of r-weak cb space with other spaces.

Theorem 3.18 Every weak cb space is r-weak cb.

Proof. Let X be a weak cb space and Pn : n ∈ N be a decreasing sequenceof s-regular closed subsets of X with

∩n∈N

Pn = ϕ. Now every s-regular closed

subset is regular closed. Hence Pn is a decreasing sequence of regular closedsubsets with

∩n∈N

Pn = ϕ. By weak cb property there exists a decreasing sequence

r-weak cb spaces 101

Zn : n ∈ N of zero-sets with∩

n∈NZn = ϕ, such that Pn ⊆ Zn for every n.

Therefore X is r-weak cb.

Corollary 3.19 Every completely regular (Hausdorff) pseudocompact space isr-weak cb.

Proof. Since every completely regular (Hausdorff) pseudocompact space is weakcb [15], hence the result follows from Theorem 3.18.

Theorem 3.20 Every Oz space is r-weak cb.

Proof. Since every Oz space is weak cb, the result follows immediately fromTheorem 3.18.

That an r-weak cb space may fail to be countably paracompact, can be shownby the following example.

Example 3.21 The Tychonoff plank T , being pseudocompact [9], is an r-weakcb space by Corollary 3.19. But T is not countably paracompact [15].

Conclusion

In this paper, a generalized weak cb space called r-weak cb space has been in-troduced with the help of s-nlsc function and its various properties have beenstudied. This space includes the well known cb, weak cb spaces, but fails to implycountably paracompactness. The results obtained are interesting in the sense thatvarious properties possessed by the cb and weak cb spaces are also possessed bythe generalized r-weak cb spaces. This study opens up the future scope of inves-tigation for the properties of the newly introduced s-nusc (s-nlsc) functions, andinterrelationship between r-weak cb and nd-spaces [10], where it is known thatevery nd-space is countably paracompact and together with weak cb it implies cbproperty.

Acknowledgement. The authors would like to thank the referees for their va-luable suggestions.

References

[1] Alo, R.A., Shapiro, H.L., Normal topological spaces, Cambridge Univ.Press, 1974.

[2] Bhattacharya, D., Some generalization of realcompact spaces, Ph. D. The-sis, 1998.

[3] Bhattacharya, D., Bhaumik, R.N., Regular Gδ-subsets and Realcompactspaces, Bull Cal. Math. Soc., 87, (1995), 39-44.


[4] Bhattacharya, D., Dey, L., r-Realcompact spaces, Comment. Math.Univ. Carolinae, 53, 2 (2012), 253-267.

[5] Blair, R.L., Spaces in which special sets are z-embedded, Canad. J. Math.,28 (1976), 673-690.

[6] Blair, R.L., Swardson, M.A., Spaces with an Oz stone-cech compactifi-cation, Topology and its Application, 36 (1990), 73-92.

[7] Christenson, C.D., Voxman, W.L., Aspects of topology, Marcel Dekker,Inc. New York and Basel, 1977.

[8] Dilworth, R.P., The normal completion of lattice of continuous functions,Trans. Amer. Math. Soc, 68 (1950), 427-438.

[9] Gillman, L., Jerison, M., Rings of Continuous Functions, UniversitySeries in Higher Math, Van Nostrand, Princeton New Jersey, 1960.

[10] Hardy, K., Juhasz, I., Normality and weak cb-property, Pac. J. Math.,vol. 64, 1976, 167-172.

[11] Hardy, K., Woods, R.G., On c-realcompact spaces and locally boundednormal functions, Pac. J. Math., 43 (1972), 647-656.

[12] Horne, J.G., Countable paracompactness and cb spaces, Notices, Amer.Math. Soc., 6 (1959), 629-636.

[13] Kelly, J.L., General topology, D. Van Nostrand, Princeton, NJ, 1955.

[14] Mack, J.E., Countably paracompactness and weak normality properties,Trans. Amer. Math. Soc., 148 (1970), 265-272.

[15] Mack, J.E., Johnson, The Dedekind completion of C(X), Pac. J. Math.,20 (1967), 231-243.



MORE PROPERTIES ON FLEXIBLE GRADED MODULES

Fida Moh’D

Department of Basic SciencesFaculty of EngineeringPrincess Sumaya University for TechnologyAmmanJordane-mail: [email protected]

Mashhoor Refai

Vice President for Academic AffairsPrincess Sumaya University for TechnologyAmmanJordane-mail: [email protected]

Abstract. In this paper, we study the structure of flexible graded modules over various

types of graded rings such as first strong and augmented graded rings. Also, we intro-

duce the notions of flexibly simple and flexibly Noetherian modules, and investigate

various properties of such modules.

Keywords: graded rings, graded modules, flexible graded modules, augmented graded

modules.

2010 Mathematics Subject Classification: 13A02.

1. Introduction

Strongly graded modules were a reasonable generalization of strongly graded rings,which play an important role in producing interesting theorems and results inGraded Ring Theory and Graded Module Theory. Of course, this is due to thenice structure of a ring or a module when it is equipped with a strong graduation.A natural question arises about whether we can extend the category of stronglygraded rings and modules in order to increase the application area of these nice re-sults. As a step on this path, first and second strongly graded rings were presentedby Refai [5] to generalize strongly graded rings. Another step was presented byRefai and Moh’D [10], where first strongly graded rings were produced to genera-lize strongly graded modules. A second generalization of strongly graded moduleswas the flexible graded modules, exhibited in [12].

104 f. moh’d, m. refai

This paper continues the work done in [12], where we further investigateabout flexible modules. In particular, we study the structure of these modulesover various types of graded rings, and observe the behavior of the submodules ofsuch modules. In addition, several characterizations related to flexible modules arelisted in different places in the article. The main result in this paper demonstratesthat over augmented graded rings, flexible and augmented modules are the same.

In Section 2, we list some important facts about flexible modules. These factswill be useful in the sequel sections.

In Section 3, we study flexible graded modules when the ring is first strong.We study various properties in this stage. In particular, we give a partial an-swer to the validity of the statement “A module is flexible iff every submodule isflexible”. Also, a partial answer of the question “whether every module can begraduated to be flexible?” is presented. On the other hand, we generalize someresults of [3]. The main result of this section asserts that for flexible modules overcommutative graded rings, every submodule can be graduated to become flexi-ble. This assertion can be interpreted as “In flexible modules over commutativerings, every discussion about a property concerning submodules can be carriedout regarding them as flexible submodules”. This fact allows us to use the nicestructure and the smoothness of flexible modules.

In Section 4, we further study the structure of flexible modules over aug-mented graded rings. The main result in this section shows that flexible modulesand augmented graded modules are the same over augmented graded rings. Thisresult allows us to build augmented graded modules through flexible modules,and avoid checking out the rather complicated conditions of augmented gradedmodules.

In Section 5, we introduce the notions of flexibly simple and flexibly Noethe-rian modules. These concepts generalize the concepts of flexible simple and fle-xible Noetherian modules [14], which were defined only for flexible modules. Weshow that in flexible modules over commutative graded rings, “flexibly simple”(resp.“flexibly Noetherian”) property implies “simple” (resp. “Noetherian”) pro-perty.

2. Preliminaries

This section presents some necessary background of graded rings and graded mo-dules considered in this paper. More details can be found in [5, 6, 7, 12]. For ageneral background of graded rings and graded modules, we advise the reader tolook in [1, 2, 3]. Throughout this paper, unless otherwise stated, G is a groupwith identity e, R =

⊕g∈G

Rg is a G−graded ring with unity 1, and M =⊕g∈G

Mg is

a G−graded left R−module. The set supp(R,G) = g ∈ G : Rg = 0 is calledthe support of R. The support of M , supp(M,G), is defined similarly. To avoidrepetition, we assume R = 0, andM = 0. One more thing, all modules consideredin this article are left modules.

more properties on flexible graded modules 105

Definition 2.1 [12] Let R be a G−graded ring, and M a G−graded R−module.Then M is said to be flexible if Mg = RgMe, for every g ∈ G.

Proposition 2.2 [12] Let R be a G−graded ring, and M a G−graded R−module.Then M is flexible iff M = RMe.

Remark 2.3 [12] Let M be a flexible G−graded R−module. Then

1. M = 0 iff Me = 0 iff e ∈ spp(M,G).

2. supp(M,G) ⊆ supp(R,G).

Definition 2.4 [12] Let R be a G−graded ring, and M be a flexible R−module.A G−graded R−submodule N of M is said to be flexible, if N is a flexible R−module.

Proposition 2.5 [12] Let R be a G−graded ring, andM be a G−graded R−module.If X is an Re−submodule of Me, then RX is a flexible R-submodule of M .

3. Flexible modules over first strongly graded rings

This section is devoted to study the structure of flexible modules over first stronglygraded rings. In addition, some different properties are considered. For instance,flexibility (i.e., to become flexible) results such as Theorem 3.8 and Proposi-tion 3.10.

Definition 3.1 [5] A G−graded ring R is said to be first strong iff RgRh = Rgh,for all g, h ∈ supp(R,G) iff RgRg−1 = Re, for all g ∈ supp(R,G) iff 1 ∈ RgRg−1 ,for all g ∈ supp(R,G).

Proposition 3.2 [5] If R is a first strongly G−graded ring, then supp(R,G) is asubgroup of G.

Lemma 3.3 [10] Let R be a first strongly G−graded ring, andM be an R−module.If N and L are Re−submodules of M , then Rg (N ∩ L) = RgN ∩Rg L, for everyg ∈ G.

The following proposition tells that on first strongly graded rings, flexibilityof a graded module is completely determined by the behavior of the supports ofthe ring and the module.

Proposition 3.4 [10] Let R be a first strongly G−graded ring, and M be aG−graded R−module. Then M is flexible iff supp(M,G) = supp(R,G).

Lemma 3.5 [10] Suppose R is a first strongly G−graded ring, and M is anR−module. If X is an R−submodule of M , then Rg X = X, for every g ∈supp(R,G).


Proposition 3.6 [14] Let R be a first strongly G−graded ring, andM be a flexibleR−module. Then every G−graded R−submodule of M is also flexible.

Proof. We have Rg−1Mg = Me, for every g ∈ supp(R,G). Let X =⊕g∈G

Xg be a

G−graded R-submodule of M . Fix g ∈ supp(R,G). By Lemma 3.5, RgX = X.So, Lemma 3.3 implies Xg = RgRg−1(X ∩ Mg) = Rg(X ∩ Me) = RgXe. Ifg /∈ supp(R,G), Proposition 3.4 yields Xg = RgXe.

Proposition 3.6 asserts that for flexible modules over first strongly gradedrings, there is no graded submodule of M , which is not flexible. In the next tworesults, if R is commutative, we prove two facts. The first fact states that we candrop the “first strong” condition from Proposition 3.6. The second fact statesthat not only are graded submodules in this proposition flexible, but also thatevery submodule is graded and flexible.

Lemma 3.7 Let R be a commutative G−graded ring, and M be a G−gradedR−module. If N is an R−submodule of RMe, then there exists an Re−submoduleX of Me such that N = RX. Further, N is a flexible G−graded R−submoduleof M .

Proof. Define the set X by X = x ∈ Me : r x ∈ N, for some r ∈ R. Then Xis an Re−submodule of Me. Moreover, N = RX, and it is a G−graded flexibleR−submodule of M , with Ng = RgX, for each g ∈ G.

Theorem 3.8 Let R be a commutative G−graded ring, and M be a flexibleG−graded R−module. Then every R−submodule of M is G−graded and flexi-ble R−submodule of M .

Proof. Apply Lemma 3.7 to M .

Remark 3.9 Theorem 3.8 shows how strong the structure of flexible modules isover commutative rings. In such modules, any property of submodules can beconsidered as a property of graded or even flexible submodules. This fact gives usthe opportunity to use the nice properties of flexible modules. Hence, a questionarises “Given an R−module M and a group G, Does a graduation of M by Gexist such that M turns into a flexible R−module?” We give a partial answer tothis question. Hopefully, we will continue searching for more answers in the nearfuture.

Although the following proposition is not related to modules over first stronglygraded rings, we prefer to put it in this section to keep the flow of the articlehomogeneous, especially since it was, in the first place, an indirect outcome ofProposition 3.6.

Proposition 3.10 Let R be a G−graded ring, and M be a free R−module. Thenthere exists a graduation ofM by G such thatM is a flexible G−graded R−module.


Proof. Let T be a basis of M . We have X = Re T is an Re−submodule of M .For each g ∈ G, set Mg = RgX. It is not difficult to see that Mg = RgMe,for each g ∈ G, and M =

∑g∈G

Mg. Let z ∈ Mh ∩ (∑

g∈G−hMg). Then there exist

rg ∈ Rg, and xg ∈ X, where g ∈ G, such that z = rh xh =

∑g∈G−h

rg xg. So,

rh xh −

∑g∈G−h

rg xg = 0. Since xg ∈ X, there exist tg ∈ T , and rg ∈ Re such that

xg = rg tg, where g ∈ G. Thus, rhrh th −

∑g∈G−h

rgrg tg = 0. Since T is linearly

independent, we have rhrh = 0, which implies z = 0. Therefore, M =

⊕g∈G

Mg.

Consequently, M is a flexible G−graded R−module.

Recall that if M is an R−module, S ⊆ R, and 0 = x ∈ M . Then x iscalled S−torsion free, if rx = 0, whenever 0 = r ∈ S. Otherwise, x is said to beS−torsion. In addition, A subset N of M is S−torsion free, if every element of Nis S−torsion free. In the upcoming work, we give a partial converse of Proposition3.6, as shown in Corollary 3.14.

Lemma 3.11 Let R be a G−graded ring, and M be a flexible R−module. If Me

contains an R−torsion free element, then supp(M,G) = supp(R,G).

Proof. Let g ∈ G. Since Me has a torsion free element, say x, we have Mg = 0iff Rg = 0. Actually, if Mg = RgMe = 0, then Rg x = 0, and hence Rg = 0.Therefore, supp(M,G) = supp(R,G).

Proposition 3.12 Let R be a G−graded ring, and M be a flexible R−modulesuch that Me contains an R−torsion free element. If g ∈ supp(R,G), then Rg

contains a regular element (an element which is not a zero divisor) iff Mg containsan R−torsion free element.

Proof. Suppose Rg contains an element rg, which is not a zero divisor. By as-sumption, let xe ∈ Me be an R−torsion free element. We have 0 = rgxe ∈ Mg.Moreover, if α ∈ R, and α (rgxe) = 0, then (α rg) xe = 0, which implies in turnα rg = 0. Thus, α = 0. Hence, the element rgxe is an R−torsion free in Mg.

For the converse, assumeMg contains an R−torsion free element. By Lemma3.11, g ∈ supp(M,G). Let xg = rg xe ∈ Mg be an R−torsion free element, whererg ∈ Rg and xe ∈ Me. Let h ∈ supp(R,G), and α ∈ Rh such that α rg = 0. Then(α rg)xe = 0 or α(rgxe) = 0. Hence, αxg = 0. Since xg is R−torsion free, weobtain α = 0. Consequently, Rg has no zero divisors in R.

In fact, Proposition 3.12 allows us to switch the discussion between torsionfree elements in M , and regular elements in R.

Proposition 3.13 Let R be a G−graded ring, and M be a flexible R−modulesuch that Mg has a torsion free element, for every g ∈ supp(M,G). Then Ris first strong iff every G−graded cyclic submodule of M , with a homogeneousgenerator, is flexible.


Proof. Assume every G−graded cyclic submodule of M , with a homogeneousgenerator, is flexible. By Lemma 3.11, supp(M,G) = supp(R,G). Let g ∈supp(R,G), and xg be an R−torsion free element in Mg. By assumption, Rxgis flexible. Fix h ∈ supp(R,G). We have Rh g−1 xg = (Rxg)h = Rh(Rxg)e =RhRg−1 xg. Since xg is R−torsion free, we get Rh g−1 = RhRg−1 . Now, put h = gin the last equation to obtain Re = RgRg−1 . Therefore, R is first strong. Theconverse follows by Proposition 3.6 .

Corollary 3.14 Let R be a G−graded ring, and M be a flexible R−module suchthat Mg has a torsion free element, for every g ∈ supp(M,G). Then R is firststrong iff every G−graded R−submodule of M is flexible.

As final applications of flexible modules over first strongly graded rings, wecurve away to homological algebra to generalize some results of [3]. The proof ofProposition 3.15 resembles the proof of the similar result of [3]. So, we will notput it in the body of the article. On the other hand, the proof of Proposition 3.16is inferred by combining Proposition 2.1 of [11], and the same result of [3], whichis valid for strongly graded rings.

Proposition 3.15 [14] Let R be a first strongly G−graded ring such that supp(R,G)is finite, M be an R−module, and N =

⊕g∈G

Ng be a flexible R−module. Then

1. For each g ∈ supp(R,G), the function φ : HomR(M,N) −→ HomRe(M,Ng)defined by φ(f) = πg f is a group isomorphism, where πg : N −→ Ng isthe canonical projection on the g−th component of N .

2. For each g ∈ supp(R,G), the function ψ : HomR(N,M) −→ HomRe(Ng,M)defined by φ(f) = f ig is a group isomorphism, where ig : Ng −→ N is thecanonical inclusion of the g−th component of N .

Proposition 3.16 [14] Let R be a first strongly G−graded ring such that supp(R,G)is finite, M be an R−module, and N =

⊕g∈G

Ng be a flexible R−module. Then

1. For each g ∈ supp(R,G), ExtnR(M,N) ∼= ExtnRe(M,Ng).

2. For each g ∈ supp(R,G), ExtnR(N,M) ∼= ExtnRe(Ng,M).

4. Flexible modules over augmented graded rings

In this section, we further study the structure of flexible modules over augmentedG−graded rings. We show in particular that over augmented graded rings, flexiblemodules and augmented modules are the same.

Definition 4.1 [7] A G−graded ring R is said to be augmented if the followingconditions hold:


1. Re =⊕g∈G

Re−g is a G−graded ring.

2. For each g ∈ G, there exists rg ∈ Rg such that Rg = Rerg. We assumere = 1.

3. If rg = 0 and rh = 0 are as in (2), then rgrh = rgh.

4. If rg = 0 and rh = 0 are as in (2), and x, y ∈ Re, then (xrg)(yrh) = xyrgh.

An rg that appears in condition (2) of Definition 4.1 is called a g−repre-sentative. The set of all selected nonzero representatives is denoted by Λ(R,G).Thus, the set Λ(R,G) may vary as the representatives vary. However, once anaugmented graded ring is under consideration, we fix Λ(R,G).

Proposition 4.4 below modifies Proposition 1.12 of [9], where it shows thataugmented graded rings are a subcategory of first strongly graded rings. We beginwith the following lemma, which we omit its proof, because it is trivial.

Lemma 4.2 Let R be an augmented G−graded ring. Then g ∈ supp(R,G) iffrg ∈ Λ(R,G).

Remark 4.3 [7] If R is an augmented G−graded ring, then

1. For every g ∈ G, Rg is a G−graded Re−module, with the graduationRg−h = Re−h rg.

2. For every g, h, g′, h′ ∈ G, Rg−h Rg′−h′ ⊆ Rgg′−hh′ .

3. Condition (4) of Definition 4.1 is equivalent to the condition

(xe−h rg)(xe−h′ rg′) = xe−hxe−h′ rgrg′ ,

for all h, h′ ∈ G, and rg, rg′ ∈ Λ(R,G).

Proposition 4.4 Every augmented G−graded ring is first strong.

Proof. Assume R is an augmented G−graded ring. Let g, h ∈ supp(R,G). ByLemma 4.2, RgRh = Rerg Rerh = ReRe rgrh = Re rgh = Rgh. Therefore, R is firststrong.

Corollary 4.5 Let R be an augmented G−graded ring. Then

1. supp(R,G) is a subgroup of G.

2. Λ(R,G) is a multiplicative group.

3. supp(R,G) is group-isomorphic to Λ(R,G).

4. Rg = rgRe = Re rg, for every g ∈ supp(R,G).


Proof. 1. Follows directly from Propositions 3.2 and 4.4.2. Apply Lemma 2.3 of [9] along with condition (1).3. Apply Lemma 2.3 and conditions (1) and (2).4. Apply (1) and condition (4) of Definition 4.1.

Remark 4.6 Corollary 4.5 gives us the right to cancel the condition “supp(R,G)is a subgroup of G” from all results in [9].

Definition 4.7 [6] LetR be an augmentedG−graded ring. AG−gradedR−moduleM is said to be augmented if the following conditions hold:

1. Mg =⊕h∈G

Mg−h is a G−graded Re−module.

2. Rg−hMg′−h′ ⊆Mgg′−hh′ , for every g, h, g′, h′ ∈ G.

Given an augmented G−graded ring, we let λ be the sum of elements ofΛ(R,G). If we set rg = 0, for every g /∈ supp(R,G), and rg ∈ Λ(R,G), otherwise,we can write λ =

∑g∈G

rg. The next proposition shows that a flexible module over

an augmented ring is completely determined by Me and λ.

Proposition 4.8 Let R be an augmented G−graded ring, and M be a G−gradedR−module. Then M is flexible iff M = λMe.

Proof. Suppose M is flexible R−module. It follows from Propositions 3.4 and4.4 that supp(M,G) = supp(R,G). Let g ∈ G. We have Mg = RgMe = rgMe.Hence, M =

⊕g∈G

(rgMe) = λMe.

Conversely, Suppose M = λMe. Then M = (∑g∈G

rg)Me =⊕g∈G

RgMe. Since

RgMe ⊆Mg, we obtain Mg = RgMe, for every g ∈ G. Therefore, M is flexible.

Corollary 4.9 Let R be an augmented G−graded ring, and M be an augmentedG−graded R−module. If supp(M,G) = supp(R,G), then M is flexible, andMg−h = rgMe−h, for every g, h ∈ G.

Proof. Assume supp(M,G) = supp(R,G). By Propositions 3.4 and 4.4, M isflexible. Proposition 4.8 implies Mg = rgMe, for all g ∈ G, where rg ∈ Λ(R,G),if g ∈ supp(R,G), and rg = 0, otherwise. Thus, Mg−h = rgMe−h, for everyg, h ∈ G.

The following proposition gives the converse of Proposition 4.8.

Proposition 4.10 Let R be an augmented G−graded ring, and M be a flexibleG−graded R−module such that Me is a G−graded Re−module. Then M is anaugmented G−graded R−module.


Proof. First, we prove thatMg is aG−graded Re−module, withMg−h = rgMe−h,for all g, h ∈ G. Propositions 3.4 and 4.4 yield supp(M,G) = supp(R,G). Letg ∈ supp(R,G) and h ∈ G. Then

∑h∈G

rgMe−h ⊆ Mg. On the other hand, let

x ∈ Mg. Then x =∑h∈G

rgme−h ∈∑h∈G

rgMe−h and hence Mg =∑h∈G

rgMe−h. Let

σ ∈ G, and x ∈ rgMe−σ ∩∑

h∈G−σrgMe−h. Then x = rgme−σ =

∑h∈G−σ

rgme−h,

where me−h ∈ Me−h, for every h ∈ G. By Corollary 4.5, rg is a unit. So we getme−σ −

∑h∈G−σ

me−h = 0. Thus, we obtain me−σ = 0, and then x = 0. Therefore,

rgMe−σ ∩∑

h∈G−σrgMe−h = 0. Consequently, Mg =

⊕h∈G

rgMe−h. The same result

obviously holds if g /∈ supp(R,G). For each g, h ∈ G, set Mg−h = rgMe−h. Letσ ∈ G. Then Re−σMg−h = rg(Re−σMe−h) ⊆ Mg−σ h. As a result, we obtain Mg

is a G−graded Re−module.Now, we show that Rg−hMσ−τ ⊆ Mgσ−hτ , for every g, h, σ, τ ∈ G. In fact,

if g, σ ∈ supp(R,G), we have Rg−hMσ−τ = rgRe−h rσMe−τ = rgrσRe−h Me−τ ⊆rgrσMe−hτ = Mgσ−hτ . The case where either g /∈ supp(R,G) or σ /∈ supp(R,G)is easy.

As a conclusion, M is an augmented G−graded R−module.

The following result is the main result in this section. It sums up bothCorollary 4.9 and Proposition 4.10 in the statement “Over augmented gradedrings, flexible modules and augmented modules are the same”. The proof is aquick application of Corollary 4.9 and Proposition 4.10.

Theorem 4.11 Let R be an augmented G−graded ring, and M be a G−gradedR−module such thatMe is a G−graded Re−module. ThenM is flexible R−moduleiff M is an augmented G−graded R−module, and supp(R,G) = supp(M,G).

5. Flexibly simple and flexibly Noetherian modules

In this section, We define flexibly simple and flexibly Noetherian modules, andgive an analogous study to that exhibited in [8]. We start by noticing that thecombination of Propositions 2.2 and 2.5 yields the fact which says “given a G−graded ring R, and a G−graded R−module M , then a G−graded submodule Nof M is flexible iff there exists an Re−submodule Ne of Me such that N = RNe”.So, regardless M is flexible or not, we deduce that RMe is the largest flexibleR−submodule of M , and 0 is the smallest flexible R−submodule of M .

Now, we move to our definition of flexibly simple modules. To avoid anyconfusion, it is important to draw the reader’s attention to the fact that Definition5.1 is a generalization of the definition of flexible simple modules [14], which isonly defined for flexible modules.

Definition 5.1 Let R be a G− graded ring, and M be a G−graded R−module.ThenM is called flexibly simple if 0 and RMe are the only flexible R−submodulesof M .


The structure of flexible modules indicates a strong relationship between fle-xibly simple R−modules and simple Re−modules, as the following propositionillustrates.

Proposition 5.2 Let R be a G− graded ring, and M be a G−graded R−module.Then M is flexibly simple iff Me is a simple Re−module.

Proof. Suppose M is a flexibly simple R−module. Let X = 0 be an Re−sub-module of Me. By Proposition 2.5, RX is a nonzero flexible R−submodule ofM . Therefore, RX = RMe, and hence X = Me. Consequently, Me is a simpleRe−module.

Conversely, suppose Me is a simple Re−module. Let N = 0 be a flexibleR−submodule ofM . We haveNe = 0 is an Re−submodule ofMe. Thus, Ne =Me,which implies that N = RMe.

In the following corollary, the definition of gr−simple modules (i.e., gradedsimple modules) can be found in [2].

Corollary 5.3 Let R be a first strongly G− graded ring, and M be a G−gradedR−module such that e ∈ supp(M,G). Then the following statements are equiva-lent:

1. RMe is gr−simple R−module.

2. Me is simple Re−module.

3. M is flexibly simple R−module.

Proof. The equivalence of (1) and (3) follows from Proposition 5.2. The impli-cation “(1) ⇒ (2)” is a well known result of Graded Module Theory [2]. For “(2)⇒ (1)”, apply Propositions 3.6 and 5.2.

In the following result, we show that nontrivial flexibly simple R−modulesare only valid over non-strongly graded rings. In fact, this is one reason that urgesmathematicians to seek well-structured graded rings, rather than strongly gradedrings, over which modules obtain nice properties. Examples of such structures arefirst and second strongly graded rings, as well as augmented graded rings.

Proposition 5.4 Suppose R is a strongly G−graded ring, and M is non-cyclicG−graded R−module. Then M is not flexibly simple. In particular, if M isflexibly simple R−module, then R is not strong.

Proof. Since R is strong, M is flexible and Me = 0. Pick 0 = x ∈ Me. SetN = Rx. We have 0 = N = M is a G−graded R−submodule of M . Moreover,N is flexible because if g ∈ G, Ng = Rg x = (Rg Re)x = Rg (Re x) = RgNe.

Next we study flexibly Noetherian R−modules. Again, Definition 5.5 gene-ralizes the definition of flexible Noetherian modules [14], which is defined only forflexible modules.


Definition 5.5 Let R be a G− graded ring, and M be a G−graded R−module.Then M is said to be flexibly Noetherian if every ascending chain of flexibleR−submodules of M terminates.

An immediate consequence of Definition 5.5 is that every flexibly simpleR−module is flexibly Noetherian.

Proposition 5.6 Let R be a G− graded ring, and M be a G−graded R−module.Then M is a flexibly Noetherian R−module iff Me is a Noetherian Re−module.

Proof. Suppose M is a flexibly Noetherian R−module. Let X(1) ⊆ X(2) ⊆ . . . bean ascending chain of Re−submodules of Me. Then by Proposition 2.5, RX(1) ⊆RX(2) ⊆ . . . is the corresponding ascending chain of flexible R−submodules ofM . By assumption, there exist n ∈ N such that RX(n) = RX(n+1) = . . .. Hence,we obtain X(n) = X(n+1) = . . .. Therefore, Me is a Noetherian Re−module.

For the converse, assume Me is a Noetherian Re−module.Let X(1) ⊆ X(2) ⊆ . . . be an ascending chain of flexible R−submodules of

M . We have X(1)e ⊆ X

(2)e ⊆ . . . is an ascending chain of Re−submodules of Me.

By assumption, there exists n ∈ N such that X(n)e = X

(n+1)e = . . .. Hence,

RX(n)e = RX(n+1)

e = . . . , or X(n) = X(n+1) = . . . .

That is, M is flexibly Noetherian.

The following corollaries generalize Corollary 4.2.6 and Remark 4.2.10 of [14],respectively. The proof of the first one follows obviously from Corollary 5.3.

Corollary 5.7 Let R be a first strongly G− graded ring, and M be a flexibleG−graded R−module. Then the following statements are equivalent:

1. M is gr−simple (resp. gr-Noetherian) R−module.

2. Me is simple (resp. Noetherian) Re− module.

3. M is flexibly simple (resp. flexibly Noetherian) R−module.

Corollary 5.8 Let R be a commutative G− graded ring, and M be a flexibleG−graded R−module. Then the following statements are equivalent:

1. M is simple (resp. Noetherian) R−module.

2. M is gr−simple (resp. gr-Noetherian) R−module.

3. Me is simple (resp. Noetherian) Re− module.

4. M is flexibly simple (resp. flexibly Noetherian) R−module.

Proof. Apply Theorem 3.8 (resp. Proposition 5.6) and Proposition 5.2.


References

[1] Cohen, M., Rowen, L., Group graded rings, Comm. in Algebra, 11 (1983),1253-1270.

[2] Dade, E.C., Group graded rings and modules, Math Z., 174 (1980), 241-262.

[3] Nastasescu, C., Strongly graded rings of finite groups, Comm. in Algebra,11 (10) (1983), 1033-1071.

[4] Nastasescu, C., Oystaeyen, Van, Graded Ring Theory, Mathematicallibrary 28, North Holand, Amsterdam, (1982).

[5] Refai, M., Various types of strongly graded rings, Abhath Al-Yarmouk Jour-nal (Pure Sciences and Engineering Series), 4 (2) (1995), 9-19.

[6] Refai, M., Augmented graded modules, Tishreen Univesity Journal for Studiesand Scientific Research, 23 (10) (2001), 231-245.

[7] Refai, M., Augmented graded rings, Turkish Journal of Mathematics,21(3)(1997), 333-341.

[8] Refai, M., Augmented Noetherian graded modules, Turkish Journal of Ma-thematics, 23(1999), 355-360.

[9] Refai, M., Moh’D, F., Characterizations of augmented graded rings,Turkish Journal of Mathematics, 29(2005), 211-220.

[10] Refai, M., Moh’D, F., First strongly graded modules, Int. J. Math.: GameTheory and Algebra, 15(4)(2006), 451-457.

[11] Refai, M., Moh’D, F., More properties on various types of strongly gradedrings, Pacific-Asian Journal of Mathematics, 1(2)(2007), 129-135.

[12] Refai, M., Moh’D, F., On Flexible graded modules, Italian Journal of Pureand Applied Mathematics, 22(2007), 125-132.

[13] Refai, M., Obeidat, M., On a strongly-supported graded rings, Mathema-tica Japonica Journal, 39(3) (1994), 519-522.

[14] Moh’D, F., On properties defined over various types of strongly graded rings,M.Sc. thesis, Yarmouk University, 2001.

Accepted: 4.11.2012


ON (N(k), ξ)-SEMI-RIEMANNIAN 3-MANIFOLDS

D.G. Prakasha

Department of MathematicsKarnatak UniversityDharwad – 580 003Indiae-mail: [email protected]

H.G. Nagaraja

Department of MathematicsCentral College CampusBangalore UniversityBangaloreIndiae-mail: [email protected]

G. Somashekhara

Department of MathematicsAcharya Institute of TechnologySoldevanahalliBangalore – 560107India

Abstract. The object of the present paper is to study 3-dimensional (N(k), ξ)-semi-

Riemannian manifolds. We study (N(k), ξ)-semi-Riemannian 3-manifolds which are

Ricci-semi-symmetric, locally ϕ-symmetric and have η-parallel Ricci tensor.

Key words and phrases: (N(k), ξ)-semi-Riemannian 3-manifold, Ricci-semi-symme-

tric, locally ϕ-symmetric, η-parallel Ricci tensor, η-Einstein manifold.

MSC(2000): 53C25, 53C50.

1. Introduction

Let (M, g) be an n-dimensional semi-Riemannian manifold [12] equipped witha semi-Riemannian metric g. If index(g)=1 then g is a Lorentzian metric and(M, g) a Lorentzian manifold [4]. If g is positive definite then g is an usualRiemannian metric and (M, g) a Riemannian manifold. The notion of (N(k), ξ)-semi-Riemannian structure was introduced and studied by Tripathi and Gupta[21] to unify N(k)-contact metric [3], Sasakian [5], [14], (ϵ)-Sasakian [17], [22],Kenmotsu [10], para-Sasakian [15], (ϵ)-para-Sasakian structures [20].

116 d.g. prakasha, h.g. nagaraja, g. somashekhara

In this paper we study 3-dimensional (N(k), ξ)-semi-Riemannian manifolds.The paper is organized as follows. Section 2 is devoted to some basic defini-tions and properties of almost contact metric, almost para contact metric and(N(k), ξ)-semi-Riemannian manifolds. Further, we prove that an (N(k), ξ)-semi-Riemannian 3-manifold is a space form if and only if the scalar curvature r ofthe manifold is equal to 6k. In Section 3, we show that a Ricci-semi-symmetric(N(k), ξ)-semi-Riemannian 3-manifold is a space-form. In Section 4, a neces-sary and sufficient condition for an (N(k), ξ)-semi-Riemannian 3-manifold to belocally ϕ-symmetric is obtained. Section 5 contains some results on (N(k), ξ)-semi-Riemannian 3-manifold with η-parallel Ricci tensor.

2. Preliminaries

Let M be an n-dimensional differentiable manifold endowed with an almost con-tact structure (ϕ, ξ, η), where ϕ is a (1, 1)-tensor field, ξ is a vector field and η isa 1-form on M satisfying

(2.1) η(ξ) = 1, ϕ2 = −I + η ⊗ ξ,

where I denotes the identity transformation. It follows from (2.1) that

(2.2) η · ϕ = 0, ϕ(ξ) = 0.

If there exists a semi-Riemannian metric g satisfying

(2.3) g(ϕX, ϕY ) = g(X,Y )− ϵη(X)η(Y ), ∀X, Y ∈ χ(M),

where ϵ = ±1, then the structure (ϕ, ξ, η, g) is called an (ϵ)-almost contact metricstructure andM is called an (ϵ)-almost contact metric manifold. For an (ϵ)-almostcontact metric manifold, we have

(2.4) η(X) = ϵg(X, ξ) and ϵ = g(ξ, ξ) ∀X ∈ χ(M).

When ϵ = 1 and index of g is 0 then M is the usual Sasakian manifold and M isa Lorentz-Sasakian manifold for ϵ = −1 and index of g is 1.

If dη(X, Y ) = g(ϕX, Y ), then M is said to have (ϵ)-contact metric structure(ϕ, ξ, η, g). For ϵ = 1 and g Riemannian, M is the usual contact metric manifold[5]. A contact metric manifold with ξ ∈ N(k), is called a N(k)-contact metricmanifold [1, 6]. If moreover, this structure is normal, that is, if

(2.5) [ϕX, ϕY ] + ϕ2[X, Y ]− ϕ[X,ϕY ]− ϕ[ϕX, Y ] = −2dη(X, Y )ξ,

then the (ϵ)-contact metric structure is called an (ϵ)-Sasakian structure and themanifold endowed with this structure is called (ϵ)-Sasakian manifold. The physicalimportance of indefinite Sasakian manifolds have been pointed out by Duggalin [9].

on (N(k),ξ)-semi-riemannian 3-manifolds 117

An (ϵ)-almost contact metric structure (ϕ, ξ, η, g) is (ϵ)-Sasakian if and only if

(2.6) (∇Xϕ)Y = g(X, Y )ξ − ϵη(Y )X, ∀X, Y ∈ χ(M),

where ∇ is the Levi-Civita connection with respect to g. Also we have

(2.7) ∇Xξ = −ϵϕX ∀X ∈ χ(M).

An almost contact metric manifold is a Kenmotsu manifold [10] if

(2.8) (∇Xϕ)Y = g(ϕX, Y )ξ − η(Y )ϕX.

By (2.8), we have∇Xξ = X − η(X)ξ.

If in (2.1), the condition ϕ2 = −I + η ⊗ ξ is replaced by

(2.9) ϕ2 = I − η ⊗ ξ

then (M, g) is called an (ϵ)-almost paracontact metric manifold equipped with an(ϵ)-almost paracontact metric structure (ϕ, ξ, η, g).

An (ϵ)-almost paracontact metric structure is called (ϵ)-para-Sasakian struc-ture [20] if

(2.10) (∇Xϕ)Y = −g(ϕX, ϕY )ξ − ϵη(Y )ϕ2X,

where ∇ is Levi-Civita connection with respect to the metric g. A manifoldendowed with an (ϵ)-para-sasakian structure is called (ϵ)-para-Sasakian manifold[20]. For ϵ = 1 and g Riemannian, M is the usual para-Sasakian manifold [15].

(N(k), ξ)-semi-Riemannian manifold

The k-nullity distribution [18] of (M, g) is the distribution

(2.11) N(k) : p → Np(k) = Z ∈ TpM : R(X, Y )Z = k(g(Y, Z)X − g(X,Z)Y ),

where k is a real number.An (N(k), ξ)-semi-Riemannian manifold consists of a semi-Riemannian mani-

fold (M, g), a k-nullity distribution N(k) on (M, g) and a non-null unit vec-tor field ξ in (M, g) belonging to N(k). Throught the paper we assume thatX, Y, Z, U, V,W ∈ χ(M), where χ(M) is the Lie algebra of vector fields in M ,unless specifically stated otherwise. Let ξ be a non null unit vector field in (M, g)and η its associated 1-form. Thus

g(ξ, ξ) = ϵ,

where ϵ = 1 or −1 according as ξ is spacelike or timelike, and

(2.12) a)g(X, ξ) = ϵη(X), b)η(ξ) = 1.


In an n-dimensional (N(k), ξ)-semi-Riemannian manifold (M, g), the followingrelations hold [21]:

R(X, Y )ξ = ϵkη(Y )X − η(X)Y ,(2.13)

R(ξ,X)Y = ϵkϵg(X, Y )ξ − η(Y )X,(2.14)

η(R(X,Y )Z) = kη(X)g(Y, Z)− η(Y )g(X,Z),(2.15)

S(X, ξ) = ϵk(n− 1)η(X),(2.16)

In a 3-dimensional Riemannian manifold we have

R(X, Y )Z = g(Y, Z)QX − g(X,Z)QY + S(Y, Z)X − S(X,Z)Y(2.17)

− r

2[g(Y, Z)X − g(X,Z)Y ],

where Q is the Ricci operator, i.e., g(QX, Y ) = S(X, Y ) and r is the scalarcurvature of the manifold. Putting Z = ξ in (2.17) and using (2.13) and (2.16),we have

ϵ(η(Y )QX − η(X)QY ) =(−ϵk +

r

2ϵ)(η(Y )X − η(X)Y ).(2.18)

Putting Y = ξ in (2.18) and then using (2.12(b)) and (2.16) (for n=3), we get

(2.19) QX =1

2(r − 2k)X − (r − 6k)η(X)ξ,

that is,

(2.20) S(X,Y ) =1

2(r − 2k)g(X,Y )− ϵ(r − 6k)η(X)η(Y ).

An (N(k), ξ)-semi-Riemannian manifold M is said to be η-Einstein if its Riccitensor S is of the form

(2.21) S(X, Y ) = ag(X, Y ) + bη(X)η(Y )

for any vector fields X, Y where a, b are functions on M . Hence from (2.20) wecan state the following:

Lemma 1 A 3-dimensional (N(k), ξ)-semi-Riemannian manifold is an η-Einsteinmanifold.

By using (2.19) and (2.20) in (2.17), we obtain

R(X,Y )Z =(r2− 2k

)g(Y, Z)X − g(X,Z)Y (2.22)

−(r2− 3k

)g(Y, Z)η(X)ξ − g(X,Z)η(Y )ξ

+ ϵη(Y )η(Z)X − ϵη(X)η(Z)Y .

An (N(k), ξ)-semi-Riemannian 3-manifold is a space of constant curvature thenit is an indefinite space form.

Remark. Relations (2.19), (2.20) and (2.22) are true for


1. An N(k)-contact metric 3-manifold [8] if ϵ = 1,

2. A Sasakian 3-manifold if k = 1 and ϵ = 1,

3. A Kenmotsu 3-manifold [7] if k = −1 and ϵ = 1,

4. An (ϵ)-Sasakian 3-manifold if k = 1 and ϵk = 1,

5. A para-Sasakian 3-manifold [2] if k = −1 and ϵ = 1,

6. An (ϵ)-para-Sasakian 3-manifold [19] if k = −ϵ and ϵk = −1.

Lemma 2 A 3-dimensional (N(k), ξ)-semi-Riemannian manifold is a space formif and only if the scalar curvature r = 6k.

Consequently, for a 3-dimensional (N(k), ξ)-semi-Riemannian manifold, wehave the following table:

M S = r =

N(k)-contact metric 12(r − 2k)g − (r − 6k)η ⊗ η 6k

Sasakian 12(r − 2)g − (r − 6)η ⊗ η 6

Kenmotsu 12(r + 2)g − (r + 6)η ⊗ η −6

(ϵ)-Sasakian 12(r − 2ϵ)g − ϵ(r − 6ϵ)η ⊗ η 6ϵ

para-Sasakian 12(r + 2)g − (r + 6)η ⊗ η −6

(ϵ)-para Sasakian 12(r + 2ϵ)g − ϵ(r + 6ϵ)η ⊗ η −6ϵ

Proof. Let a 3-dimensional (N(k), ξ)-semi-Riemannian manifold be an indefinitespace form. Then

(2.23) R(X, Y )Z = cg(Y, Z)X − g(X,Z)Y , X, Y, Z ∈ χ(M),

where c is the constant curvature of the manifold. By using the definition of Riccicurvature and (2.23) we have

(2.24) S(X, Y ) = 2cg(X, Y ).

If we use (2.24) in the definition of the scalar curvature we get

(2.25) r = 6c.

From (2.24) and (2.25) one can easily see that

(2.26) S(X,Y ) =r

3g(X,Y ).

By plugging X = Y = ξ in (2.20) and using (2.26) we obtain

(2.27) r = 6k.


Conversely, if r = 6k, then from the equation (2.22) we can easily see thatthe manifold is a space form. This completes the proof.

3. Ricci-semi-symmetric (N(k), ξ)-semi-Riemannian 3-manifolds

A semi-Riemannian manifold M is said to be Ricci semi-symmetric [13] if its Riccitensor S satisfies the condition

(3.28) R(X, Y ) · S = 0, X, Y ∈ χ(M),

where R(X, Y ) acts as a derivation on S. Ricci-semisymmetric manifold is a gene-ralization of manifold of constant curvature, Einstein manifold, Ricci symmetricmanifold, symmetric manifold and semisymmetric manifold. Ricci-semisymmetriccondition for Kenmotsu 3-manifolds, (ϵ)-para-Sasakian 3-manifolds and LP-Sasa-kian 3-manifolds are studied in [7], [19] and [16] respectively.

LetM be a Ricci-semi-symmetric (N(k), ξ)-semi-Riemannian 3-manifold. From(3.28) we have

(3.29) S(R(X, Y )U, V ) + S(U,R(X,Y )V ) = 0.

If we put X = ξ in (3.29) and use (2.14), then we get

(3.30) kg(Y, U)S(ξ, V )−ϵKη(U)S(Y, V )+kg(Y, V )S(U, ξ)−ϵKη(V )S(U, Y )=0.

By using (2.16) in (3.30) we obtain

(3.31) ϵK2kg(Y, U)η(V )− η(U)S(Y, V )− 2kg(Y, V )η(U)− η(V )S(U, Y ) = 0.

Consider that e1, e2, e3 be an orthonormal basis of the TpM , p ∈ M . Then, byputting X = U = ei in (2.2) and taking the summation for 1 ≤ i ≤ 3, we have

(3.32) ϵk8kη(V )− ϵS(V, ξ)− rη(V ) = 0.

Again, by using (2.16) in (3.32), we get

(3.33) ϵk(6k − r)η(V ) = 0,

which gives r = 6k. This implies, in view of Lemma 2, that the manifold is aspace form.

Therefore, we have the following:

Theorem 1 A Ricci-semi-symmetric (N(k), ξ)-semi-Riemannian 3-manifold is aspace form.

From Theorem 1 and the above table, we can state the following corollaries:

Corollary 1 A Ricci-semi-symmetric N(k)-contact metric 3-manifold is a ma-nifold of constant scalar curvature 6k.


Corollary 2 A Ricci-semi-symmetric Sasakian 3-manifold is a manifold of con-stant positive scalar curvature 6.

Corollary 3 [7] A Ricci-semi-symmetric Kenmotsu 3-manifold is a manifold ofconstant negative scalar curvature −6.

Corollary 4 A Ricci-semi-symmetric (ϵ)-Sasakian 3-manifold is an indefinitespace form.

Corollary 5 [2] A Ricci-semi-symmetric para-Sasakian 3-manifold is a manifoldof constant negative scalar curvature −6.

Corollary 6 [19] A Ricci-semi-symmetric (ϵ)-para-Sasakian 3-manifold is an in-definite space form.

4. Locally ϕ-symmetric (N(k), ξ)-semi-Riemannian 3-manifolds

Definition 1 An (N(k), ξ)-semi-Riemannian manifold is said to be locally ϕ-symmetric if

ϕ2(∇WR)(X, Y, Z) = 0,

for all vector fields W,X, Y, Z orthogonal to ξ. This notion was introduced forSasakian manifolds by Takahashi [17].

Now, differentiating (2.22) covariantly with respect to W , we get

(∇WR)(X,Y, Z) =1

2(∇W r)g(Y, Z)X − g(X,Z)Y − g(Y, Z)η(X)ξ

+ g(X,Zη(Y )ξ − ϵη(Y )η(Z)X + ϵη(X)η(Z)Y

− (r − 6k)

2g(Y, Z)((∇Wη)(X)ξ + η(X)∇W ξ)

− g(X,Z)((∇Wη)(Y )ξ + η(Y )∇W ξ)

+ ϵ((∇Wη)(Y )η(Z)X + (∇Wη)(Z)η(Y )X)

−ϵ((∇Wη)(X)η(Z)Y + (∇Wη)(Z)η(X)Y ).

Taking W,X, Y, Z orthogonal to ξ, we have

(4.34)(∇WR)(X, Y, Z) =

1

2(∇W r)g(Y, Z)X − g(X,Z)Y

− (r − 6k)

2g(Y, Z)(∇Wη)(X)ξ − g(X,Z)(∇Wη)(Y )ξ.

Applying ϕ2 on both sides of the above equation and using ϕ · ξ = 0, we have

(4.35) ϕ2((∇WR)(X, Y, Z)) =1

2(∇W r)g(Y, Z)ϕ2X − g(X,Z)ϕ2Y .


Now taking X,Y are orthogonal to ξ, we obtain

(4.36) ϕ2((∇WR)(X, Y, Z)) = −1

2(∇W r)g(Y, Z)X − g(X,Z)Y

Hence from (4.36), we can state the following:

Theorem 2 An (N(k), ξ)-semi-Riemannian 3-manifold is locally ϕ-symmetric ifand only if the scalar curvature r is constant.

If an (N(k), ξ)-semi-Riemannian 3-manifold is Ricci semi-symmetric, then wehave showed that r = 6k, that is r is constant.

Therefore, from Theorem (2), we have

Theorem 3 A Ricci-semi-symmetric (N(k), ξ)-semi-Riemannian 3-manifold islocally ϕ-symmetric.

5. (N(k), ξ)-semi-Riemannian 3-manifold with η-parallel Ricci tensor

Definition 2 The Ricci tensor S of an (N(k), ξ)-semi-Riemannian manifold Mis called η-parallel if it satisfies

(5.37) (∇ZS)(ϕX, ϕY ) = 0

for all vector fields X, Y and Z. The notion of Ricci-η-parallelity for Sasakianmanifolds was introduced by Kon in [11].

Now, let us consider a 3-dimensional (N(k), ξ)-semi-Riemannian manifoldwith η-parallel Ricci tensor. Then, from (2.20), we get

(5.38) S(ϕX, ϕY ) =1

2(r − 2k)[g(ϕX, ϕY )].

Differentiating (5.38) covariantly along Z, we have

(5.39) (∇ZS)(ϕX, ϕY ) =1

2dr(Z)g(ϕX, ϕY ).

If the Ricci tensor is η-parallel, then from (5.37) and (5.39) one can get

1

2dr(Z)g(ϕX, ϕY ) = 0.

From which, it follows thatdr(Z) = 0,

for all Z. This leads us to the following:


Theorem 4 Let M be an (N(k), ξ)-semi-Riemannian 3-manifold with η-parallelRicci tensor. The the scalar curvature r is constant.

In view of Theorem (2) and Theorem (4), we have the following:

Theorem 5 An (N(k), ξ)-semi-Riemannian 3-manifold with η-parallel Ricci ten-sor is locally ϕ-symmetric.

Acknowledgement. The first author (DGP) is thankful to University GrantsCommission, New Delhi, India for financial support in the form of Major ResearchProject [F.No. 39-30/2010 (SR), dated: 23-12-2012].

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[7] De, U.C., Pathak, G., On 3-dimensional Kenmotsu manifolds, Indian J.Pure Appl. Math., 35 (2)(2004), 159-165.

[8] De, U.C., Gazi, A.K., On ϕ-recurrent N(k)-contact metric manifolds,Math. J. Okayama Univ., 50 (2008), 101-112.

[9] Duggal, K.L., Space time manifolds and contact structures, Int. J. Math& Math. Sc., 13 (3) (1990), 55-553.


[10] Kenmotsu, K., A class of almost contact Riemannian manifold, TohokuMath. J., 24 (2) (1972), 93-103.

[11] Kon, M., Invariant submanifolds in Sasakian manifolds, Math. Annalen,219 (1976), 277-290.

[12] O’Neill, B., Semi-Rimannain geometry with applications to relativity,Academic Press, New York, London, 1983.

[13] Mirzoyan, V.A., Structure theorems for Riemannian Ric-semisymmetricspaces, Izv. Vyssh. Uchebn. Zaved. Mat., 6 (1992), 80-89.

[14] Sasaki, S., On differentiable manifolds with certain structures which areclosely related to almost contact structure I, Tohoku Math. J., 12 (1960),459-476.

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A NEW SIGNING ALGORITHM BASED ON ELLIPTIC CURVEDISCRETE LOGARITHMS AND QUADRATIC RESIDUEPROBLEMS

Nedal Tahat

Department of MathematicsFaculty of SciencesThe Hashemite UniversityZarqa 13133Jordane-mail: [email protected]

Emad E. Abdallah

Department of Computer Information SystemFaculty of Prince Al-Hussein Bin Abdallah IIfor Information TechnologyThe Hashemite UniversityZarqa 13115Jordane-mail: [email protected]

Abstract. In this paper we propose a new digital signature algorithm for authenticity

and integrity of a digital message. The core idea behind our approach is concentrated

on using two hard problems in the signing process. The elliptic curve discrete loga-

rithm and quadratic residue are engaged in a sophisticated manner to do the signing.

The new proposed scheme provides higher level of security than other techniques that

use a single hard problem. Clearly, Cybercriminals have to solve the two underlying

hard problems simultaneously to destroy embedded signature. Extensive experimental

results on several signed documents are performed to demonstrate the robustness of the

proposed scheme against the most common attacks on digital signatures. Moreover, the

computational complexity of the new scheme requires reasonable number of operations

in both signing and verifying algorithms.Keywords: Cryptography, Digital Signature,

Quadratic Residue, Elliptic Curve Discrete Logarithms, heuristically secure.

1. Introduction

In modern cryptography, the security and robustness of the digital signature al-gorithms are based on the difficulty of solving some hard theoretical problemssuch as factoring and discrete logarithm [2], [3], [14]. In the literature, variousdigital signature algorithms that use two theoretical hard problems [4], [5], [15]

126 n. tahat, e.e. abdallah

have been proposed. One common feature of these algorithms is that they aredepending on a number-theoretical problem and thus their implementations de-pend heavily on modular exponentiation which is known to be time consumingwith high computational complexity.

Elliptic curves are used in cryptography by Koblitz [7] and Miller [10] to over-come the costly modular exponentiation. The cryptographic schemes with securitylie on the so-called elliptic curve discrete logarithm (ECDLP) [7], [10] are provedto provide longer security and better efficiency than both integer factorizationsystem and discrete logarithm systems. The ECDLP has become a turning pointof the rigorous development of cryptographic schemes [1], [9], [6], [13], [11], [12].

Motivated by the need for secure digital signature scheme,we propose a robustsignature approach using the ECDLP and quadratic residue problem (QRP) hardproblems. The new scheme offers better security than all other schemes basedon either ECDLP or QRP. This is because the probability of solving two hardproblems simultaneously by adversaries is believed to be negligible. Moreover,the proposed scheme does not involve any modular exponentiation operation inall algorithms. The remainder of this paper is organized as follows. In Section 2,we briefly review some background material and describe the ECDLP and QRPhard problems. In Section 3 we introduce the proposed approach and describe indetail the key generation, the signing and the verification algorithms. In Sections4 and 5, we present some security analysis and the performance evaluation ofthe proposed approach. Finally, we conclude and point out future directions inSection 5.

1.1. Elliptic curve

In this section we describe some elementary tools on elliptic curves and define thetwo underlying hard problems ECDLP and QRP.

Definition 1.1 Let K be a field of characteristic neither of 2 nor 3, then anelliptic curve can be expressed as:

(1.1) y2 = x3 + ax+ b

where a, b ∈ K and 4a3 + 27b2 = 0. The set E(K) consists of all point (x, y),x, y ∈ K which satisfy the defining equation (1.1) together with a special pointO called the point at infinity. Let G be a point on the elliptic curve defined inin equation (1.1) and if n is the smallest positive integer satisfying the equationnG = O, then we say that G has an order n and is called the base point. See [6],[7], [8], [10] for complete discussion on how to add and to multiply elliptic curvespoints.

• ECDLP: Let G and C be two elliptic curve points on equation (1.1). Thenfind a positive integer k such that kG = C.

• QRP: Let p, q are two strong primes of large size and γ is an integer. Thencompute an integer γ such that γ ≡ β2 (mod pq)

a new signing algorithm based on elliptic curve discrete ... 127

2. The new signature scheme

The proposed signature scheme consists of four main phases: initialization, al-gorithm for key generation, algorithm for signing messages, and algorithm forsignature verification.

2.1. Initialization

• The field K = Fp of order p, where p is be a large prime number and p− 1have two prime factors p and q.

• Two coefficient a, b ∈ Fp that define the equation y2 = x3 + ax + b(mod p)over Fp.

• n = pq, so that n/(p − 1) is a root points of elliptic curve construct acirculating subgroup. G is a generating element for subgroup and its rankequals n.

• h(.) is a secure hash function.

2.2. Algorithm for generating keys

Step 1 Pick randomly two integers α and β from Zn

Step 2 Compute an integer k such that kα4β ≡ −1(modn)

Step 3 Calculate T = kG

The signer publishes his public keys as (p,G, n, T ) and his corresponding pri-vate keys as (α, β, k).

2.3. Algorithm for signing message

Suppose the verifier wants the signer’s signature on his message m. The signerthen:

Step 1 Select an integer r ∈ Zn such that gcd(r, n) = 1

Step 2 Compute R = rβ2G = (x1, x2), where u ≡ x1(modn)

Step 3 Calculate s ≡ 12(r + h(m)u2β2)(modn)

Step 4 Calculate v ≡ 12α2β(r − h(m)u2β2)(modn)

The original signer then produces (R, s, v) as a signature of message M .

2.4. Algorithm for verifying signature

Verifier tests the validity of the signature (R, s, v) by checking the following steps


Step 1 Compute W1 = s2G+ V 2T (modn)

Step 2 Compute W2 = h(m)u2R (modn)

Accept the signature as valid if and only if W1 = W2

Theorem 2.1 If the algorithms for keys and signing message are run smoothlythen the validation of signature is correct.

Proof. Squaring s and v

s2 ≡ 1

4(r2 + h(m)2u4β4 + 2rh(m)u2β2) (modn)

v2 ≡ 1

4α4β(r2 + h(m)2u4β4 − 2rh(m)u2β2) (modn)

we have

s2G =1

4(r2 + h(m)2u4β4 + 2rh(m)u2β2)G (modn)

v2T =1

4α4β(r2 + h(m)2u4β4 − 2rh(m)u2β2)kG (modn)

=1

4kα4β(r2 + h(m)2u4β4 − 2rh(m)u2β2)G (modn)

= −1

4(r2 + h(m)2u4β4 − 2rh(m)u2β2)G (modn)

Thus

W1 = s2G+ v2T

=1

4(r2 + h(m)2u4β4 + 2rh(m)u2β2)G− 1

4(r2 + h(m)2u4β4 − 2rh(m)u2β2)G

= h(m)u2rβ2G

= h(m)u2R

= W2

We accept the signature if W1 = W2. With the knowledge of the signer’spublic key (n, T ) and the signature (R, s, v) of m, the verifier can authenticatethe message m because the verifier can be convinced that the message was reallysigned by the signer. Else, the signature (R, s, v) is invalid.

3. Security analysis

One of the known models of security in cryptography is heuristic or ad-hoc model.We show that our scheme is heuristically secure by considering several possibleattacks by an adversary (Adv). For every attack, we give valid reasons of whythis attack fails.


Attack 1: Adv wishes to obtain secret keys (α, β, k) using all information avai-lable in the system. In this case, Adv needs to solve T = kG (modn) andkα4β ≡ −1(modn) which are clearly infeasible due to the difficulty of solvingthe ECDLP and factoring problem .

Attack 2: Adv tries to derive the signature (R, s, v) for a given message m byfixing the value of two integers in order to find the remaining one. In this case,Adv randomly fixes either (R, s) or (R, v) or (s, v) to find s, v or R respectivelyto satisfy W1 = W2. Obviously, this is as difficult as solving QRP and ECDLPsimultaneously. For example, say Adv fixes the value (R, s) and tries to figureout the value of v . Adv then needs to solve the following equations that can bereduced from

(2.1) s2G+ v2T = h(m)u2R

Adv start by computing

γ = v2T

where γ is known and can be calculated easily. Note that, solving the aboveequation is as hard as solving ECDLP. But, even if ECDLP is solvable then theabove equation is reducible to the next equation as below:

(2.2) λ ≡ kv2(modn)

where λ is known but solving this equation is hard as solving the QRP.

Attack 3: Adv may also try collecting t valid signatures (Rj, sj, vj) on messageMj where j = 1, 2, ..., t and attempts to find secret keys of the signature scheme.In this case, Adv has s equations as follows:

s21 + v21k = r1h(M1)u21β

2

s22 + v22k = r2h(M2)u22β

2

...

s2t + v2t k = rth(Mt)u2tβ

2

In the above s equations, there are (t+ 2) variables that is k, β and rj, wherej = 1, 2, ..., t which are unknown by the Adv. Hence, k and β remain hard to detectbecause Adv can generate infinite solutions of the above system of equations butcannot figure out which one is correct.

Attack 4: It is assumed that Adv is able to solve ECDLP problem. In this case,Adv knows k and rβ2 ≡ ξ (modn) but cannot figure out the values of r and βbecause breaking of QRP is difficult. Now from s2G+ v2T = h(m)u2R, Adv willhave

(2.3) s2 + v2k ≡ rh(m)u2β2(modn)


Adv can launch Attack 2 but will not be successful due to the hardness of breakingQRP. Since Adv knows k, Adv may try to obtain α and β from equation

(2.4) kα4β ≡ −1(mod n)

but still fail although Adv does know the factorization of n.

Attack 5: It is assumed that Adv is able to solve FAC problem. That means,he knows the prime factorization p and q. In this case, Adv will learns nothingabout k, α and β from the equation (2.4). Adv also has no information about kand (r, β) from equation (2.3) because there are three unknowns in the equation.

4. Performance evaluation

In this section we investigate the efficiency and the performance of our proposedscheme in terms of number of keys, computational complexity and communicationcosts. The complete lists of the notations that we used to analyze the performanceare shown in Table 1.

Table 1: Lists of the notations that we used to analyze the performance of theproposed scheme

Notation Description

SK Number of secret keysPK Number of public keysTmul The time complexity for executing the modular multiplicationTadd The time complexity for executing the modular additionTexp The time complexity for executing the modular exponentiationTec−add The time complexity for executing the addition

of two elliptic curve pointsTec−mul The time complexity for executing the multiplication

on elliptic curve pointsTsqr The time for modular square computationTr The time complexity for selecting a random integerTh The time complexity for performing a one-way hash function h

The performance of our new signature scheme is summarized as follows: Thenumber of keys in this scheme is given by PK = 4, and SK = 3. The signer needsTec−mul+6Tmul+6Tsqr+Texp+Tr+Th time complexity to create a signature on anymessage, m.The signature validation verifier needs 3Tec−mul + 3Tsqr + Tmul + Th.Finally the communication costs and the parameters for signing message andverifying signature are respectively given by 3|n| and 2|n|.


To describe the efficiency performance in terms of Tmul, we use the conversionproposed in [8]. It converts various operations units to the time complexity forexecuting the modular multiplication.

Texp ≈ 240Tmul; Tec−mul ≈ 29Tmul; Tec−add ≈ 0.12Tmul

Assuming the time complexity for Th and Tr are negligible, we found that, oursigning message requires 275Tmul + 6Tsqr time complexity and our verifying sig-nature requires 88Tmul + 3Tsqr time complexity. The widely used RSA signaturescheme needs 240Tmul in both signature message and in its verifying signature.clearly our new proposed scheme is more efficient than RSA and yet our schemeis based on two hard problems which offer a longer security than RSA.

5. Conclusion

In this paper, we presented a new computationally inexpensive digital signaturescheme based on the ECDLP and QRP problems. The new scheme offers higherlevel of security than other algorithms that based on ECDLP or QRP. The perfor-mance of the proposed method was evaluated through extensive experiments thatclearly showed a perfect resiliency against a wide range of attacks including thekey-only attack, ECDLP and QRP attacks, the chosen-message and feed attacks.Furthermore, it requires only minimal and acceptable number of operations inboth signing and verifying algorithm.

References

[1] Chung, Y.F., Huang, K.H., Lai, F., Chen, T.S., ID-based digital sig-nature scheme on elliptic curve cryptosystem, Computer Standards and In-terfaces, 29 (6) (2007), 601-604.

[2] Diffie, W., Hellman, M., New directions in cryptography, IEEE Trans.Info. Theory, IT (31) (1976), 441-445.

[3] ElGamal, T., A public key cryptosystem and a signature scheme based ondiscrete logarithms, IEEE Info. Theory, IT (31) (1985), 479-472.

[4] Ismail, S.E., Yahya, A.H., A new version of ElGamal signature scheme,Sains Malaysian, 35 (2) (2006), 69-72.

[5] Ismail, S.E., Yahya, A.H., Two new signature schemes based on the in-teractability of factoring a large composite number. In the Proceeding of theFirst International Conference on Quantitative Sciences and Its Applications,UUM, Kedah, 2005, 1-8.

[6] Johnson, D., Menezes, A., Vanston, S., The Elliptic Curve Digital Sig-nature Algorithm, International Journal of Information Security, 1 (1) (2001),36-63.


[7] Koblitz, N., Elliptic curve cryptosystems, Mathematics of Computation,48 (177) (1987), 203-209.

[8] Koblitz, N., Menezes, A., Vanstone, S., The state of elliptic curvecryptography, Design, Code Cryptography, 19 (2000), 173 -193.

[9] Liu, D., Huang, D., Dai, Y., Luo, P., New Schemes for Sharing Pointson an Elliptic Curve, Computers and Mathematics with Application, 56(2008), 1556-1561.

[10] Miller, V., Uses of elliptic curves in cryptography, Advances in cryptology-Crypto ’85, LNCS, 218 (1986), 417426.

[11] Nikooghadam, M., Bonyadi, R.M., Malekian, E., A. Zakerolhos-seini, A., A protocol for digital signature based on the elliptic curve discretelogarithm problem, Journal of Applied Sciences, 8 (10) (2008), 1919-1925.

[12] Popescu, C., An identification scheme based on the elliptic curve discretelogarithm problem, In the Proceedings of The 4th International Conferenceon High-Performing Computing in the Asia-Region, 2 (2000), 624-625.

[13] Rabah, K., Elliptic curve ElGamal encryption and signature schemes,Information Technology Journal, 13 (3) (2005), 299-309.

[14] Rivest, R., Shamir, A., Adleman, L., A method for obtaining digi-tal signature and public key cryptosystem, Communication of the ACM, 21(1978), 120-126.

[15] Tahat, N., Ismail, S.E., Ahmad, R.R., A new signature scheme basedon factoring and discrete logarithms, Journal of Mathematics and Statistics,4 (4) (2008), 222-225.



Motto:The science wouldn’t be so good today,if yesterday we hadn’t thought about today.

Grigore C. Moisil

ECCENTRICITY, SPACE BENDING, DIMENSION

Marian Nitu

DumbravitaStr. Kos Karoly Nr. 10, Cod Postal 307160Judetul TimisRomaniae-mail: [email protected]

Florentin Smarandache

University of New Mexico705 Gurley Ave. Gallup, NM 87301USAe-mail: [email protected]

Selariu Mircea

Str. HarnicieiNr.10, Ap.6, Cod Postal 300507TimisoaraRomaniae-mail: [email protected]

Abstract. The main goal of this paper is to present new transformations, previouslynon-existent in traditional mathematics, that we call centric mathematics (CM) butthat became possible due to the new born eccentric mathematics, and, implicitly, to thesupermathematics (SM).

As shown in this work, the new geometric transformations, namely conversion or

transfiguration, wipe the boundaries between discrete and continuous geometric forms,

showing that the first ones are also continuous, being just apparently discontinuous.

Abbreviations and annotations

C I Circular and Centric, E I Eccentric and Eccentrics,

F I Function, M I Mathematics,

CE I Circular Eccentric, F CE I FCE,

CM I Centric M, EM I Eccentric M,

SM I Super M, F CM I FCM,

F EM I FEM, F SM I FSM

134 m. nitu, fl. smarandache, m.e. selariu

Introduction: conversion or transfiguration

In linguistics a word is the fundamental unit to communicate a meaning. It canbe composed by one or more morphemes. Usually, a word is composed by a basicpart, named root, where one can attach affixes. To define some concepts and toexpress the domain where they are available, sometimes more words are needed:two, in our case. In this paper several new concepts are introduced and they arerelated to SuperMathematics (SM).

The principal new idea in this paper is that it introduces a new mathematicaltransformation with a large significance in the fields of Physics, previously non-existent in the original classical mathematics, named herein as centric mathemat-ics (CM). They became possible thanks to this new mathematics called EccentricMathematics (EM) and to the Super Mathematics (SM), which are puts togetherwith (CM) with (EM). The (CM) is now a particular case of a linear numericeccentricity for s = 1 in (SM).

Supermathematical conversion

The concept is the easiest and methodical idea which reflects a finite of one ormore series of attributes, where these attributes are essentials.

The concept is a minimal coherent and usable information, relative to anobject, action, property or a defined event.

According to the Explicatory Dictionary, the conversion is, among manyother definitions/meanings, defined as “changing the nature of an object”.

Next, we will talk about this thing, about transforming/changing/converting,previously impossible in the ordinary classic mathematics, now named alsoCENTRIC (CM), of some forms in others, and that became possible due to thenew born mathematics, named ECCENTRIC (EM) and to the new built-in ma-thematical complements, named temporarily also SUPERMATHEMATICS (SM).

We talk about the conversion of a circle into a square, of a sphere into a cube,of a circle into a triangle, of a cone into a pyramid, of a cylinder into a prism, ofa circular torus in section and shape into a square torus in section and/or form,etc. (Fig.1).

Supermathematical Conversion (SMC) is an internal pry for the mathematicaldictionary enrichment, which consists in building-up of a new denomination, withone or more new terms (two in our case), by assimilating some words from thecurrent language in a specialized domain, as Mathematics, with the intention toname and adequate the new operations that became possible only due to thenew born eccentric mathematics, and implicity to supermathematics. Becausepreviously mentioned conversions could not be made until today in MC, but onlyin SM, we need to call them as SUPERMATHEMATICAL conversion (SMC).

eccentricity, space bending, dimension 135

In [14] the continuous transformation of a circle into a square was named alsoeccentric transformation, because, in that case, the linear numeric eccentricity svaries/grows from 0 to 1, being a slide from centric mathematics domain MC →s = 0 to the eccentric mathematics, ME(s = 0) → s ∈ (0, 1], where the circularform draws away more and more from the circular form until reaching a perfectsquare (s = ±1).

Eccentric transformation

s Î [0, 12]

s s

ParametricPlot[Evaluate[Table[

(1 - 0.08 ) Cos[ ] / Sqrt[1 - ( Sin[ ]) ],

(1 - 0.08 ) Sin[ ]/Sqrt[1 - ( Cos[ ]) ], , 0,1], , 0,2Pi]]

s t s t

s t s t s t

2

2


(1 - 0.05 ) Cos[ ]/Sqrt[1 - ( Sin[ ]) ] ,

(1 - 0.08 ) Sin[ ]/Sqrt[1 - ( Cos[ ]) ]

, 0,1], , 0,2Pi]]

s t s t

s t s t

s t

2

2

Figure 1: Conversion or transfiguration in 2D of a circle into a square and/or intoa rectangle IECCENTRIC TRANSFORMATION


In the same work, the reverse transformation, of a square into a circle, wasnamed as centering transformation. Same remarks are valid also for transforminga circle into a rectangle and a rectangle into a circle (Fig. 2). Most modernphysicists and mathematicians consider that the numbers represent the reality’slanguage. The truth is that the forms are those which generate all physical laws.

Centering transformation

Centering transformationParametricPlot[Evaluate[Table[

(1 + 0.08 ) Cos[ ] / Sqrt[1 - ( Sin[ ]) ],

, 0,1], , 0,2.05Pi]]

s t s t

s t

2

(1 + 0.08 ) Sin[ ]/Sqrt[1 - ( Cos[ ])] ,s t s t2


, 0,1], , 0,2.05Pi]]

(1 + 0.08 ) Cos[ ]/Sqrt[1 - ( Sin[ ]) ],

(1 + 0.05 ) Sin[ ]/Sqrt[1 - ( Cos[ ]) ],

s t s t

s t s t

2

2

s t

Figure 2: Conversion or transfiguration in 2D of a square and/or a rectangle intoa circle ICENTERING TRANSFORMATION


s = 0 ? 0,4 ? 0,7 ? ? å = 0

ParametricPlot3D[Cos[u]Cos[v], Sin[u]Cos[v],

Sin[v], u, 0,2Pi, v, / , / ]-Pi Pi

2 2

CSM

S C

ParametricPlot3D[Cos[ ] Cos[ ] (Sqrt[1 - (Sin[ ])^2]Sqrt[1 - (Sin[ ])^2])

Sin[ ] Cos[ ]/(Sqrt[1 - (Cos[ ]) ]Sqrt[1 - (Sin[ ]) ]),Sin[ ]/Sqrt[1 - (Cos[ ])^2], , 0,2Pi, ,-Pi, Pi

t u t

u

t u t u

u u t u

2 2

Figure 3: The conversion of a sphere into a cube

s = 0 ? 0,4 ? 0,7 ? 1 ? å = 0

ParametricPlot3D[ Sin[ ], Cos[ ],2 , , 0,2Pi, , 0,1]

v u v u v

u v

CSM

S C

ParametricPlot3D[ Sin[ ]/Sqrt[1 - (0.98Cos[ ]) ],v Cos[ ]/Sqrt[1 - (0.98Sin[ ])^2] , 2 , , 0,2Pi, , 0,1]

v u u

u u v u v

2

Figure 4: The Conversion of a cone into a pyramid


Look what the famous Romanian physicist Prof. Dr. Liviu Sofonea in Repre-sentative Geometries and Physical Theories, Ed. Dacia, Cluj-Napoca, p. 24, in1984, in the chapter named Mathematical geometry and physical geometry wrote:In the centric mathematical geometry one does what can be done, how can be done,with what can be done, and in supermathematical geometry we can do what mustbe done, with what must be done, as we will proceed. In the supermathematicalgeometry, between the elements of the ’CM scaffold’, one can introduce as manyother constructive elements as we want, which will give an infinitely denser scaffoldstructure, much more durable and, consequently, higher, able to offer an unseenhigh level and an extremely deep description and gravity.

s = 0 ? 0,4 ? 0,7 ? 1 ? å = 0

ParametricPlot3D[Sin[ ], Cos[ ],0.5 ,

, 0,2Pi, , 0, Pi]

u u v

u v

ParametricPlot3D[Cos[ - ArcSin[0.98Sin[ ]]],

Cos[ - Pi/2 + ArcSin[0.98Sin[-Pi/2]]],2 ,

, 0,2Pi, , 0,1

u u

u v

u v

CSM

S C

Figure 5: The Conversion or transfiguration of a cylinder into a prism


s = 0 ? 0,4 ? 0,7 ? 1 ? å = 0

ParametricPlot3D[(3 + Cos[ ])Cos[ ],(3 + Cos[ ])Sin[ ], Sin[ ], , 0,2Pi, , 0,2Pi

v u

v u v u v

ParametricPlot3D[

(3 + Cos[ ]/Sqrt[1 - (Sin[ ]) ]) Cos[ ]/Sqrt[1 - (Sin[ ]) ],

, (3 + Cos[ ]/Sqrt[1 - (Sin[ ]) ]) Sin[ ]/Sqrt[1 - (Cos[ ]) ],Sin[ ]/Sqrt[1 - (Cos[ ])^2], , 0,2Pi, , 0,2Pi]

v v u u2 2

2 2v v u u

v v u v

Figure 6: The conversion or transfiguration of the circular thorus into a squarethorus, both in form and in section

The fundamental principles of the geometry are, according to their topologicaldimensions: the corps (3), the line (2), and the point (0).

The elementary principles of geometry are the point, the line, the space, thecurve, the plane, the geometrical figures (such as the segment, triangle, square,rectangle, rhombus, the polygons, the polyhedrons, etc. the arcs, circle, ellipse,hyperbola, the scroll, the helix, etc.) both in 2D and in 3D spaces.

With the fundamental geometrical elements are defined and built all the formsand geometrical structures of the objects:

• Discrete (discontinuous) statically forms, or directly starting from a finite(discrete) set of points statically bonded with lines and planes.

• Continuous (or dynamical, mechanical) forms, starting from a single pointand considering its motion, therefore the time, and obtaining in this waycontinuous forms of curves, as trajectories of points or curves, in the plane(2D) or in the space ( 3D)


Consequently, one has considered, and still is considering, the existence of twogeometries: the geometry of discontinuous, or discrete geometry, and the geometryof the continuum.

As both objects limited by plane surfaces (cube, pyramid, prism), apparentlydiscontinuous, as those limited by different kinds of continuous surfaces (sphere,cone, cylinder) can be described with the same parametric equations, the firstones for numerical eccentricity s = ±1 and the last ones for s = 0, it results thatin SM there exists only one geometry, the geometry of the continuum.

In other words, the SM erases the boundaries between continuous and dis-continuous, as SM erased the boundaries between linear and nonlinear, betweencentric and eccentric, between ideal/perfection and real, between circular andhyperbolic, between circular and elliptic, etc.

Between the values of numerical eccentricity of s = 0 and s = ±1, there existan infinity of values, and for each value, an infinity of geometrical objects, which,each of them has the right to a geometrical existence.

If the geometrical mathematical objects for s ∈ [0∨±1] belong to the centricordinary mathematics (CM) (circle→ square, sphere→ cube, cylinder → prism,etc.), those for s ∈ (0,±1) have forms, equations and denominations unknownin this centric mathematics ( CM). They belong to the new mathematics, theeccentric mathematics (EM) and, implicitly, to the supermathematics (SM), whichis a reunion of the two mathematics: centric and eccentric, that means SM = MC∪ ME.

Concluding remarks

The principal new idea in this paper is that it introduces a new mathematicaltransformation with a large significance in the fields of Physics, previously inex-istent in the original classical mathematics named here as centric mathematics(CM); and now they became possible thanks to this new mathematics, called Ec-centric Mathematics (EM), and to the Super Mathematics (SM), which are puttogether: (CM) with (EM). The (CM) is now a particular case of a linear numericeccentricity for s = 1 in (SM).

In this paper the authors prove that these new geometric transformations,named Conversions or Transfigures, eliminate the borders between the discreteand continuous forms, showing that the first ones are also continuous but onlyapparently continuous. They mean: the conversion of a circle in a square, of asphere in a cube, of a circle in a triangle, of a cone in a pyramid, of a cylinderin a prism, of a torus with circular section in a torus with a square section, etc.Also, they consider this eccentricity in the formation and deformation of the space.The authors claim that all of these transformations are possible because of theeccentricity considered as 4-th up to n-th dimension of the space to completethe usual accepted (x, y, z) dimension. This is the reason why they consider theeccentricity as a dimension of the formation or deformation of the space.


The extension of a three dimensional space to a n-dimensional space becamepossible if the linear constant eccentricity e and the angle eccentricity epsilonwhich are the polar coordinates of the eccentricity E (e, epsilon) became of oneor multiple variables considered eccentricities too.

References

[1] Smarandache, Fl., Selariu, M.E., Immediate calculation of some pois-son type integrals using supermathematics circular EcCENTRIC functions ,arXiv: 0706.4238v1 [math.GM]

[2] Selariu, M.E., Functii circulare excentrice, Com. I Conferinta Nationalade Vibratii ın Constructia de Masini, Timisoara, 1978, 101–108.

[3] Selariu, M.E., Functii circulare excentrice si extensia lor, Bul. St. Tehn.I.P. ”TV” Timisoara, Seria Mecanica, Tom 25 (39), Fasc. 1 (1980), 189–196.

[4] Selariu, M.E., The definition of the elliptic eccentric with fixed eccenter,A V-a Conf. Nat. Vibr. Constr. Masini, Timisoara, 1985, 175–182.

[5] Selariu, M.E., Elliptic eccentrics with mobile eccenter, A V-a Conf. Nat.Vibr. Constr. Masini, Timisoara, 1985, 183–188.

[6] Selariu, M.E., Circular eccentrics and hyperbolics eccentrics, Com. a V-aConf. Nat. V.C.M. Timisoara, 1985, 189-194.

[7] Selariu, M.E., Eccentric Lissajous figures, Ccom. a V-a Conf.Nat. V.C.M.Timisoara, 1985, 195-202.

[8] Selariu, M.E., Supermatematica, Com. VII Conf. Internat. Ing. Manag.Tehn., TEHNO’95 Timisoara, vol. 9 (1995), Matematica Aplicata, 41-64.

[9] Selariu, M.E., Smarandache stepped functions, Revista Scientia grande.

[10] Selariu, M.E., Supermatematica. Fundamente, Editura Politehnica, Timi-soara, 2007.

[11] Selariu, M.E., Supermatematica. Fundamente, Vol. I. Editia a 2-a, Edi-tura Politehnica, Timisoara, 2012.

[12] Selariu, M.E., Supermatematica. Fundamente, Vol. II. Editia a 2-a,Editura Politehnica, Timisoara, 2012.

[13] Selariu, M.E., Smarandache, Fl., Nitu, M., Cardinal functions andintegral functions, International Journal Of Geometry, vol. I, (2012), no. 1,5-14.


[14] Selariu, M.E., Spatiul matematicii centrice si spatiul matematicii excen-trice, www.cartiAZ.ro, the 6th page, 2012.

[15] Selariu, M.E., Matematica atomica. Metoda determinarii succesive acifrelor consecutive ale unui numar, www.cartiAZ.ro, the 4th page, 2012.



STRONGLY DUO AND DUO RIGHT S-ACTS

Mohammad Roueentan

Majid Ershad

Department of Mathematics

College of scienceShiraz UniversityShiraz 71454Irane-mails: [email protected], [email protected]

[email protected]

Abstract. New kinds of acts, namely strongly duo and duo acts over a monoid are

introduced and investigated. This leads to the study of the relation between these kinds

of acts and other classes of acts, such as injective, projective and multiplication. It is

shown that a projective act is duo if and only if it is a multiplication act. In addition

if S is commutative, then a cyclic S-act is strongly duo if and only if it is cyclic quasi-

injective.

Keywords: duo, strongly duo, right S-act.

2000 Mathematics Subject Classification: 20M30.


In this paper, S is a monoid and an S-act AS (or A) is a unitary right S-act. LetA be a right S-act and let B be a subact of A. We say that B is fully invariant iff(B) ⊆ B for every endomorphism f of A and A is called duo if every subact ofA is fully invariant. This notion generalizes the concept of right duo semigroups(semigroups for which every right ideal is two sided) see [1]. Also a right S-actA is called strongly duo if for every subact B of A the trace of B in A is equalto B, i.e., tr(B,A) =

∪f∈Hom(B,A)

f(B) = B (see the properties of trace in [4]

page 146). It is clear that every strongly duo act is duo. In this paper severalequivalent conditions to being strongly duo are given. For instance, it is shownthat a right S-act A is strongly duo if and only if it is duo and is quasi-injectiverelative to all inclusions from its cyclic subacts. Also we prove that a projectiveright S-act A is duo if and only if it is multiplication (i.e., every subact of A isof the from AI for some right ideal I of S). It is proved that if SS is strongly

144 m. roueentan, m. ershad

duo, then all right S-acts are torsion free (divisible) and S is left reversible. Toobtain a characterization for strongly duo acts, we need the following definition(see Definition 1.4.20 of [4]).

Definition 1.1 Suppose S is a monoid and A is a right S-act. For an elementa ∈ A we define the annihilator of a as ann(a):=(s, t) ∈ S × S : as = at.

Theorem 1.2 Let S be a monoid and let A be a right S-act. Then the followingare equivalent:

(i) A is strongly duo.

(ii) Every subact of A is strongly duo.

(iii) Every finitely generated subact of A is strongly duo.

(iv) If B,C are subacts of A and B is a homomorphic image of C, then B ⊆ C.

(v) If ann(a) ⊆ ann(b) for some a, b ∈ A, then b ∈ aS.

Proof. (i)−→(ii) and (ii)−→(iii) are clear.

(iii)−→(i) Suppose B is a subact of A and f : B −→ A is a homomorphism.Let b be an element of B and let C = bS ∪ f(b)S. If g = f |bS, then, clearly,f(b) = g(b) ∈ tr(bS, C). By assumption, C is strongly duo and so f(b) ∈tr(bS, C) = bS. It follows that tr(B,A) = B.

(i)−→(iv) If B,C are two subacts of A and f : C −→ B is an epimorphism,then B = Im(f) ⊆ tr(C,A) = C.

(iv)−→(v) Suppose ann(a) ⊆ ann(b) for some a, b ∈ A. Define f : aS −→ bSby f(as) = bs for every s ∈ S. Clearly, f is a well-defined epimorphism and sobS ⊆ aS by assumption.

(v)−→(i) Suppose B is a subact of A and f ∈ Hom(B,A). If b ∈ B, thenann(b) ⊆ ann(f(b)) and hence f(b) ∈ bS ⊆ B by assumption. Consequently,tr(B,A) = B.

The following corollary shows that, there are acts which are not strongly duo.

Corollary 1.3 Suppose S is a monoid and A is a strongly duo right S-act. If SS

is a subact of A, then AS = SS.

Proof. By part (ii) of the previous theorem, SS is strongly duo. Also every cyclicright S-act is a homomorphic image of SS. Thus the result holds by part (iv) ofTheorem 1.2.

Definition 1.4 A right S-act A is called completely ordered if for any two sub-acts B,C of A, either B ⊆ C or C ⊆ B.

strongly duo and duo right S-acts 145

Proposition 1.5 Suppose S is a monoid and A is a completely ordered right S-act. If A satisfies the descending chain condition on cyclic subacts, then A isstrongly duo. In particular if S is a finite principal right ideal monoid, then SS isstrongly duo.

Proof. For elements a, b ∈ A with condition ann(a) ⊆ ann(b), we show thatb ∈ aS. If b /∈ aS, then by assumption aS ⊆ bS and so a = bs for somes ∈ S. Since bsS ⊇ bs2S ⊇ ..., by the hypothesis there exists n ∈ N such that(bsn)S = (bsn+1)S and so bsn = bsn+1t for some t ∈ S. Thus asn−1 = asnt. Sinceann(a) ⊆ ann(b), bsn−1 = bsnt. Repeat this argument to obtain a = ast. Thusb = bst = (bs)t = at ∈ aS, which is a contradiction. Therefore, bS ⊆ aS and AS isstrongly duo by Theorem 1.2. In case that S is a principal right ideal monoid, SS

is completely ordered. Hence the finite condition on S implies that SS is stronglyduo.

Corollary 1.6 Suppose S is a monoid which satisfies the descending chain con-dition on principal right ideals. If A is a completely ordered right S-act, then Ais strongly duo.

Proof. We show that A satisfies the descending chain condition on cyclic subacts.If a, b ∈ A and bS ⊆ aS, then there exists s ∈ S such that bS = asS. Thus everydescending chain of cyclic subacts of A has the form, aS ⊇ as1S ⊇ as1s2S ⊇ ....Now consider the descending chain, S ⊇ s1S ⊇ s1s2S ⊇ .... By assumptionthere exists n ∈ N such that s1...snS = s1...sn+1S. It follows that A satisfies thedescending chain condition on cyclic subacts and by the previous proposition isstrongly duo.

The following proposition reveals some similarities between strongly duo actsand cohopfian modules.

Proposition 1.7 If S is a monoid and A is a strongly duo right S-act, then everymonomorphism f : A −→ A is an epimorphism.

Proof. Suppose f : A −→ A is a monomorphism and define g : f(A) −→ Aby g(f(a)) = a for every a ∈ A. Since f is a monomorphism, g is a well-definedhomomorphism and clearly g(f(A)) = A. Since A is strongly duo, g(f(A)) ⊆tr(f(A), A) = f(A) and so f(A) = A, that is f is an epimorphism.

Now, we study the properties of duo acts. As the following lemma shows agood source of duo acts is provided by multiplication acts. By [3] an S-act A iscalled multiplication if every subact of A is of the form AI, for some right ideal Iof S. Recall that a monoid S is said to be left reversible if every two right idealsof S have a nonempty intersection. Also for a right S-act A and subact B of A,the factor act A/B is defined as A/ρB, where ρB is the Rees congruence (see [4]).

Lemma 1.8 Over a monoid S the following statements hold:


(i) A right S-act A is duo if and only if for each endomorphism f of A and eachelement a of A, f(a) = at for some t ∈ S. In particular, if S is commutativeand A is a duo right S-act, then End(A) is a commutative monoid.

(ii) Let B ⊆ C be subacts of a right S-act A. If B and C/B are fully invariantsubacts of A and A/B respectively, then C is fully invariant in A.

(iii) If SS is duo, then for any two elements s, t ∈ S, st = tx for some x ∈ S.

(iv) SS is duo if and only if every right ideal of S is a two sided ideal.

(v) If SS is duo, then S is left reversible.

(vi) Every multiplication right S-act is duo.

Proof. (i) Note that if A is duo, then f(aS) ⊆ aS for all f ∈ End(A) and a ∈ A.In particular, if S is commutative and A is a duo right S-act, then the first partof (i) implies that End(A) is commutative.

(ii) Let f ∈ End(A) and define f : A/B −→ A/B by f(a) = f(a). As B isfully invariant in A, f is a well-defined homomorphism and f(C) = f(C/B) andso f(C) ⊆ C.

(iii) For every s ∈ S define λs : S −→ S by λs(t) = st for every t ∈ S.Thus by part (i), λs(t) ∈ tS for every t ∈ S and the result follows. (iv) and (v)are obvious by part (i). (vi) If B is a subact of a multiplication right S-act A,then B = AI for some right ideal I of S and so for every endomorphism f of A,f(B) = f(AI) = f(A)I ⊆ AI = B.

Corollary 1.9 Suppose T is a proper submonoid of a monoid S. Then ST theright T -act S is not duo.

Proof. Suppose s ∈ S\T and define f : ST −→ ST by f(x) = sx for every x ∈ S.If ST is duo, then by the previous lemma, s = f(1) = 1t for some t ∈ T and sos ∈ T , which is a contradiction.

Proposition 1.10 Suppose S is a monoid and A is a completely ordered right S-act. If SS is duo and A satisfies the ascending chain condition on cyclic subacts,then A is duo.

Proof. Let f : A −→ A be an endomporphism of A and let a1 ∈ A. Suppose forsome elements a2, a3, ... of A, f(a1) = a2 , f(a2) = a3,.... Since A is completelyordered, for every n ∈ N, either an ∈ an+1S or an+1 ∈ anS. If for every n ∈ N,anS ⊂ an+1S, then we have the ascending chain a1S ⊂ a2S ⊂ a3S... , which isa contradiction. Thus there exists the smallest n ∈ N such that an+1S ⊆ anS.Hence an−1 ∈ anS, an−2 ∈ an−1S, ..., a1 ∈ a2S and so a1 ∈ anS. If a1 = ans forsome s ∈ S, then f(a1) = f(an)s = an+1s. Since an+1 ∈ anS, an+1 = ant for somet ∈ S and so f(a1) = ants. By Lemma 1.8.(iii), ts = sx for some x ∈ S and hencef(a1) = ansx = a1x. Now the result follows by Lemma 1.8.(i).

Now, the behavior of duo acts under taking subacts and homomorphic imagesis considered.


Definition 1.11 Let S be a monoid. Then an S-act A is called (cyclic) quasi-injective if for every (cyclic) subact B of A and for every homomorphism f ∈Hom(B,A), there exists a homomorphism g ∈ Hom(A,A) which extends f , i.e.,g|B = f . Also A is called quasi-projective if for a given epimorphism g : A −→ Cand homomorphism f : A −→ C, there exists a homomorphism h : A −→ A suchthat gh = f .

Proposition 1.12 Suppose S is a monoid and A is a duo right S-act. Then thefollowing statements hold:

(i) If A is quasi-injective, then every subact of A is duo and quasi-injective.

(ii) If A is quasi-projective, then every homomorphic image of A is duo andevery Rees factor of A is quasi-projective.

Proof. (i) Suppose B is a subact of A and C is a subact of B. Let f be anendomorphism ofB. Since A is quasi-injective, f can be lifted to an endomorphismf of A. Thus f(C) = f(C), which is contained in C because A is duo. Also it iseasy to see that B is quasi-injective.

(ii) Let B be a homomorphic image of A and let g be an endomorphism of B.Suppose B = A

ρfor some right congruence ρ on A and C

ρis a subact of B. Since A

is quasi-projective, there exists an endomorphism g∗ of A such that π g∗ = g π,where π is the natural epimorphism. Since A is duo, g∗(C) ⊆ C and so g(C

ρ) ⊆ C

ρ.

Thus B is duo. By a similar proof as part (ii) of Lemma 1.8, we can concludethat every Rees factor of A is quasi-projective.

It is easy to see that every homomorphic image of any multiplication rightS-act is multiplication. Also by Proposition 1.12.(ii), if SS is duo, then everycyclic right S-act is duo. Thus, by Lemma 1.8, we have the following result.

Corollary 1.13 Over a monoid S the following statements are equivalent:

(i) SS is duo.

(ii) SS is multiplication.

(iii) Every cyclic right S-act is multiplication.

(iv) Every cyclic right S-act is duo.

Proposition 1.14 Suppose S is a monoid and A is a right S-act. Then thefollowing hold:

(i) If every countably generated subact of A is duo, then A is duo.

(ii) If A is duo and cyclic quasi-injective, then A is strongly duo.


Proof. (i) Suppose a ∈ A and f ∈ End(A). Let B = aS ∪ f(a)S ∪ f(f(a))S ∪ ....Clearly, B is a countably generated subact of A and g := f |B is an element ofEnd(B). Thus, by Lemma 1.8.(i) and by assumption, g(a) = as for some s ∈ S.Hence f(a) = g(a) = as for some s ∈ S and by Lemma 1.8.(i) A is duo.

(ii) Let B be a subact of A. Since A is cyclic quasi-injective, for every b ∈ Band every homomorphism f : B −→ A, there exists f : A −→ A such thatf(bS) = f(bS). Hence the duo condition on A implies that f(b) = f(b) ∈ bS ⊆ B,proving that A is strongly duo.

Note that, by Propositions 1.12.(i), 1.14.(ii) and Corollary 1.3, over everymonoid S there exist right S-acts which are not strongly duo (duo). To investigatethe relation between projective acts and duo acts we need the concept of the ”dualbasis” for acts. In [3], Khaksari et al generalized the concept of dual basis frommodules to acts over commutative monoids. But it is easy to see that the condition”commutativity” is not necessary for this result. Regarding this observation wehave the following theorem.

Theorem 1.15 Let S be a monoid and let A be a projective right S-act. Then Ais duo if and only if A is multiplication.

Proof. If A is multiplication, then by Lemma 1.8.(vi), A is duo. Conversely,suppose A is duo and B is a subact of A. Let A∗ = Hom(A, S). Since A isprojective, by Theorem 1 of [3], there exists a subset T = (xα : fα) : α ∈ Λof A × A∗ such that for every x ∈ A, x = xαfα(x), where (xα, fα) ∈ T . Let I bethe right ideal generated by the elements of the form fα(x) for x ∈ B and α ∈ Λ.We claim that B = AI. If x ∈ B ⊆ A, then x = xαfα(x) for some α ∈ Λ and(xα, fα) ∈ T and hence x ∈ AI. Now suppose x ∈ B, a ∈ A and α ∈ Λ. Thusλa fα ∈ End(A), where λa : S −→ A is defined by λa(s) = as for every s ∈ S.Consequently, afα(x) = λa(fα(x)) ∈ λa(fα(xS)) ⊆ xS because A is duo. HenceAI ⊆ B and so B = AI.

Corollary 1.16 Suppose S is an idempotent monoid (i.e., I2 = I for every rightideal I of S) and A is a projective right S-act. Then the following statements areequivalent:

(i) A is multiplication.

(ii) A is strongly duo.

(iii) A is duo.

Proof. (i)←→(iii) holds by the previous theorem.

(i)−→(ii) Suppose for some right ideal I of S, B = AI is a subact of A. Thenfor every f ∈ Hom(B,A), f(B) = f(AI) = f(AI2) = f(AI)I ⊆ AI = B and soA is strongly duo.

(ii)−→(iii) is clear.


Lemma 1.17 Suppose S is a monoid and SS is strongly duo. If I is a projectiveright ideal of S, then I2 = I.

Proof. By Theorem 1 of [3], Itr(I, S) = I for every projective right ideal I of S.Since SS is strongly duo, tr(I, S) = I and so I2 = I.

A monoid S is called right hereditary if all right ideals of S are projective.

Theorem 1.18 Suppose S is a commutative monoid. Then the following hold:

(i) A right S-act A is strongly duo if and only if it is duo and cyclic quasi-injective. In particular, a cyclic right S-act A is strongly duo if and only ifit is cyclic quasi-injective.

(ii) If E(S) is the injective envelope of the right S-act SS, then SS is injectiveif and only if E(S) is a duo right S-act.

(iii) If S is right hereditary, then SS is strongly duo if and only if I2 = I forevery right ideal I of S.

Proof. (i) By Proposition 1.14.(ii), every duo cyclic quasi-injective right S-act isstrongly duo. Now let A be a strongly duo right S-act. Clearly, A is duo. We showthat A is cyclic quasi-injective. For this, let f : aS −→ A be a homomorphismwhere a ∈ A. Then f(a) ∈ aS because A is strongly duo. Thus f(a) = asfor some s ∈ S. Define f : A −→ A by f(x) = xs for every x ∈ A. Since Sis commutative, f is a well-defined homomorphism which is an extension of f .Now the result follows by the definition of cyclic quasi-injectivity. Also over acommutative monoid S, every cyclic right S-act is duo. Thus a cyclic right S-actA is strongly duo if and only if it is cyclic quasi-injective.

(ii) If E(S) is a duo right S-act, then by Proposition 1.14.(ii), E(S) is stronglyduo and by Corollary 1.3, SS = E(S) and so SS is injective. The converse isobvious because S is commutative and so it is duo.

(iii) The necessity follows by Lemma 1.17. Conversely, by Theorem 2 of[3], every right ideal I of S is multiplication and so it is strongly duo by Corol-lary 1.16.

Remark. Note that, in the previous theorem, the condition ”cyclic quasi-injec-tivity” is necessary. Indeed, if S = (Z, .), then SS is clearly duo, but by Theorem1.2, SS is not strongly duo. In fact SS is not cyclic quasi-injective.

A right S-act A is called torsion free if for any a, b ∈ A and for any rightcancelable element c ∈ S the equality ac = bc implies a = b.

Proposition 1.19 Suppose S is a monoid and SS is strongly duo. Then thefollowing hold:

(i) Every left cancelable element of S is invertible.


(ii) All right S-acts are divisible (torsion free).

(iii) Every right cancelable element of S is invertible.

(iv) If S is left (right) cancelative, then S is a group.

Proof. (i) If c is a left cancelable element of S, then ann(c) ⊆ ann(s) for everys ∈ S. Thus by Theorem 1.2, s ∈ cS for every s ∈ S and therefore S = cS.Consequently cx = 1 for some x ∈ S. Also since SS is duo, by Lemma 1.8.(iii),1 = cx = xy for some y ∈ S, which implies c is invertible.

(ii) By (i) all right S-acts are torsion free. Also by (i) SS is divisible andhence all right S-acts are divisible by Proposition 3.2.2 of [4].

(iii) is clear by part (ii) and Theorem 4.6.1 of [4] and Lemma 1.8.(iii).

(iv) holds by parts (i) and (iii).

Proposition 1.20 Suppose S is a monoid and Ai : i ∈ I is a family of rightS-acts. Then the following hold:

(i) If∏i∈I

Ai

(⨿i∈I

Ai

)is duo, then for every i ∈ I, Ai is duo.

(ii) If∏i∈I

Ai

(⨿i∈I

Ai

)is strongly duo, then for every i ∈ I, Ai is strongly duo.

Proof. (i) Suppose A =∏i∈I

Ai is duo. For a fixed element j ∈ I, let fj ∈ End(Aj).

Define f : A −→ A, by f(aii∈I) = bii∈I , where

(1.1) bi =

ai, i = j;

fj(aj), i = j,

for every element aii∈I of A =∏i∈I

Ai. Clearly f ∈ End(A) and by Lemma

1.8.(i), f(aii∈I) = aii∈Is = aisi∈I for some s ∈ S. Thus, for every, aj ∈ Aj,fj(aj) = ajs, for some s ∈ S, and so Aj is duo by Lemma 1.8.(i). The proof for

A =⨿i∈I

Ai is straightforward and will be omitted.

(ii) is obvious by Theorem 1.2.

In general, if Ai : i ∈ I is a family of duo (strongly duo) right S-acts, then∏i∈I

Ai and⨿i∈I

Ai are not duo (strongly duo). See the following example.


Example 1.21

(i) If S is a monoid and θS is the one element right S-act, then clearly θS isduo (strongly duo). But we can easily see that θS ⊔ θS is not duo (stronglyduo).

(ii) Suppose S is a commutative monoid which is not a group and SS is prin-cipally weakly injective. By Theorem 1.18.(i), SS is strongly duo and soit is duo, but (S × S)S is not duo (strongly duo). To see this, definef : (S × S)S −→ (S × S)S by f(x, y) = (y, x), for every (x, y) ∈ S × S. It isclear that f is a homomorphism. Thus if (S × S)S is duo, then by Lemma1.8.(i), for every (x, y) ∈ S × S, f(x, y) = (x, y)s for some s ∈ S. It impliesS = xS = Sx for every x ∈ S, and hence S is a group, a contradiction.

2. Strongly duo and duo acts over monoids with a zero element

In this section, we study the behavior of duo and strongly duo acts under theformation of 0-decomposition of an act. We assume that S is a monoid with azero. If A is a right S-act, then A is called centered if A has a unique zero element.Let A be a centered right S-act with a zero element θA. By a 0-decomposition ofA we mean an expression A =

∪i∈I

Ai of A such that for every i, Ai is a centered

subact of A and for all distinct i, j ∈ I, Ai ∩ Aj = θA. In what follows, we will

use the notation A =0⨿

i∈IAi for this concept. By [2], over monoids with a zero

element, every centered right S-act has a 0-decomposition.

Proposition 2.1 Suppose S is a monoid and A is a right S-act. If A =0⨿

i∈IAi is

a 0-decomposition of A, then the following hold:

(i) For every i ∈ I, Ai is a fully invariant subact of A if and only ifHom(Ai, Ak) = 0 for all distinct i, k ∈ I.

(ii) If A is duo, then for every i ∈ I, Ai is duo.

Proof. (i) Suppose i ∈ I and Ai is a fully invariant subact of A. For k = iin I, let f ∈ Hom(Ai, Ak) and suppose jk : Ak −→ A is the inclusion mapand πi : A −→ Ai is the canonical projection. Clearly f = jk f πi is anendomorphism of A and so by assumption f(Ai) ⊆ Ai. Since f(Ai) = f(Ai) ⊆ Ak,f(Ai) ⊆ Ai ∩ Ak = 0 and hence f = 0. Conversely, suppose i ∈ I and for everyk = i in I,Hom(Ai, Ak) = 0. We show that Ai is a fully invariant subact ofA. For this let f ∈ End(A) and suppose ji : Ai −→ A is the inclusion mapand πi : A −→ Ai is the canonical projection. It is clear that for every k = iin I, πk f ji ∈ Hom(Ai, Ak) = 0, where πk : A −→ Ak is the canonicalprojection. Thus f(Ai) ⊆

∪k∈I

(πkfji)(Ai) ⊆ πifji(Ai) ⊆ Ai.


(ii) Suppose i ∈ I and fi ∈ End(Ai). If πi : A −→ Ai is the canonicalprojection and ji : Ai −→ A is the inclusion map, then f = ji fi πi ∈ End(A).Thus for every subact B of Ai, fi(B) = f(B) ⊆ B because A is duo.

Theorem 2.2 Suppose S is a monoid and A is a right S-act. If A =0⨿

i∈IAi is a

0-decomposition of A, then A is duo if and only if for every i ∈ I, Ai is duo andHom(Ai, Ak) = 0 for all distinct i, k ∈ I.

Proof. The necessity follows by the previous proposition. Conversely, supposeB is a subact of A and f ∈ End(A). Let πk : A −→ Ak denote the canonicalprojection and jk : Ak −→ A denote the inclusion map for every k ∈ I. Thusπk f jk(Ak∩B) ⊆ Ak∩B because πk f jk ∈ End(Ak) and Ak is duo for everyk ∈ I. Also by assumption, πk f ji(Ak ∩ B) = 0 for all distinct k, i ∈ I. SinceB =

∪k∈I

(Ak ∩B), f(B) ⊆∪k∈I

f(Ak ∩B) ⊆∪k∈I

πkfjk(Ak ∩B) ⊆∪k∈I

(Ak ∩B) ⊆ B,

i.e., B is fully invariant, which implies A is duo.

Theorem 2.3 Suppose S is a commutative monoid and A is a right S-act. If

A =0⨿

i∈IAi is a 0-decomposition of A, then A is strongly duo if and only if Ai

0⊔ Aj

is strongly duo for all distinct i, j ∈ I.

Proof. The necessity follows by Theorem 1.2. Conversely, suppose for every

i = j ∈ I, Ai

0⊔ Aj is strongly duo. Since Ai

0⊔ Aj is duo, by Proposition 2.1.(i),

Hom(Ai, Aj) = 0 for every i = j ∈ I. Also by Theorem 1.2, for every i ∈ I, Ai isstrongly duo and so is duo. Hence by Theorem 2.2, A is duo. Now we show that Ais cyclic quasi-injective. Suppose aS is a cyclic subact of A and f ∈ Hom(aS,A).

If a ∈ Ai and f(a) ∈ Aj for some i = j ∈ I, then a, f(a) ∈ Ai

0⊔ Aj. Since

ann(a) ⊆ ann(f(a)), by Theorem 1.2, f(a) ∈ aS ⊆ Ai and so f(a) ∈ Ai∩Aj = 0.Define f : A −→ A by f(x) = θA for every x ∈ A. Clearly f is an extension of fand so in this case, A is cyclic quasi-injective. Now suppose a, f(a) ∈ Ai for somei ∈ I. Since Ai is strongly duo, Ai is cyclic quasi-injective by Theorem 1.18.(i).Hence there exists f : Ai −→ Ai such that f(a) = f(a). Define g : A −→ A by

(2.1) g(x) =

f(x), x ∈ Ai;

θA, x /∈ Ai.

It is easy to see that g is a well-defined homomorphism and is an extension of f .It follows that A is cyclic quasi-injective and the proof is complete by Theorem1.18.(i).

Definition 2.4 Suppose S is a monoid and A is a right S-act. We say that Ais weakly duo if every non-zero subact of A contains a non-zero subact which isfully invariant in A.


Theorem 2.5 Let S be a monoid and A be a quasi-projective right S-act. Thefollowing are equivalent:

(i) A is a duo right S-act.

(ii) Every factor of A is weakly duo.

Proof. (i)−→ (ii) By Proposition 1.12.(ii), every factor of A is duo and so isweakly duo.

(ii)−→(i) Suppose B is a non-zero subact of A such that B is not fully in-variant. By the hypothesis, there exists a non-zero subact C ⊂ B which is fullyinvariant in A. Let D =

∪L : L is a fully invariant subact of A and L ⊂ B.

Clearly D is a fully invariant subact of A and D = B. Thus B/D is a non-zerosubact of A/D and so by assumption there exists a subact E ⊆ B such that E/Dis a non-zero fully invariant subact of A/D. Thus D ⊂ E ⊆ B. By Lemma1.8.(ii), E is fully invariant in A and by the choice of D we must have B = E, acontradiction.

By the previous theorem we give the following result.

Corollary 2.6 If S is a monoid, then SS is duo if and only if every cyclic rightS-act is weakly duo.

Theorem 2.7 If S is a monoid and A is a projective right S-act, then the fol-lowing are equivalent:

(i) A is duo.

(ii) A is multiplication.

(iii) Every factor of A is weakly duo.

Proof. The result follows by Theorems 1.15 and 2.5.

Recall that a subact B of an S-act A is called essential in A, if B ∩C = 0 foreach 0 = C ≤ A (see [5]).

Proposition 2.8 If S is a monoid and A is a duo right S-act, then for everymonomorphism f : A −→ A, f(A) is an essential subact of A.

Proof. Suppose f : A −→ A is a monomorphism and 0 = B is a subact of A suchthat B ∩ f(A) = 0. Since A is duo, f(B) ⊆ B, and so f(B) ⊆ B ∩ f(A). Thusf(B) = 0 and so B = 0, a contradiction.

Acknowledgment. The authors would like to thank the referee for providingvaluable comments and suggestions.


References

[1] Anjaneyulu, A., Structure and ideal theory of duo semigroups, SemigroupForum, 22 (1981), 257-276.

[2] Clifford, A.H., Preston, G.B., The Algebraic Theory of Semigroups,vols. 1 and 2, Amer. Math. Soc. Surveys, no. 7, 1961-1967.

[3] Khaksari, A., Moghimi, Gh., Jahanpanah Bavaryani, S., Dual basisprojective system, International Journal of Algebra, 5 (5) (2011), 251-253.

[4] Kilp, M., Knauer, U., Mikhalev, A.V., Monoids, Acts and Categories,With Application to Wreath Product, Berlin. New York, 2000.

[5] Song, G.T., Goldie’s theorem for semigroups, Semigroup Forum, 47 (1993),182-195.



ON AUTOMATIC BOUNDEDNESS OF LINEAR OPERATORSON CONVEX BORNOLOGICAL SPACES

Abdelaziz Tajmouati

Sidi Mohamed Ben Abdellah UniversityFaculty of Sciences Dhar El MarhazFezMoroccoe-mail: [email protected]

Abstract. In this paper, using the notion of separating space of a linear operator

defined on a bornological vector space introduced in [5], we give some useful criteria

to study the automatic boundedness of operators. In particular, we give necessary and

sufficient conditions in order that operators should be bounded (Theorem 3.1 and Theo-

rem 4.1).

Keywords and phrases: bornological vector space, separating space, Mackey conver-

gence, linear operator, automatic boundedness.

1. Introduction

In [4], Sinclair studied the necessary conditions for continuity of homomorphisms,derivations and pair of operators acting on a Banach space.

The aim of the present paper is to extend some of this results in case ofbornological vector space (bvs) and consequently obtains some techniques to an-swer the boundedness problem for linear operators.

We extend naturally the notion of separating space of some linear operator Sbetween (bvs) X and (bvs) Y (see Definition 3.1 below). The notion of separatingspace characterizes the continuity of linear operators

The nice properties of the notion of separating space is for an linear operatorT acting on (bvs) X then, T is bounded if, and only if, its separating space isreduce a zero (see Theorem 3.1 below).

In the following we introduce some techniques which are necessary to studythe boundedness of homomorphisms, derivations and pair of linear operators. Par-ticulary, we give the characterization of bounded operators acting in bornologicalquotient (see Theorem 4.1 below).

156 a. tajmouati

2. Preliminaries

Recall that a bornology on a set X is a family B of subset X such that B is acovering of X, hereditary under inclusion and stable under finite union. The pair(X,B) is called bornological set.

A subfamily B′of B is said to be base of bornology B, if every element of B

is contained in an element of B′.

Let X and Y be two bornological set, a map X of Y is called bounded if theimage of every bounded subset of X is bounded in Y .

A bornology B on K-vector space E is said to be vector bornology on E ifthe maps (x, y) 7→ x+ y and (λ, x) 7→ λ.x are bounded.

We called a bornological vector space(b.v.s) any pair (X,B) consisting of avector space E and a vector bornology B on E.

A vector bornology on a vector space is called a convex vector bornology if itis stable under the formation of convex hull.

A bornological vector space is said a convex bornological vector space (cbvs)if it bornology is convex.

A (b.v.s) space is called of type M1 if, it satisfies the countability conditionof Mackey:

For every sequence of bounded (Bk)k in E, there exists a sequence

of scalars (λk)k≥0 such that∞∪k=0

λkBk is bounded in E.

Observe that every Banach algebra is a multiplicative convex bornologicalcomplete algebra of type M1. Also, if E is unital topological algebra with conti-nuous inverse such that it is F -space, then E is a multiplicative convex bornolo-gical complete algebra of type M1.

A sequence (xn)n≥0 in bornological vector space (b.v.s) E is said Mackey-convergent to 0 (or converge bornological to 0) if there exists a bounded setB ⊂ E such that

∀ε > 0 ∃n0 ∈ N n ≥ n0 implies xn ∈ εB.

If E is (cbvs), then (xn)n≥0 is Mackey-convergent to 0 if there exists a boundeddisk B ⊂ E such that (xn)n≥0 ⊂ EB and (xn)n≥0 converges to 0 in EB, where(EB, pB) is the vector space spanned by B and endowed with the semi-norm pBgauge of B.

Let E be a (bvs), E is said separated if there is not a non-zero bounded linein E, equivalently, every sequence Mackey-convergent its limit is unique.

A (bvs) E is separated if, and only if, for every bounded disk B the space(EB, pB) is a normed space.

A setB in a (bvs) E is saidM -closed (or b-closed) if every sequence (xn)n≥0⊆BMackey convergent in E its limit belongs in B.

Let E a separated (bvs) and let F be a subspace of E, the bornologicalquotient space E/F is separated, if and only if, F is b-closed in E.

on automatic boundedness of linear operators ... 157

Let E be a (cbvs) and A is a disk in E. A is called a completant disk if thespace (EA, pA) spanned by A and semi-normed by the gauge of A is a Banachspace.

A (cbvs) E is called a complete convex bornological vector space if its borno-logy has a base consisting of completant disks.

Let E be a (bvs) and A ⊂ E, the bornological closure (briefly b-closure orM-closure) of A denoted A is the intersection of all bornological closed subsets ofE containing A. A (cbvs) E satisfies M -closed properties if for any subset A of Ewe have A = A(1) where A(1) is the set of limits in Mackey-sense of the sequencesbelonging in A.

For bornology with a nets see [1], [2]. Recall that every bornology of a (cbvs)having a countable base has a net.

Theorem 2.1 (bornological closed graph theorem) Let (E,B) be a complete (cbvs)and (E ′,B′

) be a (cbvs) such that B′has a net. Then, every linear map u : E → E ′

with a bornological closed graph in E × E′is bounded.

Theorem 2.2 (bornological isomorphism) Let E be a complete (cbvs) and let Fbe a (cbvs) where its bornology has a net. Then every bijective bounded linear mapu : F → E is a bornological isomorphism.

For details, see [1], [2], [3], or [6].

3. Separating space

Definition 3.1 [5] Let X and Y two (bvs), let T a linear map between X and Y .We called separating space of T the subset of Y denotes by σ(T )

σ(T ) = y ∈ Y/∃(xn)n ⊂ X : xn −→M 0 and T (xn) −→M y.

Proposition 3.1 Let X and Y two (cbvs) of type M1. Then, every separatingspace of linear map T : X −→ Y is a b-closed subspace in Y .

Proof. Evidently, σ(T ) is a subspace vector of Y . σ(T ) is b-closed, indeed.Let (yk)k≥0 a sequence in σ(T ) converging to y in Y . We prove that y ∈ σ(T ).

For every k ∈ N∗, yk ∈ σ(T ). Then, there is a sequence (xn,k)n ⊂ X such that

xn,k −→M 0 and T (xn,k) −→M y.

Since xn,k −→M 0, then there is a bounded disc Bk in X:

(xn,k)n ⊂ XBkand lim

n−→+∞pBk

(xn,k) = 0.

(Bk)k≥0 is a sequence of bounded disc in X which is of type M1, hence thereis (λk)k ⊂ R∗+ and there is circled bounded B such that Bk ⊂ λkB, thereforepB ≤ pBk

, for any k ∈ N. (B can be disked, if not we take the disked hull of B).

158 a. tajmouati

Consequently, there exists a bounded disked B in X such that

(xn,k)n,k ⊂ XB,

we have

∀k ≥ 1 ∃Nk ∈ N, n ≥ Nk =⇒ pB(xn,k) ≤1

k.

On other hand, (T (xn,k))n converges bornological to yk.Again, by the same argument, there exists a bounded disked B′ in Y such

that [T (xn,k)− yk]n ⊂ YB′ , we have

∀k ≥ 1 ∃N ′k ∈ N, n ≥ N ′

k =⇒ pB′ (T (xn,k)− yk) ≤1

k.

Consequently, there exists a sequence (zk)k ⊂ XB such that (T (zk)− yk)k ⊂ YB′

and, for every k ∈ N∗, we have

pB(zk) <1

kand pB′ (T (zk)− yk) <

1

k.

Since yk →M y, then there is a bounded disked C in Y such that

(yk − y)k≥1 ⊂ YC and pC(yk − y) <1

k.

Let D be the disked hull of B′ ∪ C. We have pD ≤ pB′ , pD ≤ pC , YB′ ⊂ YD

and YC ⊂ YD.On other hand, for every k ∈ N∗, we have

T (zk)− y = (T (zk)− y) + (yk − y).

This gives

(T (zk)− y)k≥1 ⊂ YD and limk−→+∞

pD(T (zk)− y) = 0.

We conclude that, y ∈ σ(T ).

Proposition 3.2 Let X and Y two (cbvs) and let T : X → Y be a linear map.Then, we have

i) σ(T ) = 0 if, and only if, the graph of T is b-closed.

ii) Let R and S be two linear operators of X into Y , if TR = ST , then

S(σ(T )) ⊂ σ(T ).

Proof. Let G(T ) the graph of T .

i) Suppose that σ(T ) = 0.


Let (xn, T (xn))n≥0 ⊂ G(T ) such that (xn, T (xn)) −→M (x, y) ∈ X × Y .Then, xn −→M x and T (xn) −→M y. Since, T (xn − x) = T (xn) − T (x), thenT (xn − x) −→M y − T (x).

Consequently, (y − T (x)) ∈ σ(T ). Hence, y = T (x).Conversely, suppose that G(T ) is b-closed, and let y ∈ σ(T ). There exists

(xn)n≥0 ⊂ X such that:

xn −→M 0 and T (xn) −→M y.

Since, (xn, T (xn))n≥0 ⊂ G(T ) and (xn, T (xn))n −→M (0, y). Then, (0, y) ∈ G(T ).Therefore, y = 0.

ii) Let y ∈ σ(T ). There exists (xn)n≥0 ⊂ X such that xn −→M 0 andT (xn) −→M y. R being bounded, then R(xn) −→M 0.

On other hand, TR(xn) = ST (xn) and S is bounded then TR(xn) −→M S(y).Consequently, S(y) ∈ σ(T ), i.e., S(σ(T )) ⊂ σ(T ).

Theorem 3.1 Let X be a complete (cbvs) and Y a (cbvs) such that its bornologyhas a net. Let T : X → Y be a linear map. Then T is bounded if, and only if,σ(T ) = 0.

Proof. Applique the b-closed graph theorem’s and Proposition 3.2.

4. Characterization of bounded operators

Proposition 4.1 Let X and Y two (cbvs) of type M1 and Z a separated (bvs).Suppose that X is complete and the bornology of Y has a net. Let S : X → Y belinear and R : Y → Z be bounded linear map. Then

i) RS is bounded if, and only if, R(σ(S)) = 0.

ii) [Rσ(S)](1) = σ(RS).

For the proof. we shall need the following lemma.

Lemma 4.1 Let X be a (bvs) and F be a vector subspace of E. Let φ : E −→E/F the canonical surjection. Then, For every sequence (xn)n≥0 ⊂ E such that(φ(xn))n≥0 bornological converges to 0 in E/F , there exists a sequence (yn)n≥0 ⊂ Fsuch that (xn − yn)n≥0 bornological converges to 0 in E.

Proof. Let (xn)n≥0 ⊂ E such that (φ(xn))n borbological converges to 0 in E/F .The exists increasing sequence (ϵn)n ⊂ R∗+ converging to 0 and a bounded disckedB in E such that

φ(xn) ∈ ϵnφ(B), ∀ n ∈ N.Let (zn)n ⊂ B such that φ(xn) = ϵnφ(zn) for every n. Then, (xn−ϵnzn)n ⊂ F.

We set yn = xn − ϵnzn, ∀ n ∈ N. Then

xn − yn = ϵnzn ∈ ϵnB, ∀ n ∈ N.

Therefore, (xn − yn)n bornological converges to 0.

160 a. tajmouati

Proof of Proposition 4.1

i) Suppose that RS is bounded.Let y ∈ σ(S). Then there exists (xn)n≥0 ⊂ X such that xn −→M 0 and

S(xn) −→M y.R being bounded, then RS(xn) −→M R(y).On other hand, by hypothesis RS is bounded then, RS(xn) −→M 0.Z is separated, thus R(y) = 0, i.e: Rσ(S) = 0.Conversely, suppose that R(σ(S)) = 0.Let Q : Y −→ Y/σ(S) the canonical quotient map.Consider R0 : Y/σ(S) −→ Z defined by: R0(y + σ(S)) = R(y).Clearly, R0 is well defining and we have R = R0Q. Therefore, R0QS = RS.Since R0 is bounded, it suffice to show that QS is bounded and for this we

prove that σ(QS) = 0 (Theorem 3.1).Let y + σ(S) ∈ σ(QS). Then there exists (xn)n≥0 ⊂ X such that

xn −→M 0 and QS(xn) −→M Q(y) = y + σ(S).

Then, Q(S(xn)− y) −→M 0 in Y/σ(S).By Lemma 4.1, there exists a sequence (yn)n≥0 ⊂ σ(S) such that:

S(xn)− y − yn −→M 0 in Y.

Thus, for every k ∈ N∗, there exists (xn,k)n≥0 ⊂ X such that:

xn,k −→M 0 and S(xn,k) −→M yk.

Now, as already showed in proposition 3.2, we conclude that there exists twobounded disked B and B′ in X and Y respectively and a sequence (zn)n≥0 ⊂ XB

such that(S(xn)− yn)n≥0 ⊂ YB′

and, for every k ∈ N∗, we have

pB(zk) <1

kand pB′(S(xz)− yk) <

1

k.

SinceS(xn − zn)− y = (S(xn)− y − yn) + (y − S(zn)),

the sequence (S(xn − zn))n≥0 bornological converges to y. Since (xn − zn)n≥0

bornological converges to 0. Then, y ∈ σ(S).Consequently, σ(QS) = 0.

ii) We shows that [Rσ(S)](1) = σ(RS).We have R(σ(S)) ⊂ σ(RS), indeed.Let y ∈ σ(S). There exists (xn)n≥0 ⊂ X such that

xn −→M y and S(xn) −→M y.


R being bounded, then R(S(xn)) −→M R(y). This gives, R(y) ∈ σ(RS).Hence, σ(RS) is b-closed (Proposition 3.2). Then

[Rσ(S)](1) ⊂ σ(RS).

Next, to show the converse inclusion, consider the canonical quotient map:

Q0 : Z −→ Z/[Rσ(S)](1)

such thatQ0(z) = z + [Rσ(S)](1).

Then, Q0 is bounded. Thus Q0R is bounded.On other hand, Q0[Rσ(S)] = 0 where 0 is the class of 0. Then, by Propo-

sition 3.2, Q0RS is bounded. Therefore,

Q0σ(RS) = 0.

Thusσ(RS) ⊂ [Rσ(S)](1).

The proof is complete.

Remark 1 In the conditions of Proposition 4.1, the subspace S−1[σ(S)] isb-closed in X.

Proof. S−1[σ(S)] = Ker(QS) = (QS)−1(0). Since σ(S) is b-closed and Y/σ(S)is separeted, then, S−1[σ(S)] is b-closed in X.

Theorem 4.1 Let X and Y two (cbvs) of type M1 and S : X → Y be linearmap. Suppose that X is complete and the bornology of Y has a net.

Let X0 and Y0 two subspaces b-closed of X and Y respectively such thatS(X0) ⊂ Y0.

Let S0 : X/X0 −→ Y/Y0 defined by:

S0(x+X0) = S(x) + Y0.

Then, S0 is bounded if, and only if, σ(S) ⊂ Y0.

Proof. Suppose that S0 is bounded. Let y ∈ σ(S). There exists (xn)n≥0 ⊂ Xsuch that : xn −→M 0 and S(xn) −→M y. Then

S0(xn +X0) = S(xn) + Y0 −→M y + Y0 and S0(xn +X0) −→M S0(X0).

Y0 being b-closed, then Y/Y0 is separated. Consequently, y + Y0 = S0(X0) ⊂ Y0.Thus, y ∈ Y0.

Conversely, suppose that σ(T ) ⊂ Y0. Consider the canonical quotient mapQ : Y −→ Y/Y0. Q is bounded by the definition of the quotient bornology, andwe have

Q(σ(S)) ⊂ Q(Y0) = 0.

162 a. tajmouati

Then, QS is bounded, (by Proposition 3.1). Since S(X0) ⊂ Y0, we have:

QS(X0) = 0.

On other hand,

S0(x+X0) = S(x) + Y0 = QS(x).

Then, S0 is bounded.

Remark 2 In the conditions of Proposition 4.2, we have

σ(S|X0) ⊂ Y0 ∩ σ(S),

where S/X0 is the restriction of S in X0.

Proposition 4.2 Let X and Y two (cbvs) of type M1 such that X is completeand the bornology of Y has a net. Let Z1, Z2, ..., Zn the complete (cbvs) and

T1, T2, ..., Tn the bounded linear maps of Zj into X such that X =n∑

j=1

TjZj. Let

S : X −→ Y be bounded linear map. Then, we have

[σ(ST1) + · · ·+ σ(STn)](1) = σ(S).

Proof. The proof is establish in two parts.1) Suppose that, for every j ∈ 1, 2, ..., n, STj is bounded.Let T : Z −→ X defined by

T (z1, ..., zn) = T1(z1) + · · ·+ Tn(zn)

such that

Z = Z1 ⊕ Z2 ⊕ . . .⊕ Zn.

Since for j = 1, ..., n, Tj is bounded, by definition the bornology of Z, T isbounded.

On other hand we have

ST (z) = S(T1(z1) + · · ·+ Tn(zn)) = ST1(z1) + · · ·+ STn(zn).

Then, ST is bounded.Now, we prove again that S is bounded:Let x ∈ X. Then x = T (z), where z ∈ Z. Thus, S(x) = ST (z).

Let B a bounded in X =n∑

j=1

TjZj. Then, there exists B1, B2, ..., Bn bounded

in Zj such that

B ⊂ T1B1 + T2B2 + · · ·+ TnBn

Therefore,

S(B) ⊂ ST1(B1) + ST2(B2) + · · ·+ STn(Bn).


Since STj is bounded, for every j ∈ 1, 2, ..., n,n∑

j=1

STj(Bj) is bounded in Y .

Thus, S(B) is bounded in Y . Consequently, S is bounded. Then, σ(S) = 0.Since, σ(STj) = 0, ∀j = 1, 2, ..., n. Thus :[

n∑j=1

σ(STj)

](1)

= σ(S)

2) General Case:We have:

σ(STj) ⊂ σ(S), ∀j = 1, 2, . . . , n.

Indeed, let y ∈ σ(STj). There is a sequence (znj )n≥0 ⊂ Zj such that

znj −→M 0 et STj(znj ) −→M y.

Since, by hypothesis, Tj are bounded, then

Tj(znj ) −→M 0 and S[Tj(z

nj )] −→M y.

Then, y ∈ σ(S), i.e., σ(STj) ⊂ σ(S). This gives

n∑j=1

σ(STj) ⊂ σ(S).

Since σ(S) is b-closed, we have[n∑

j=1

σ(STj)

](1)

⊆ σ(S).

Next, we shows the converse inclusion.

Let Q : Y −→ Y/W , where W =

[n∑

j=1

σ(STj)

](1)

. Then, Q is bounded.

On other hand, we have

Q[σ(STj)] ⊂ Q(W ) = 0.

Then,Q[σ(STj)] = 0.

By Proposition 3.2, we conclude that QSTj are bounded. Therefore, by thefirst case it results that QS is bounded. Then, Q[σ(S)] = 0, i.e: σ(S) ⊂ W.

Acknowledgement. The author would like to thank the referee for helpfulsuggestions and remarks.

164 a. tajmouati

References

[1] HOGBE-NLEND, H., Bornologies and functional analysis, Amsterdam1977.

[2] HOGBE-NLEND, H., Thorie des bornologies et applications, LectureNotes, 213, Springer-Verlag, 1971.

[3] POPA, N.,B-closed Graph Theorem, (CRAS) Paris-Ser, A-B 273 (1971),A 294-A 297.

[4] A.M. SINCLAIR.Automatic continuity of linear operators, Lecture notesseries, 21, Cambridge Univ. Press.

[5] TAJMOUATI, A., On Boundedness and continuity of Jordan, ordinary andquadratic produc in Alternative semi-prime algebras, Italian Journal of Pureand Applied Mathematics, 30 (2013), 269-278.

[6] WEALBROECK, L., Topological vector spaces and algebras. Springer Lec-ture Notes in Math., 230, 1971.

Accepted: 3.05.2013


STABILITY OF NONLINEARNABLA FRACTIONAL DIFFERENCEEQUATIONS USING FIXED POINT THEOREMS

J. Jagan Mohan

Department of MathematicsBirla Institute of Technology and Science PilaniHyderabad CampusHyderabad – 500078Andhra PradeshIndiaemail: [email protected]

N. Shobanadevi

Fluid Dynamics DivisionSchool of Advanced SciencesVellore Institute of Technology UniversityVellore – 632014Tamil NaduIndiaemail: [email protected]

G.V.S.R. Deekshitulu

Department of MathematicsJNTU KakinadaKakinada – 533003Andhra PradeshIndiaemail: [email protected]

Abstract. Difference equations are often used to analyze sampled data systems, in

which stability problems are considered to be important. This is evident from a large

number of research papers dedicated to it. However, stability results for nonlinear

nabla fractional difference equations are not yet reported. The present article discusses

stability of nonlinear nabla fractional difference equations of Riemann–Liouville and

Caputo type, using fixed point theory.

Keywords: fractional order, difference equation, zero solution, fixed point, stability.

AMS Subject Classification: 39A10, 39A99.

1. Introduction

Fractional calculus deals with the study of fractional order integrals and deriva-tives and their diverse applications [11]. The analogous theory for discrete frac-tional calculus was initiated and a series of papers continuing this research hasappeared [3], [4], [5], [6], [7], [8], [9], [10], [12], [13].

166 j. jagan mohan, n. shobanadevi, g.v.s.r. deekshitulu

Qualitative properties of fractional difference equations assume importancein the absence of closed form solutions. One such qualitative property, whichhas wide applications, is the stability of solutions. Motivated by the applica-tion of fixed point theory in the study of stability of differential equations, weuse Krasnoselskii fixed point theorem, Schauder fixed point theorem and discreteArzela–Ascoli theorem to discuss the stability of nonlinear nabla fractional differ-ence equations.

The present article is organized as follows. Section 2 contains basic definitionsand results concerning nabla discrete fractional calculus. In Sections 3 and 4, weestablish sufficient conditions for stability of nonlinear nabla fractional differenceequations of Riemann–Liouville and Caputo type. We illustrate our main resultsthrough few examples in Section 5.

2. Nabla discrete fractional calculus

Throughout the article, we consider the discrete time scale [2]

T = Na = a, a+ 1, a+ 2, ...,

where a ∈ R is fixed. For any function f : Na → R, the backward difference ornabla operator is defined as ∇f(t) = f(t)− f(t− 1) for t ∈ Na+1.

Definition 2.1. For any real numbers α and t, the α rising function is defined by

(2.1) tα =Γ(t+ α)

Γ(t), t ∈ R \ ......,−2,−1, 0, 0α = 0.

Lemma 2.1. For any real numbers a and b, the quotient expansion of two gammafunctions at infinity is given by

Γ(t+ a)

Γ(t+ b)= ta−b

[1 +O

(1t

)], |t| → ∞.

Regarding the rising factorial function we observe the following properties.

Lemma 2.2. Assume the following factorial functions are well defined. For anypositive real numbers t, α and β,

1. ∇tα = αtα−1.

2. tα(t+ α)β = tα+β.

3. tα+β =t∑

j=0

C(t, j)(t− j)αjβ.

4. If α < t ≤ r, then t−α ≥ r−α.

5. If α ≥ β, then t−α ≤ t−β.

stability of nonlinear nabla fractional difference equations ...167

6. If 0 < α < 1, then (t− αβ)αβ ≥[(t− β)β

]α.

7. If α > 1, then[(t+ β)−β

]α<

Γ(1 + αβ)

[Γ(1 + β)]α(t+ αβ)−αβ.

Proof. The proofs of (1), (2), (3) are straightforward. The proof of (4) followsfrom Eulers infinite product

Γ(t) =1

t

∞∏n=1

(1 + 1

n

)t(1 + t

n

) .Consider

t−α =Γ(t− α)

Γ(t)=

t

t− α

∞∏n=1

(1 +

1

n

)−α( n+ t

n+ t− α

)≥ r

r − α

∞∏n=1

(1 +

1

n

)−α( n+ r

n+ r − α

)=

Γ(r − α)

Γ(r)= r−α.

The proof of (5) follows from Lemma 2.1. Consider

t−α

t−β=

Γ(t− α)

Γ(t− β)= tβ−α

[1 +O

(1t

)]=

1

tα−β

[1 +O

(1t

)]≤ 1.

For the proofs of (6) and (7), we refer [4], [5].

Definition 2.2. (Nabla Fractional Sum [8], [13]) Let f : Na → R and α > 0 begiven. Then the αth-order nabla fractional sum of f is given by

(2.2) ∇−αa f(t) =

1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1 for t ∈ Na

where ρ(s) = s− 1. Also, we define the trivial sum by ∇−0a f(t) = f(t) for t ∈ Na.

Definition 2.3. (R-L Nabla Fractional Difference [8], [13]) Let f : Na → R, α > 0be given and let N ∈ N be chosen such that N − 1 < α ≤ N . Then, the αth-orderRiemann–Liouville nabla fractional difference of f is given by

(2.3) ∇αaf(t) = ∇N∇−(N−α)

a f(t) for t ∈ Na+N .

For α = 0, we get ∇0af(t) = f(t) for t ∈ Na.

Definition 2.4. (Caputo Nabla Fractional Difference [12]) Let f : Na → R, α > 0be given and let N ∈ N be chosen such that N − 1 < α ≤ N . Then, the αth-orderCaputo nabla fractional difference of f is given by

(2.4) ∇αa∗f(t) = ∇−(N−α)

a

[∇Nf(t)

]for t ∈ Na+N .

For α = 0, we set ∇0a∗f(t) = f(t) for t ∈ Na.


Theorem 2.3. [7], [13] (Power Rule) Let α > 0 and µ > −1. Then, for t ∈ Na,we have

1. ∇−αa (t− a)µ =

Γ(µ+ 1)

Γ(α+ µ+ 1)(t− a)α+µ.

2. ∇αa (t− a)µ =

Γ(µ+ 1)

Γ(µ− α + 1)(t− a)µ−α.

Let f(t, r) : Na ×R → R, u(t) : Na → R and 0 < α < 1. Consider a nonlinearnabla fractional difference equation of Riemann–Liouville type together with aninitial condition of the form

∇αa−1u(t) = f(t, u(t)), t ∈ Na+1,(2.5)

∇−(1−α)a−1 u(t)

∣∣∣t=a

= u(a) = u(0).(2.6)

From [13], u(t) is a solution of the initial value problem (2.5)–(2.6) if and only ifu(t) has the following representation

(2.7) u(t) =(t− a+ 1)α−1

Γ(α)u0 +

1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1f(s, u(s)), t ∈ Na.

If we consider a nonlinear nabla fractional difference equation of Caputo typetogether with an initial condition of the form

∇αa∗u(t) = f(t, u(t)), t ∈ Na+1,(2.8)

u(a) = u0.(2.9)

Then from [13], u(t) is a solution of the initial value problem (2.8)–(2.9) if andonly if u(t) has the following representation

(2.10) u(t) = u0 +1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1f(s, u(s)), t ∈ Na.

Definition 2.5. The solution u = φ(t) of the initial value problem (2.5)–(2.6) or(2.8)–(2.9) is said to be

1. stable, if given ϵ > 0 and t0 ≥ 0, there exists δ = δ(ϵ, t0) such that|u0 − φ(t0)| < δ ⇒ |u(t, x0, t0)− φ(t)| < ϵ for all t ≥ t0.

2. attractive, if there exists η = η(t0) such that |u0 − φ(t0)| < η impliesu(t, x0, t0) → φ(t) as t → ∞.

3. asymptotically stable if it is stable and attractive.

Definition 2.6. The space l∞a is the set of real sequences defined on the set ofpositive integers where any individual sequence is bounded with respect to theusual supremum norm. Clearly, l∞a is a Banach space under the supremum norm.


Definition 2.7. A set Ω of sequences in l∞a is uniformly Cauchy (or equi–Cauchy),if for every ϵ > 0, there exists a positive integer N such that |u(i)− u(j)| < ϵ,whenever i, j > N for any u = u(t) in Ω.

Theorem 2.4. (Discrete Arzela-Ascoli’s theorem) A bounded, uniformly Cauchysubset Ω of l∞a is relatively compact.

Theorem 2.5. (Krasnoselskii’s fixed point theorem) Let S be a nonempty, closed,convex and bounded subset of the Banach space X and let A : X → X andB : S → X be two operators such that

1. A is a contraction with constant L < 1,

2. B is continuous, BS resides in a compact subset of X,

3. [x = Ax+By, y ∈ S] =⇒ x ∈ S.

Then the operator equation Ax+Bx = x has a solution in S.

Theorem 2.6. (Schauder fixed point theorem) Let Ω be a closed, convex andnonempty subset of a Banach space X. Let T : Ω → Ω be a continuous mappingsuch that TΩ is a relatively compact subset of X. Then T has at least one fixedpoint in Ω. That is, there exists an x ∈ Ω such that Tx = x.

3. Riemann–Liouville type fractional difference equation

Let l∞a be the set of all real sequences u = u(t)∞t=a with norm ∥u∥ = supt∈Na

|u(t)|,

then l∞a is a Banach space. Define the operators

Pu(t) =(t− a+ 1)α−1

Γ(α)u0 +

1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1f(s, u(s)).(3.1)

Au(t) =(t− a+ 1)α−1

Γ(α)u0.(3.2)

Bu(t) =1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1f(s, u(s)).(3.3)

Lemma 3.1. Let the following condition be satisfied:

(I) There exist constants β1 ∈ (α, 1) and L1 ≥ 0 such that

(3.4) |f(t, u(t))| ≤ L1(t− a)−β1 , for t ∈ Na+1.

Define

(3.5) S1 = u(t) : |u(t)| ≤ (t− a)−γ1 for t ∈ Na+n1+1


where γ1 = (−1/2)(α− β1) and n1 ∈ N such that

(3.6)|u0|Γ(α)

(n1 − γ1)(1/2)(α+β1)−1 +

L1Γ(1− β1)

Γ(1 + α− β1)(n1 − γ1)

−γ1 ≤ 1.

Then the operator B is continuous and BS1 is a compact subset of R fort ∈ Na+n1+1.

Proof. Clearly, γ1 > 0. Using Lemma 2.1, for t ∈ Na+1,

(t− a)−γ1 =Γ(t− a− γ1)

Γ(t− a)= (t− a)−γ1

[1 +O

( 1

t− a

)].

Further, (t−a)−γ1→0 as t→∞. Then, there exists n1 ∈ N such that (t−a)−γ1 → 0for (t − a) > n1. Clearly γ1 < 1 − (1/2)(α + β1) and n1 − γ1 < t − a. Using

Lemma 2.2, we get (n1 − γ1)(1/2)(α+β1)−1 ≤ (n1 − γ1)

−γ1 ≤ (t − a)−γ1 → 0 and(n1 − γ1)

−γ1 ≤ (t − a)−γ1 → 0. Thus, we have the inequality (3.6) which impliesthat the set S1 exists. Now, we show that B maps S1 in S1. Clearly, S1 is a closed,bounded, and convex subset of R. Applying condition (I), Theorem 2.3, Lemma2.2 and (3.4), we have

|Bu(t)| ≤ 1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1|f(s, u(s))|

≤ 1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1L1(s− a)−β1

= L1∇−αa (t− a)−β1

=L1Γ(1− β1)

Γ(1 + α− β1)(t− a)α−β1

=L1Γ(1− β1)

Γ(1 + α− β1)(t− a− γ1)

−γ1(t− a)−γ1

≤ L1Γ(1− β1)

Γ(1 + α− β1)(n1 − γ1)

−γ1(t− a)−γ1

≤ (t− a)−γ1 ,

implies BS1 ⊂ S1 for t ∈ Na+n1+1. Next, we show that B is continuous on S1. Letϵ > 0 be given. Then, there exists T1 ∈ N and T1 ≥ n1 such that, for t ∈ Na+T1+1,

(3.7)L1Γ(1− β1)

Γ(1 + α− β1)(t− a)α−β1 <

ϵ

2.

Since (t− a) > T1 ≥ n1 and 0 < γ1 < β1 − α < 1, we have

(t− a)α−β1 ≤ (t− a)−γ1 → 0,


which implies the validity of (3.7). Let un be a sequence such that un → u. Fort ∈ a+ n1 + 1, a+ n1 + 2, ..., a+ T1, applying the continuity of f , we have

|Bun(t)−Bu(t)|

≤ 1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1|f(s, un(s))− f(s, u(s))|

≤[ 1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1][

maxs∈a+1,a+2,...,a+T1

|f(s, un(s))− f(s, u(s))|]

=(t− a)α

Γ(α + 1)

[max

s∈a+1,a+2,...,a+T1|f(s, un(s))− f(s, u(s))|

]→ 0 as n → ∞.

For t ∈ Na+T1+1,

|Bun(t)−Bu(t)| ≤ 1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1[|f(s, un(s))|+ |f(s, u(s))|]

≤ 2L1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1(s− a)−β1

= 2L1∇−αa (t− a)−β1 =

2L1Γ(1− β1)

Γ(1 + α− β1)(t− a)α−β1 < ϵ.

Thus, for all t ∈ Na+n1+1, |Bun(t)−Bu(t)| → 0 as n → ∞ implies B is continuous.Finally, we show that BS1 is relatively compact. Let t1, t2 ∈ Na+T1+1 such thatt2 > t1. Then, we have

|Bu(t1)−Bu(t2)|

=∣∣∣ 1

Γ(α)

t1∑s=a+1

(t1 − ρ(s))α−1f(s, u(s))− 1

Γ(α)

t2∑s=a+1

(t2 − ρ(s))α−1f(s, u(s))∣∣∣

≤ 1

Γ(α)

t1∑s=a+1

(t1 − ρ(s))α−1|f(s, u(s))|+ 1

Γ(α)

t2∑s=a+1

(t2 − ρ(s))α−1|f(s, u(s))|

≤ L1Γ(1− β1)

Γ(1 + α− β1)(t1 − a)α−β1 +

L1Γ(1− β1)

Γ(1 + α− β1)(t2 − a)α−β1 < ϵ.

Thus, Bu : u ∈ S1 is a bounded and uniformly Cauchy subset implies BS1 isrelatively compact. Hence the proof.

Lemma 3.2. Assume that condition (I) holds. Then a solution of (2.5) is in S1

for t ∈ Na+n1+1.

Proof. Clearly, A is a contraction mapping with the constant 0, implies condition(1) of Theorem 2.5 holds. Using Lemma 3.1, the operator B is continuous andBS1 is a compact subset of R, implies condition (2) of Theorem 2.5 holds. Also


Pu = Au + Bu and u is the solution of (2.5)–(2.6) if it is a fixed point of theoperator P . Now we prove condition (3) of Theorem 2.5. Let us suppose, for afixed v ∈ S1, u = Au+Bv. Applying condition (I) and (3.6), we have

(3.8)

|u(t)| ≤ |Au(t)|+ |Bv(t)|

≤ (t− a+ 1)α−1

Γ(α)|u0|+

1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1|f(s, v(s))|

≤ (t− a)α−1

Γ(α)|u0|+

L1Γ(1− β1)

Γ(1 + α− β1)(t1 − a)α−β1

=|u0|Γ(α)

(t− a)−γ1(t− a− γ1)1/2(α+β1)−1

+L1Γ(1− β1)

Γ(1 + α− β1)(t− a)−γ1(t− a− γ1)

−γ1

=[ |u0|Γ(α)

(n1 − γ1)1/2(α+β1)−1

+L1Γ(1− β1)

Γ(1 + α− β1)(n1 − γ1)

−γ1](t− a)−γ1

≤ (t− a)−γ1 ,

for t ∈ Na+n1+1. Thus, u ∈ S1. According to Theorem 2.5, u is a fixed point of Pimplies u is the solution of (2.5)–(2.6). Hence the proof.

Theorem 3.3. Assume that condition (I) holds, then the zero solution of (2.5) isattractive.

Proof. Using Lemma 3.1, the solutions of (2.5) exist in S1 and tend to 0 ast → ∞. Hence the proof.

Theorem 3.4. Assume that the following condition is satisfied:

(II) There exist constants β2 ∈ (α, 1) and L2 ≥ 0 such that

(3.9) |f(t, u(t))− f(t, v(t))| ≤ L2(t− a)−β2|u− v|, for t ∈ Na+1.

Then, the solutions of (2.5) are stable provided that

(3.10) c = L2Γ(1− β2) < 1.

Proof. Let u(t) be the solution of (2.5)–(2.6), and let u(t) be the solution of (2.5)

with the initial condition u(0) = u0. Applying condition (II), we have


|u(t)− u(t)| ≤ (t− a+ 1)α−1

Γ(α)|u0 − u0|

+ 1Γ(α)

t∑s=a+1

(t− ρ(s))α−1|f(s, u(s))− f(s, u(s))|

≤ (t− a+ 1)α−1

Γ(α)|u0 − u0|+

L2

Γ(α)

t∑s=a+1

(t− ρ(s))α−1(s− a)−β2 |u− u|

=(t− a+ 1)α−1

Γ(α)|u0 − u0|+ L2∇−α

a (t− a)−β2 |u− u|

=(t− a+ 1)α−1

Γ(α)|u0 − u0|+

L2Γ(1− β2)

Γ(1 + α− β2)(t− a)α−β2 |u− u|

≤ (2)α−1

Γ(α)|u0 − u0|+

L2Γ(1− β2)

Γ(1 + α− β2)(1)α−β2 |u− u|

= α|u0 − u0|+ L2Γ(1− β2)|u− u|< α|u0 − u0|+ c|u− u|,

which implies that

(3.11) |u− u| < α

1− c|u− u0|.

For any given ε > 0, let δ = (1−c)α

ϵ. Then, |u− u0| < δ implies |u− u| < ε, whichshows that the solutions of (2.5) are stable. This completes the proof.

Theorem 3.5. Assume that conditions (I) and (II) hold, then the zero solutionof (2.5) is asymptotically stable provided that (3.10) holds.


(III) There exist constants β3 ∈(α, 1+α

2

), γ2 =

12(1−α) and L3 ≥ 0 such that

(3.12) |f(t, u(t))| ≤ L3(t− a+ γ2)−β3|u(t)| for t ∈ Na+1.

Then, the zero solution of (2.5) is attractive.

Proof. Set S2 = u(t) : |u(t)| ≤ (t − a)−γ2 for t ∈ Na+n2+1, where n2 ∈ N suchthat

(3.13)|u0|Γ(α)

(n2 − γ2)−γ2 +

L3Γ(1− β3 − γ2)

Γ(1 + α− β3 − γ2)(n2 − γ2)

α−β3 ≤ 1.

Clearly β3 − α > 0. Using Lemma 2.1, for t ∈ Na+1,

(t− a)α−β3 =Γ(t− a+ α− β3)

Γ(t− a)= (t− a)−(β3−α)

[1 +O

( 1

t− a

)].

Further, (t − a)−γ2 → 0 as t → ∞. Then, there exists n2 ∈ N such that

(t − a)α−β3 → 0 for (t − a) > n2. Clearly, β3 − α < γ2 and n2 − γ2 < t − a.


Using Lemma 2.2, we get (n2 − γ2)−γ2 ≤ (n2 − γ2)

α−β3 ≤ (t − a)α−β3 → 0 and

(n2−γ2)α−β3 ≤ (t−a)α−β3 → 0. Thus, we have the inequality (3.13) which implies

that the set S2 exists. Now, we prove condition (3) of Theorem 2.5. Suppose, fora fixed v ∈ S2 and for all u ∈ R, u = Au+Bv. Applying condition (III), we have

|u(t)| ≤ |Au(t)|+ |Bv(t)|

≤ |u0|Γ(α)

(t− a+ 1)α−1 +1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1|f(s, v(s))|

≤ |u0|Γ(α)

(t− a+ 1)α−1 +1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1L3(s− a+ γ2)−β3 |v(s)|

≤ |u0|Γ(α)

(t− a+ 1)α−1 +L3

Γ(α)

t∑s=a+1

(t− ρ(s))α−1(s− a+ γ2)−β3(s− a)−γ2

=|u0|Γ(α)

(t− a+ 1)α−1 +L3

Γ(α)

t∑s=a+1

(t− ρ(s))α−1(s− a)−β3−γ2

≤ |u0|Γ(α)

(t− a)α−1 +L3Γ(1− β3 − γ2)

Γ(1 + α− β3 − γ2)(t− a)α−β3−γ2

=|u0|Γ(α)

(t− a)−γ2−γ2 +L3Γ(1− β3 − γ2)

Γ(1 + α− β3 − γ2)(t− a)α−β3−γ2

=|u0|Γ(α)

(t− a)−γ2(t− a− γ2)−γ2

+L3Γ(1− β3 − γ2)

Γ(1 + α− β3 − γ2)(t− a)−γ2(t− a− γ2)

α−β3

=[ |u0|Γ(α)

(t− a− γ2)−γ2 +

L3Γ(1− β3 − γ2)

Γ(1 + α− β3 − γ2)(t− a− γ2)

α−β3

](t− a)−γ2

≤[ |u0|Γ(α)

(n2 − γ2)−γ2 +

L3Γ(1− β3 − γ2)

Γ(1 + α− β3 − γ2)(n2 − γ2)

α−β3

](t− a)−γ2

≤ (t− a)−γ2

for t ∈ Na+n2+1. Thus, condition (3) of Theorem 2.5 holds. The proof of condition(2) of Theorem 2.5 is similar to that of Lemma 3.1, and we omit it. Therefore, byTheorem 2.5, P has a fixed point u ∈ S2 implies u is the solution of (2.5)–(2.6).Moreover, all functions in S2 tend to 0 as t → ∞, then the zero solution of (2.5)tends to zero as t → ∞, which shows that the zero solution of (2.5) is attractive.This completes the proof.

Theorem 3.7. Assume that conditions (II) and (III) hold, then the zero solutionof (2.5) is asymptotically stable provided that (3.10) holds.


(IV) There exist constants η ∈ (0, 1), β4 ∈(α, 2+αη

2+η

)and L4 ≥ 0 such that

(3.14) |f(t, u(t))| ≤ L4(t− a+ 1)−β4

∣∣∣u(t+ β4 − α

2

)∣∣∣η for t ∈ Na+1.


Then, the zero solution of (2.5) is attractive.

Proof. Set

(3.15) S3 = u(t) : |u(t)| ≤ (t− a)−γ3 for t ∈ Na+n3+1

where γ3 = (1/2)(β4 − α) and n3 ∈ N satisfies that

(3.16)|u0|Γ(α)

(n3 − γ3)β4−γ3−1 +

L4Γ(1− β4 − γ3η)

Γ(1 + α− β4 − γ3η)(n3 − γ3)

−γ3 ≤ 1

Clearly γ3 > 0. Using Lemma 2.1, for t ∈ Na+1,

(t− a)−γ3 =Γ(t− a− γ3)

Γ(t− a)= (t− a)−γ3

[1 +O

( 1

t− a

)].

Further, (t − a)−γ3 → 0 as t → ∞. Then, there exists n3 ∈ N such that(t − a)−γ3 → 0 for (t − a) > n3. Clearly γ3 < 1 + γ3 − β4 and n3 − γ3 < t − a.

Using Lemma 2.2, we get (n3 − γ3)β4−γ3−1 ≤ (n3 − γ3)

−γ3 ≤ (t − a)−γ3 → 0 and(n3 − γ3)

−γ3 ≤ (t − a)−γ3 → 0. Thus, we have the inequality (3.16) which im-plies that the set S3 exists. Now, we prove condition (3) of Theorem 2.5 only,and the remaining part of the proof is similar to that of Theorem 3.6. Since

η ∈ (0, 1), β4 ∈(α, 2+αη

2+η

)and γ3 = (1/2)(β4 − α), then γ3, γ3η, α + γ3 ∈ (0, 1)

and β4+γ3η ∈ (α, 1). Suppose, for a fixed v ∈ S3 and for all u ∈ R, u = Au+Bv.Applying condition (IV), Lemma 2.2 and (3.16), we have

|u(t)| ≤ |Au(t)|+ |Bv(t)|

≤ |u0|Γ(α)

(t− a+ 1)α−1 +1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1L4(s− a+ 1)−β4

∣∣∣v(s+ β4 − α

2

)∣∣∣η≤ |u0|

Γ(α)(t− a+ 1)α−1 +

L4

Γ(α)

t∑s=a+1

(t− ρ(s))α−1(s− a+ γ3η)−β4

((s− a+ γ3)

−γ3)η

≤ |u0|Γ(α)

(t− a+ 1)α−1 +L4

Γ(α)

t∑s=a+1

(t− ρ(s))α−1(s− a+ γ3η)−β4(s− a+ γ3η)

−γ3η

≤ |u0|Γ(α)

(t− a)α−1 +L4

Γ(α)

t∑s=a+1

(t− ρ(s))α−1(s− a+ γ3η)−β4−γ3η

≤ |u0|Γ(α)

(t− a)α−1 +L4

Γ(α)

t∑s=a+1

(t− ρ(s))α−1(s− a)−β4−γ3η

=|u0|Γ(α)

(t− a)β4−γ3−γ3−1 +L4Γ(1− β4 − γ3η)

Γ(1 + α− β4 − γ3η)(t− a)α−β4−γ3η

≤ |u0|Γ(α)

(t− a)β4−γ3−γ3−1 +L4Γ(1− β4 − γ3η)

Γ(1 + α− β4 − γ3η)(t− a)α−β4


=|u0|Γ(α)

(t− a)β4−γ3−γ3−1 +L4Γ(1− β4 − γ3η)

Γ(1 + α− β4 − γ3η)(t− a)−γ3−γ3

=|u0|Γ(α)

(t− a)−γ3(t−a−γ3)β4−γ3−1+

L4Γ(1−β4−γ3η)

Γ(1+α−β4−γ3η)(t−a)−γ3(t−a−γ3)

−γ3

=[ |u0|Γ(α)

(t− a− γ3)β4−γ3−1 +

L4Γ(1− β4 − γ3η)

Γ(1 + α− β4 − γ3η)(t− a− γ3)

−γ3](t− a)−γ3

≤[ |u0|Γ(α)

(n3 − γ3)β4−γ3−1 +

L4Γ(1− β4 − γ3η)

Γ(1 + α− β4 − γ3η)(n3 − γ3)

−γ3](t− a)−γ3

≤ (t− a)−γ3

for t ∈ Na+n3+1. Thus, condition (3) of Theorem 2.5 holds. Hence the proof.

4. Caputo type fractional difference equation

Let l∞a be the set of all real sequences u = u(t)∞t=a with norm ∥u∥ = supt∈Na

|u(t)|,

then l∞a is a Banach space. Define the operator

(4.1) Tu(t) = u0 +1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1f(s, u(s)),

Obviously, u(t) is a solution of (2.8)–(2.9) if it is a fixed point of the operator T .

Lemma 4.1. Assume that the following condition is satisfied:

(V) there exist constants γ4, L5 > 0 such that

(4.2)∣∣∣u0 +

1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1f(s, u(s))∣∣∣ ≤ L5(t− a)−γ4 for t ∈ Na+1.

Then there exists a solution for (2.8).

Proof. Define the set

(4.3) S4 = u(t) : |u(t)| ≤ L5(t− a)−γ4 for t ∈ Na+1.

Clearly, S4 is a closed, bounded and convex subset of R. Using Lemma 2.1, wehave

(t− a)−γ4 =Γ(t− a− γ4)

Γ(t− a)= (t− a)−γ4

[1 +O

( 1

t− a

)]for t ∈ Na+1. Then, (t − a)−γ4 → 0 as t → ∞. To prove that T has a fixedpoint, first we show that T maps S4 in S4. For t ∈ Na+1, from condition (V ),we have |Tu(t)| ≤ L5(t − a)−γ4 implies TS4 ⊂ S4. Next, we show that T iscontinuous on S4. Let ϵ > 0 be given. Then there exists a N1 ∈ Na+1 such that

t > N1 and L5(t − a)−γ4 <ϵ

2. Let un be a sequence such that un → u. For

t ∈ a+ 1, a+ 2, a+ 3, ..., N1, applying the continuity of f , we have


|Tun(t)− Tu(t)| ≤ 1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1|f(s, un(s))− f(s, u(s))|

≤[ 1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1]

maxs∈a+1,a+2,a+3,...,N1

|f(s, un(s))− f(s, u(s))|

=Γ(t− a+ α)

Γ(α + 1)Γ(t− a)max

s∈a+1,a+2,a+3,...,N1|f(s, un(s))− f(s, u(s))| → 0

as n → ∞.

For t ∈ N1 + 1, N1 + 2, ..., we have

|Tun(t)− Tu(t)| ≤∣∣∣ 1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1(f(s, un(s))− f(s, u(s))∣∣∣

≤ 2L5(t− a)−γ4 < ϵ.

Thus, for all t ∈ Na+1, |Tun(t) − Tu(t)| → 0 as n → ∞, which means that T iscontinuous. Finally, we show that TS4 is relatively compact. Let t1, t2 ∈ Na+1

and t2 > t1 ≥ N1, we have

|Tu(t2)− Tu(t1)| ≤∣∣∣ 1

Γ(α)

t2∑s=a+1

(t2 − ρ(s))α−1f(s, u(s))∣∣∣

+∣∣∣ 1

Γ(α)

t1∑s=a+1

(t1 − ρ(s))α−1f(s, u(s))∣∣∣(4.4)

≤ L5(t2 − a)−γ1 + L5(t1 − a)−γ1 < ϵ.

Therefore, Tu : u ∈ S4 is a bounded and uniformly Cauchy subset. Hence, byTheorem 2.4, TS4 is relatively compact. According to Theorem 2.6, T has a fixedpoint in S4 which is the solution of (2.8) - (2.9). Hence the proof.

Theorem 4.2. Assume that condition (V) holds. Then the zero solution of (2.8)is attractive.

Proof. By Lemma 4.1, the solutions of (2.8) exist and are in S4. But all functionsin S4 tend to 0 as t → ∞. Then, the zero solution of (2.8) tend to zero as t → ∞.This completes the proof.


(VI) there exist constants γ5 ∈ (α, 1) and L6 > 0 such that

(4.5) |f(t, u(t))− f(t, v(t))| ≤ L6(t− a)−γ5 |u− v| for t ∈ Na+1.

Then, the solutions of (2.8) are stable provided that

(4.6) c = L6Γ(1− γ5) < 1.


Proof. Let u(t) be the solution of (2.8)–(2.9), and let u(t) be the solution of (2.8)

satisfying the initial condition u(0) = u0. Applying condition (VI), we have

|u(t)− u(t)| ≤ |u0 − u0|+1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1|f(s, u(s))− f(s, u(s))|

≤ |u0 − u0|+L6

Γ(α)

t∑s=a+1

(t− ρ(s))α−1(s− a)−γ5 |u− u|

= |u0 − u0|+ L6∇−αa (t− a)−γ5 |u− u|

= |u0 − u0|+L6Γ(1− γ5)

Γ(1 + α− γ5)(t− a)α−γ5 |u− u|

≤ |u0 − u0|+ L6Γ(1− γ5)|u− u|

< |u0 − u0|+ c|u− u|

which implies that

(4.7) |u− u| < α

1− c|u− u0|.

For any given ε > 0, let δ = (1−c)α

ϵ. Then, |u− u0| < δ implies |u− u| < ε, whichshows that the solutions of (2.8) are stable. This completes the proof.

Theorem 4.4. Assume that conditions (V) and (VI) hold. Then, the zero solutionof (2.8) is asymptotically stable provided that (4.6) holds.


(VII) there exist constants γ6 ∈ (α, 1) and L7 > 0 such that

(4.8)∣∣∣ (t− a)α

Γ(1− α)u0 + f(t, u(t))

∣∣∣ ≤ L7(t− a)−γ6 for t ∈ Na+1.

Then, there exists a solution for (2.8).

Proof. Define the set

(4.9) S5 =

u(t) : |u(t)| ≤ L7Γ(1− γ6)

Γ(1 + α− γ6)(t− a)α−γ6

for t ∈ Na+1.

From the definition, clearly, S5 is a closed, bounded and convex subset of R. First,we show that T maps S5 in S5. From condition (VII), we have


|Tu(t)| =

∣∣∣∣∣u0 +1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1f(s, u(s))

∣∣∣∣∣=

∣∣∣∣∣ 1

Γ(α)

t∑s=a+1

(t−ρ(s))α−1 (s−a)α

Γ(1−α)u0+

1

Γ(α)

t∑s=a+1

(t−ρ(s))α−1f(s, u(s))

∣∣∣∣∣=

∣∣∣∣∣ 1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1[ (s− a)α

Γ(1− α)u0 + f(s, u(s))

]∣∣∣∣∣≤ L7

Γ(α)

t∑s=a+1

(t− ρ(s))α−1(s− a)−γ6

= L7∇−αa (t− a)−γ6

=L7Γ(1− γ6)

Γ(1 + α− γ6)(t− a)α−γ6

for t ∈ Na+1 implies TS5 ⊂ S5. The remaining proof of TS5 to be relativelycompact is similar to that of Lemma 4.1, and we omit it. By Theorem 2.6, T hasa fixed point in S5 which is the solution of (2.8)–(2.9). Hence the proof.

Theorem 4.6. Assume that condition (VII) holds. Then the zero solution of (2.8)is attractive.

Theorem 4.7. Assume that conditions (VI) and (VII) hold. Then the zero solu-tion of (2.8) is asymptotically stable provided that (4.6) holds.


(VIII) there exist constants β5 >1

1− α,

α

β5 − 1< γ7 <

1

β5

and L8 > 0,

such that

(4.10)∣∣∣ (t− a)α

Γ(1− α)u0 + f(t, u(t))

∣∣∣ ≤ L8|u(t+ γ7)|β5 for t ∈ Na+1.

Then, there exists a solution for (2.8) provided that

(4.11)L8Γ(1− β5γ7)

Γ(1 + α− β5γ7)

Γ(1 + β5γ7)

[Γ(1 + γ7)]β5≤ 1.

Proof. From β5 >1

1− α, we have that

α

β5 − 1<

1

β5

which implies that γ7 exists.

In addition, γ7 <1

β5

means that Γ(1 − β5γ7) > 0 and Γ(1 + α − β5γ7) > 0,

α

β5 − 1< γ7 implies that α− β5γ7 < −γ7. Define the set

(4.12) S6 =u(t) : |u(t)| ≤ (t− a)−γ7 for t ∈ Na+1

.


We show that T maps S6 in S6. Applying condition (VIII), Lemma 2.2 and (4.11),we have

|Tu(t)| =

∣∣∣∣∣u0 +1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1f(s, u(s))

∣∣∣∣∣=

∣∣∣∣∣ 1

Γ(α)

t∑s=a+1

(t− ρ(s))α−1[ (s− a)α

Γ(1− α)u0 + f(s, u(s))

]∣∣∣∣∣≤ L8

Γ(α)

t∑s=a+1

(t− ρ(s))α−1|u(s+ γ7)|β5

≤ L8

Γ(α)

t∑s=a+1

(t− ρ(s))α−1[(t− a+ γ7)−γ7 ]β5

≤ L8

Γ(α)

Γ(1 + β5γ7)

[Γ(1 + γ7)]β5

t∑s=a+1

(t− ρ(s))α−1(t− a+ β5γ7)−β5γ7

≤ L8

Γ(α)

Γ(1 + β5γ7)

[Γ(1 + γ7)]β5

t∑s=a+1

(t− ρ(s))α−1(t− a)−β5γ7

= L8Γ(1 + β5γ7)

[Γ(1 + γ7)]β5∇−α

a (t− a)−β5γ7

=L8Γ(1− β5γ7)

Γ(1 + α− β5γ7)

Γ(1 + β5γ7)

[Γ(1 + γ7)]β5(t− a)α−β5γ7

≤ L8Γ(1− β5γ7)

Γ(1 + α− β5γ7)

Γ(1 + β5γ7)

[Γ(1 + γ7)]β5(t− a)−γ7 ≤ (t− a)−γ7

for t ∈ Na+1 implies TS6 ⊂ S6. The remaining part of the proof is similar to thatof Lemma 4.1, so we omit it.

Theorem 4.9. Assume that the condition (VIII) and (4.11) hold, then the zerosolution of (2.8) is attractive.

Theorem 4.10. Assume that the condition (VI) and (VIII) hold, then the zerosolution of (2.8) is asymptotically stable provided that (4.6) and (4.11) hold.

5. Examples

Example 5.1.

(5.1) ∇0.5−1u(t) = (0.3)t−0.25 sin(u(t)), ∇−0.5

−1 u(t)∣∣∣t=0

= u(0), t ∈ N1.


Here, f(t, u(t)) = (0.3)t−0.25 sin(u(t)), t ∈ N1. Clearly, |f(t, u(t))| ≤ (0.3)t−0.25

implies condition (I) holds. Further, |f(t, u(t)) − f(t, v(t))| ≤ (0.3)t−0.25|u − v|implies condition (II) is satisfied. Here L2 = 0.3 and β2 = 0.25 and we have

c = L2Γ(1 − β2) = (0.3)Γ(0.75) ≈ (0.3)(1.2254) < 1, implies that the inequality

(3.10) holds. Hence, the solution of (5.1) is asymptotically stable by Theorem 3.5.

Example 5.2.

(5.2) ∇0.5−1u(t) = (0.1)(t+ 1)t−0.6u(t), ∇−0.5

−1 u(t)∣∣∣t=0

= u(0), t ∈ N1.

Here f(t, u(t)) = (0.1)(t+1)−0.6u(t), t ∈ N1, where β3 = 0.6 and α = 0.5. Clearly,

β3 ∈(α,

1 + α

2

), γ2 =

1

2(1− α) = 0.25 and L3 = 0.1 such that

|f(t, u(t))| = |(0.1)(t+ 1)−0.6u(t)| ≤ (0.1)(t+ 0.25)−0.6|u(t)|

which implies that condition (III) is satisfied. Further,

|f(t, u(t))− f(t, v(t))| ≤ (0.1)(t+ 1)−0.6|u− v| ≤ (0.1)(t)−0.6|u− v|,

implies that condition (II) is satisfied. Here, L2 = 0.1, β2 = 0.6 and α = 0.5.Then, we have c = L2Γ(1 − β2) = (0.1)Γ(0.4) ≈ (0.1)(2.2181) < 1, implies thatthe inequality (3.10) holds. Hence, the solution of (5.2) is asymptotically stableby Theorem 3.7.

Example 5.3.

(5.3) ∇0.5−1u(t) = (0.2)(t+1.5)−0.6[u(t+0.05)]0.3, ∇−0.5

−1 u(t)∣∣∣t=0

= u(0), t ∈ N1.

Here, f(t, u(t)) = (0.2)(t + 1.5)−0.6[u(t + 0.05)]0.3, where β4 = 0.6, η = 0.3,

L4 = 0.2 and α = 0.5. Clearly, η ∈ (0, 1), β4 ∈(α,

2 + αη

2 + η

)such that

|f(t, u(t))| = |(0.2)(t+ 1.5)−0.6[u(t+ 0.05)]0.3| ≤ (0.2)(t+ 1)−0.6|u(t+ 0.05)|0.3

which implies condition (IV) is satisfied. Hence, the zero solution of (5.3) isattractive by Theorem 3.8.

Example 5.4.

(5.4) ∇0.50∗ u(t) = (0.2)t−0.75sin(u(t)), u(0) = 0, t ∈ N1.

Using (2.10), we get

u(t) =(0.2)

Γ(0.5)

t∑s=1

(t− ρ(s))−0.5s−0.75 sin(u(s)).


Now, consider∣∣∣ (0.2)Γ(0.5)

t∑s=1

(t− ρ(s))−0.5s−0.75 sin(u(s))∣∣∣ ≤ (0.2)

Γ(0.5)

t∑s=1

(t− ρ(s))−0.5s−0.75 ≤ t−0.25

implies that (V) holds. Further, |f(t, u(t))− f(t, v(t))| ≤ (0.2)t−0.75|u− v| implies

condition (VI) is satisfied. Here, L6 = 0.2, γ5 = 0.75 and we have c = L6Γ(1−γ5) =0.2Γ(0.25) ≈ (0.2)(3.6256) < 1, implies (4.6) holds. Hence, the zero solution of(5.4) is asymptotically stable by Theorem 4.4.

Example 5.5.

(5.5) ∇0.50∗ u(t) = (0.2)t−0.75 sin(u(t))− u0

t0.5

Γ(0.5), u(0) = u0, t ∈ N1.


u(t) = u0 +1

Γ(0.5)

t∑s=1

(t− ρ(s))−0.5[(0.2)s−0.75 sin(u(s))− u0

s0.5

Γ(0.5)

].

Now, consider∣∣∣u0t0.5

Γ(0.5)+ f(t, u(t))

∣∣∣ ≤ (0.2)t−0.75| sin(u(s))| ≤ (0.2)t−0.75 ≤ t−0.75

implies that (VII) holds. Further, |f(t, u(t))− f(t, v(t))| ≤ 0.2t−0.75|u− v| impliescondition (VI) is satisfied. Here, L6 = 0.2, γ5 = 0.75 and we have c = L6Γ(1−γ5) =0.2Γ(0.25) ≈ (0.2)(3.6256) < 1, implies (4.6) holds. Hence, the zero solution of(5.5) is asymptotically stable by Theorem 4.7.

Example 5.6.

(5.6) ∇0.10∗ u(t) = (0.5)[u(t+ 0.2)]2 − u0

t0.5

Γ(0.5), u(0) = u0, t ∈ N1.


u(t) =1

Γ(0.5)

t∑s=1

(t− ρ(s))−0.5[(0.5)[u(s+ 0.2)]2 − u0

s0.5

Γ(0.5)

].

Now, consider∣∣∣u0t0.5

Γ(0.5)+ f(t, u(t))

∣∣∣ = ∣∣∣(0.5)[u(t+ 0.2)]2∣∣∣ ≤ (0.5)|u(t+ 0.2)|2

implies that (VIII) holds. Here, L8 = 0.5, β5 = 2 and γ7 = 0.2 and we have

L8Γ(1− β5γ7)

Γ(1 + α− β5γ7)

Γ(1 + β5γ7)

[Γ(1 + γ7)]β5=

(0.5)Γ(0.6)

Γ(0.7)

Γ(1.4)

[Γ(1.2)]2≈ 0.6039 < 1,


implies (4.11) is satisfied. Hence, the zero solution of (5.6) is attractive by Theo-rem 4.9.

Acknowledgements. For this work, the author J. Jagan Mohan was supportedby the University Grants Commission Start Up Grant, Government of India.

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Accepted: 6.05.2013


EXP-FUNCTION METHOD USING MODIFIED RIEMANN–LIOUVILLE DERIVATIVE FOR SINGULARLY PERTURBEDBOUSSINESQ EQUATIONS OF FRACTIONAL-ORDER

Qazi Mahmood Ul Hassan

Syed Tauseef Mohyud-Din1

Department of MathematicsFaculty of SciencesHITEC University Taxila CanttPakistan

Abstract. This paper witnesses the combination of an efficient transformation and

Exp-function method to construct generalized solitary wave solutions of the nonlinear

singularly perturbed sixth-order Boussinesq equations of fractional-order. Computa-

tional work and subsequent numerical results re-confirm the efficiency of proposed al-

gorithm. It is observed that suggested scheme is highly reliable and may be extended

to other nonlinear differential equations of fractional order.

Keywords: perturbed sixth-order Boussinesq equations, fractional calculus, exp-

function method, modified Riemann-Liouville derivative.

1. Introduction

The subject of factional calculus [1], [2] is a rapidly growing field of research, atthe interface between chaos, probability, differential equations, and mathematicalphysics. In recent years, nonlinear fractional differential equations (NFDEs) havegained much interest due to exact description of nonlinear phenomena of manyreal-time problems. The fractional calculus is also considered as a novel topic [3],[4]; has gained considerable popularity and importance during the recent past.It has been the subject of specialized conferences, workshops and treatises orso, mainly due to its demonstrated applications in numerous seemingly diverseand widespread fields of science and engineering. Some of the areas of present-day applications of fractional models [5]–[8] include fluid flow, solute transportor dynamical processes in self-similar and porous structures, diffusive transportakin to diffusion, material viscoelastic theory, electromagnetic theory, dynamicsof earthquakes, control theory of dynamical systems, optics and signal processing,bio-sciences, economics, geology, astrophysics, probability and statistics, chemical

1Corresponding author. E-mail: [email protected]

186 q.m. ul hassan, s.t. mohyud-din

physics, and so on. As a consequence, there has been an intensive developmentof the theory of fractional differential equations, see [1]–[8] and the referencestherein. Recently, He and Wu [9] developed a very efficient technique which iscalled exp-function method for solving various nonlinear physical problems. Thethrough study of literature reveals that Exp-function method has been appliedon a wide range of differential equations and is highly reliable. The exp-functionmethod has been extremely useful for diversified nonlinear problems of physicalnature and has the potential to cope with the versatility of the complex nonlineari-ties of the problems. The subsequent works have shown the complete reliabilityand efficiency of this algorithm. He et.al. [10]–[11] used this scheme to find pe-riodic solutions of evolution equations; Mohyud-Din [12-13] extended the samefor nonlinear physical problems including higher-order BVPs; Oziz [14] tried thisnovel approach for Fisher’s equation; Wu et.al. [15], [16] for the extension ofsolitary, periodic and compacton-like solutions; Yusufoglu [17] for MBBN equa-tions, Zhang [18] for high-dimensional nonlinear evolutions; Zhu [19], [20] for theHybrid-Lattice system and discrete m KdV lattice; Kudryashov [21] for exact soli-ton solutions of the generalized evolution equation of wave dynamics; Momani [22]for an explicit and numerical solutions of the fractional KdV equation; The basicmotivation of this paper is the development of an efficient combination comprisingan efficient transformation, exp-function method using Jumarie’s derivative ap-proach [23]–[26] and its subsequent application to construct generalized solitarywave solutions of the nonlinear singularly perturbed sixth-order Boussinesq equa-tions of fractional-order [27]–[28]. It is to be highlighted that Ebaid [29] provedthat c = d and p = q are the only relations that can be obtained by applyingexp-function method to any nonlinear ordinary differential equation.

Theorem 1 [29] Suppose that u(r) and(u(γ))λ

are respectively the highest orderlinear term and the highest order nonlinear term of a nonlinear ODE, where r andγ are both positive integers. Then the balancing procedure using the Exp-function

ansatz; U (η) =

∑dn=−c an exp (nη)∑qm=−p bm exp (mη)

, leads to c = d and p = q,∀r, s, λ ≥ 1 .

Theorem 2 [29] Suppose that u(r) and u(s)uk are respectively the highest orderlinear term and the highest order nonlinear term of a nonlinear ODE, where r, sand Ω are all positive integers. Then the balancing procedure using the Exp-function ansatz leads to c = d andp = q,∀r, s, k ≥ 1.

Theorem 3 [29] Suppose that u(r) and(u(s))Ω are respectively the highest orderlinear term and the highest order nonlinear term of a nonlinear ODE, wherer, sand Ω are all positive integers. Then the balancing procedure using the Exp-function ansatz leads to c = d andp = q,∀r, s ≥ 1, ∀Ω ≥ 2.

Theorem 4 [29] Suppose that u(r) and (u(s))Ωuλ are respectively the highest or-der linear term and the highest order nonlinear term of a nonlinear ODE, wherer, s,Ω and λ are all positive integers. Then the balancing procedure using theExp-function ansatz leads to c = d andp = q, ∀r, s,Ω, λ ≥ 1.

exp-function method using modified riemann-liouville derivative ...187

2. Jumarie’s fractional derivative

Jumarie’s fractional derivative is a mo1dified Riemann-Liouville derivative definedas [27]–[30]

(1) Dαt f (x) =

1

Γ (−α)

∫ x

0

(x− t)−α−1 (f (t)− f (0)) dt, α ≤ 0,

1

Γ (−α)

d

dx

∫ x

0

(x− t)−α (f (t)− f (0)) dt, 0 ≤ α ≤ 1

[fα−n (x)n]n, n ≤ α ≤ n+ 1, n ≥ 1

where f : R → R, x → f(x) denotes a continuous (but not necessarily differen-tiable) function.Some useful formulas and results of Jumarie’s modified Riemann–Liouville derivative were summarized in [27]–[28].

Dαxc = 0, α ≥ 0, c = constant(2)

Dαx [cf (x)] = cDα

xf (x)α ≥ 0, c = constant(3)

Dαxx

β =Γ (1 + β)

Γ (1 + β − α)xβ−α, β ≥ α ≥ 0,(4)

Dαx [f (x) g (x)] = [Dα

xf (x) g (x) + f (x) [Dαx g (x)] .(5)

Dαxf (x (t)) = f

′

x (x) .xα (t) .(6)

3. Exp-function method [28]–[31]

We consider the general nonlinear FPDE of the type

(7) P (u, ut, ux, uxx uxxx, ..., Dαt u,D

αxu,D

αxxu, ...) = 0, 0 < α ≤ 1,

where Dαt u,D

αxu,D

αxxu are the modified Riemann-Liouville derivative of u with

respect to t, x, xx, respectively.Using a transformation [32]

(8) η = kx+ωtα

Γ (1 + α)+ η0, k, ω, η0 are all constants with k, ω = 0

we can rewrite equation (7) in the following nonlinear ODE;

(9) Q(u, u′, u′′, u′′′, uiv) = 0,

where the prime denotes derivative with respect to η.According to the Exp-function method, we assume that the wave solution can

be expressed in the following form

(10) u (η) =

∑dn−c an exp [nη]∑qm−p bm exp [mη]


where p, q, c and d are positive integers which are known to be further determined,an and bm are unknown constants. We can rewrite equation (4) in the followingequivalent form

(11) u (η) =ac exp (cη) + · · ·+ a−d exp (−dη)

bp exp (pη) + · · ·+ b−q exp (−qη).

This equivalent formulation plays an important and fundamental part for findingthe analytic solution of problems. To determine the value of q and p by using [25],

(12) p = c, q = d

4. Solution procedure

In this section, we apply the exp-function method to construct generalized soli-tary solutions for Burger’s Equations of fractional-order. Numerical results arevery encouraging. In this section, we apply exp-function method to constructgeneralized solitary of the singularly perturbed sixth-order Boussinesq equations.Numerical results are very encouraging.

Example 4.1 Consider the singularly perturbed sixth-order Boussineques equa-tion where a and k are arbitrary constants.

utt = uxx + (p(u))xx + βuxxxx + δuxxxxxx

Taking β = 1, δ = 1 and p (u) = 3u2, the model equation is given as

(13) utt = uxx + 3(u2)xx + uxxxx

with the initial conditions

u (x, 0) =2ak2ekx

(1 + aekx), ut (x, 0) =

2ak3√1 + k2

(1− aekx

)ekx

(1 + aekx)3

Using (8), equation (13) can be converted to an ordinary differential equation

(14) (ω2 + k2)u′′ + 6k2(u′2 + u′′u)uu′′′ + k4u(iv) = 0,

where the prime denotes the derivative with respect to η. The solution of theequation (13) can be expressed in the form, equation (11). To determine thevalue of c andp, by using [25],

(15) p = c, q = d

Case 4.1.1. We can freely choose the values of c and d, but we will illustrate thatthe final solution does not strongly depend upon the choice of values of candd.For simplicity, we set p = c = 1 and q = d = 1 equation (11) reduces to

(16) u (η) =a1 exp [η] + a0 + a−1 exp [−η]

b1 exp [η] + a0 + b−1 exp [−η].


Substituting equation (16) into equation (14), we have

(17)1

A

c5 exp (5η) + c4 exp (4η) + c3 exp (3η) + c2 exp (2η)+c1 exp (η) + c0 + c−1 exp (−η) + c−2 exp (−2η)+c−3 exp (−3η) + c−4 exp (−4η) + c−5 exp (−5η)

= 0

A = 5 (b1 exp (η) + b0 + b−1 exp (−η))6 where ci (i = −5,−4, ..., 4, 5)are constants obtained by Maple 16.

Equating the coefficients of exp (nη) to be zero, we obtain

(18)

(c−5 = 0, c−4 = 0, c−3 = 0, c−2 = 0, c−1 = 0, c0 = 0,

c1 = 0, c2 = 0, c3 = 0, c4 = 0, c5 = 0

)By solving (18), we get different solutions of (13).

1st Solution set:

a−1 =−1

24

b20 (K4 − ω2 +K2)

K2b1, a0 =

1

6

b0 (5K4 + ω2 −K2)

K2,(19)

a1 = −1

6

b1 (K4 − ω2 +K2)

K2, b−1 =

1

4

b20b1, b0 = b0, b1 = b1

Therefore, we obtained the following generalized solitary solution u (x, t) of equa-tion (13)

(20)

u (x, t) =

(−124

b20(K4−ω2+K2)e−xK+ωtα

Γ(1+α)

K2b1+ 1

6

b0(5K4+ω2−K2)K2 − 1

6

b1(K4−ω2+K2)exK+ωtα

Γ(1+α)

K2

)14

b20e−xK+ωtα

Γ(1+α)

b1+ b0 + b1e

xK+ωtβ

Γ(1+α)

Figure 4.1(a) α = 0.25 Figure 4.1(b) α = 1

Case 4.1.2. If p = c = 2 and q = d = 1, then equation (11) reduces to

(21) u (η) =a2 exp [2η] + a1 exp [η] + a0 + a−1 exp [−η]

b2 exp [2η] + b1 exp [η] + a0 + b−1 exp [−η].


Proceeding as before, we obtain

1st Solution set:

(22)

a−1 =

−1

24

b20 (K4 − ω2 +K2)

K2b1, a0 =

1

6

b0 (5K4 + ω2 −K2)

K2,

a1 = −1

6

b1 (K4 − ω2 +K2)

K2, b−1 =

1

4

b20b1, b0 = b0, b1 = b1

,

Hence we get the generalized solitary wave solution of equation (13) as follows(23)

u (x, t) =

(−124

b20(K4−ω2+K2)e−(xK+ωtα)

K2b1+ 1

6

b0(5K4+ω2−K2)K2 − 1

6

b1(K4−ω2+K2)exK+ωtα

K2

)14

b20e−(xK+ωtα)

b1+ b0 + b1exK+ωtα

Figure 4.1(c) α = 0.25 Figure 4.1(d) α = 1

We get the same soliton solutions which clearly illustrate that final solution doesnot strongly depends upon these parameters.

5. Conclusion

In this paper, we applied exp-function method to construct generalized solitarysolutions of the nonlinear fractional order singularly perturbed sixth-order Boussi-nesq equations. It is observed that the Exp-function method is very convenient toapply and is very useful for finding solutions of a wide class of nonlinear problems.

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COMMON FIXED POINT FOR SELF AND NONSELF-MAPSTHROUGH AN IMPLICIT RELATION

D. Surekha

Department of MathematicsHyderabad Institute of Technology & ManagementRR District, Hyderabad (A.P.)Indiae-mail: [email protected]

T. Phaneendra

Applied Analysis DivisionSchool of Advanced SciencesVIT UniversityVellore-632014, Tamil NaduIndiae-mail: [email protected]

Abstract. Common fixed point theorems for three self and nonself-maps have been

proved through the notions of property E.A., orbital completeness and weak compatibi-

lity under an implicit relation. The results of Singh and Mishra (1997), Singh and Asish

Kumar (2006), Khan and Dolmo (2007) and Imdad and Ali (2008) are then particular

cases.

Keywords: property E.A. weakly compatible maps, implicit relation, orbitally com-

plete metric space.

2010 Mathematics Subject Classification: 54H25.


Let (X, d) be a metric space. Self-maps S and A on X are compatible [2] if

(1.1) limn→∞

d(SAxn, ASxn) = 0

whenever ⟨xn⟩ ∞n=1 ⊂ X is such that

(1.2) limn→∞

Sxn = limn→∞

Axn = p for some p ∈ X.

194 t. phaneendra, d. surekha

A point x ∈ X is a coincidence point of self-maps S and A on X if Sx = Ax.Self-maps which commute at their coincidence points are called weakly compa-tible [3]. Thus S and A are weakly compatible if SAx = ASx whenever x ∈ X issuch that Ax = Sx. It is obvious that every compatible pair is weakly compatible.One can find examples from [3] for weakly compatible maps which are not com-patible. Let R+ be the space of nonnegative real numbers. A contractive modulusϕ : R+ → R+ with the choice ϕ(0) = 0 and ϕ(t) < t for t > 0 is upper semicon-tinuous (abbriviated as usc) if for each t0 ≥ 0, lim sup

n→∞ϕ(tn) ≤ ϕ(t0) whenever

⟨tn⟩∞n=1 ⊂ R+ is such that limn→∞

tn = 0.

With these notations, Singh and Mishra, [11] proved the following result:

Theorem 1.1 Let S, T and A be self-maps on X satisfying the inclusions

(1.3) S(X) ⊂ A(X) and T (X) ⊂ A(X)

and the contractive-type condition

(1.4)

d(Sx, Ty)

≤ ϕ

(max

d(Ax,Ay), d(Sx,Ax), d(Ty,Ay),

d(Ty,Ax) + d(Sx,Ay)

2

)for all x, y ∈ X,

where ϕ is a nondecreasing, usc contractive modulus. Suppose that one of S(X),T (X) and A(X) is a complete subspace of X. If

(a) (A, S) and (A, T ) are weakly compatible,

then the three maps S, T and A will have a unique common fixed point.

From the definition of compatibility, it is clear that self-maps S and A arenoncompatible on X if (1.2) holds good but lim

n→∞d(Sxn, Axn) is either = 0 or

+∞ for some ⟨xn⟩ ∞n=1 ⊂ X. It may be noted from [4] that both compatible and

noncompatible maps are included in the class of maps with (1.2). Self-maps Sand A on X are said to satify the property E.A. [1] if (1.2) holds good for some⟨xn⟩∞n=1 ∈ X, where the common limit p is called a tangent point. It is knownthat weak compatibility and property E.A. are independent.

With these ideas, the following theorem was proved in [10]:

Theorem 1.2 Let S, T and A : Y → X satisfy the inequality

(1.5)

d(Sx, Ty)

< max

d(Ax,Ay), αd(Ax, Sx), αd(Ay, Ty),

d(Ax, Ty) + d(Sx,Ay)

2

,

for all x, y ∈ X with 0 < α < 1

and both conditions

common fixed point for self and nonself-maps ... 195

(b) S(Y ) ⊂ A(Y ),

(c) T (Y ) ⊂ A(Y ),

where Y an arbitrary nonempty subset of X. Suppose one of the pairs (S,A) and(T,A) satisfies the property E.A. on Y. Then there is a coincidence point commonto S, T and A in Y. Further, if Y=X and (a) of Theorem 1.1 holds good, then S,T and A will have a unique common fixed point in X.

In this paper, we prove the generalizations of Theorem 1.1 and Theorem 1.2using implicit relations.

2. Main results

Given x0 ∈ X and S, T and A, self-maps on X, if there exist points x1, x2, x3, ...in X such that

(2.1) Sx2n−2 = Ax2n−1, Tx2n−1 = Ax2n, for n = 1, 2, 3, ...

then the sequence ⟨Axn⟩∞n=1 ⊂ X is an (S, T,A)-orbit or simply an orbit at x0.The metric space X is (S, T,A)-orbitally complete or orbitally complete [7], [8] atx0 if every Cauchy sequence in some orbit at x0 ∈ X converges in X. It easilyfollows that every complete metric space is orbitally complete at each of its points.However the converse is not true (cf. [7], [8]).

Let R6+ denote the space of 6-tuples of nonnegative real numbers in this paper,

and ψ : R6+ → R, lower semicontinuous in each coordinate variable such that

(C1) ψ is nondecreasing in the fifth and sixth coordinate variables,

(C2) For every l ≥ 0,m ≥ 0, there is a constant 0 ≤ ω < 1 such that

(2.2) minψ(l,m,m, l, l +m, 0), ψ(l,m, l,m, 0, l +m) ≤ 0 ⇒ l ≤ ωm,

(C3) ψ(l, l, 0, 0, l, l) > 0 for all l > 0.

Such implicit-type relations were first introduced by Popa [9], which covers severalcontractive conditions in proving fixed point theorems.

Theorem 2.1 Let S, T and A be self-maps on X satisfying the implicit-typeinequality

(2.3) ψ(d(Sx, Ty), d(Ax,Ay), d(Sx,Ax), d(Ty,Ay), d(Ty,Ax), d(Sx,Ay)) ≤ 0

for all x, y ∈ X.

Suppose that either (S,A) or (T,A) satisfies the property E.A. and that one of thefollowing statements holds good:


1. A(X) is orbitally complete at some x0 ∈ X,

2. S(X) ⊂ A(X),

3. T (X) ⊂ A(X).

Then there is a coincidence point u common to S, T and A. Further, if

(a) (S,A) or (T,A) is weakly compatible,

then the point of coincidence of S, T and A with respect to u will be their uniquecommon fixed point.

Proof. Suppose that (S,A) satisfies the property E.A. Then (1.2) holds good forsome ⟨xn⟩∞n=1 ⊂ X. We claim that q = lim

n→∞Txn = p. Writing x = y = xn in

(2.3), we find that

ψ(d(Sxn, Txn), d(Axn, Axn), d(Axn, Sxn), d(Axn, Txn),

d(Axn, Txn), d(Axn, Sxn)) < 0.

Applying the limit as n→ ∞ in this, using (1.2) and the lower semicontinuity ofψ, we get

ψ(d(p, q), 0, 0, d(p, q), d(p, q), 0) ≤ 0,

which gives (2.2) with l = d(p, q) andm = 0. Therefore, by (C2), we get d(p, q) ≤ 0or p = q. Thus

(2.4) limn→∞

Sxn = limn→∞

Txn = limn→∞

Axn = p for some p ∈ X.

While if (T,A) satisfies the property E.A., there is some ⟨yn⟩∞n=1 ⊂ X such that

(2.5) limn→∞

Tyn = limn→∞

Ayn = r for some r ∈ X.

Then (2.3) with x = y = yn gives

ψ(d(Syn,T yn), d(Ayn, Ayn), d(Ayn, Syn), d(Ayn,T yn), d(Ayn,Tyn), d(Ayn, Syn))< 0.

As n → ∞, this along with (2.5), s = limn→∞

Sxn and the lower semicontinuity of

ψ imply thatψ(d(s, r), 0, d(r, s), 0, 0, d(r, s)) ≤ 0.

So that (C2) with l = d(s, r) and m = 0 gives d(s, r) ≤ 0 or s = r, proving (2.4).

Case 2.1: Suppose that A(X) orbitally complete at x0 ∈ X. Then (2.4) impliesthat p ∈ A(X) so that p = Au for some u ∈ X. From (2.3) with x = u andy = pn, we would get

ψ(d(Su, Tpn), d(Au,Apn), d(Au, Su), d(Apn, Tpn), d(Au, Tpn), d(Apn, Su)) < 0.


Employing the limit as n→ ∞ and using (2.4), this gives

ψ(d(Su,Au), 0, d(Au, Su), 0, 0, d(Au, Su)) ≤ 0,

which in view of (2.2) with l = d(Au, Su) and m = 0 together with (C2) givesd(Au, Su) = 0 or

(2.6) Au = Su = p.

Again from (2.3) with x = u = y and (2.6), it follows that

ψ(d(Su, Tu), d(Au,Au), d(Au, Su), d(Au, Tu), d(Au, Su), d(Au, Tu)) < 0

or ψ(d(p, Tu), 0, 0, d(p, Tu), 0, d(p, Tu)) < 0 so that from (C2) with l = d(p, Tu)and m = 0, we get d(Ap, Tu) = 0 or Tu = p, that is

(2.7) Au = Su = Tu = p.

Case 2.1: Suppose that (2.1) holds good. Then ⟨Sxn⟩ ∞n=1 ⊂ S(X) and hence

lies in S(X) so that p ∈ S(X) ⊂ A(X). It follows that u is a common coincidencefrom Case (2.1).

Case 2.1: Suppose that (2.1) holds good. Then as above, we get that ⟨Txn⟩ ∞n=1

converges in T (X) so that p ∈ S(X) ⊂ A(X) and (2.7) follows from Case (2.1).

To prove the second part of Theorem 2.1, we suppose that (S,A) is weaklycompatible. From (2.6), we find that SAu = ASu or Ap = Sp. But then from(2.3) with x = y = p, we get

ψ(d(Sp, Tp), d(Ap,Ap), d(Ap, Sp), d(Ap, Tp), d(Ap, Tp), d(Ap, Sp)) < 0 or

ψ(d(Sp, Tp), 0, 0, d(Sp, Tp), d(Sp, Tp), 0) < 0,

which, in view of (C2) with l = d(Sp, Tp) and m = 0, gives d(Sp, Tu) = 0 orSp = Tp, that is

(2.8) Ap = Sp = Tp.

If (T,A) is weakly compatible, from (2.7), it follows that Tp = Ap. Then takingx = y = p in (2.3) and using this and (C2) with l = d(Sp, Tp) and m = 0 andsimplifying, we get (2.8).

Now, p is a fixed point of S. In fact, from (2.3) with x = u and y = p, we get

ψ(d(Su, Tp), d(Au,Ap), d(Au, Su), d(Ap, Tp), d(Au, Tp), d(Ap, Su)) < 0 or

ψ(d(p, Sp), d(p, Sp), 0, 0, d(p, Sp), d(Sp, p)) < 0.

Again using (C2) in this, we get d(p, Sp) = 0. Hence p is a common fixed pointof S, T and A, in view of (2.8).


Finally, suppose that q is also a common fixed point of S, T and A with p = qso that d(p, q) > 0, then writing x = p and y = q in (2.3), we obtain

ψ(d(p, q), d(p, q), d(p, p), d(q, q), d(p, q), d(q, p)) < 0,

which is a contradiction to the choice (C3). This shows that p = q, that is thecommon fixed point of S, T and A is unique.

The following example emphasizes that none of the conditions (2.1), (2.1)and (2.1) can be dropped in Theorem 2.1 to ensure a common coincidence pointfor the three self-maps:

Example 2.1 Let

ψ(l1, l2, l3, l4, l5, l6) = l1 − qmax

l2, l3, l4,

l5 + l62

,

where 0 ≤ q < 1. Then ψ satisfies (C1) and

ψ(l,m,m, l, l +m, 0) = l − qmaxl,m) = ψ(l,m, l,m, 0, l +m)

so that from (2.2) we get

(2.9) l ≤ qmaxl,m.

It may be noted that (2.9) is trivial if l = 0. Therefore let l > 0.Again, if maxl,m = l, (2.9) would give a contradiction that l ≤ ql < l,

since q < 1. Thus maxl,m = m so that l ≤ qm), proving (C2).Finally, ψ(l, l, 0, 0, l, l) = (1 − q)l ≤ 0, which is a contradiction to the choice

of ψ if l > 0, and (C3) follows.

Let X = R+ with d(x, y) = |x− y| for all x, y ∈ X. Define

Sx =

1

2(x = 0)

x

2(x > 0) ,

Tx =

1

2(x = 0)

2x

3(x > 0)

and Ax =

1 (x = 0)

3x

4(x > 0) .

Note that S(X) = T (X) = A(X) = (0,∞). But S(X) = T (X) = [0,∞) so that(2.1) and (2.1) are not satisfied. For x0 = 0, we choose

x1 =2

3, x2n = 3

(16

27

)n

, x2n+1 = 2

(16

27

)n

for n ≥ 1,

the orbit is 1

2,4

3,8

9, ...,

9

4

(16

27

)n

,3

2

(16

27

)n

, ...

.


While, for x0 > 0, choose

x1 =2

3x0, x2n =

(16

27

)n

x0, x2n+1 =2

3

(16

27

)n

x0, n = 1, 2, 3, ...,

so that its orbit isx02,

(2

3

)2

x0, ...,3

4

(16

27

)n

x0,1

2

(16

27

)n

x0, ...

.

In each case, the orbit converges to 0; (S,A) and (T,A) satisfy the property E.A.However, since the limit 0 does not belong to A(X), the subspace A(X) cannotnot be orbitally complete at each x0. Thus, (2.1) fails and the three maps donot have a common coincidence point, though X is complete. In other words, atleast one of (2.1), (2.1) and (2.1) is necessary in Theorem 2.1 to obtain a commoncoincidence point.

Remark 2.1 Set

ψ(l1, l2, l3, l4, l5, l6) = l1 − ϕ

(max

l2, l3, l4,

l5 + l62

),

where ϕ is a nondecreasing usc contractive modulus. Then (C1) holds good.Ignoring the triviality l = 0, we assume that l > 0, and m ≥ 0 so that

ψ(l,m,m, l, l +m, 0) = l − ϕ(maxl,m) = ψ(l,m, l,m, 0, l +m)

and from (2.2) we get

l − ϕ(maxl,m) ≤ 0 or l ≤ ϕ(maxl,m),

which would give a contradiction that l ≤ ϕ(l) < l if maxl,m = l. Thusmaxl,m = m so that l ≤ ϕ(m) and hence l ≤ qm for some 0 < q < 1 proving(C2). Finally the choice of ϕ gives

ψ(l, l, 0, 0, l, l) = l − ϕ(max

l, 0, 0, l+l

2

)= l − ϕ(l) > 0,

and hence (C3).The choice of ψ reveals that (1.4) is a special case of (2.3). Given x0 ∈ X,

using (1.3), one can define an orbit ⟨Axn⟩∞n=1 at x0 with the choice (2.1). But,from [11], ⟨Axn⟩∞n=1 is a Cauchy sequence in A(X). Condition (1) holds goodwhenever if A(X) is a complete subspace of X and hence Axn → p as n → ∞for some p ∈ A(X) so that p = Au for some u ∈ X. Then ⟨Sx2n−2⟩∞n=1 and⟨Tx2n−1⟩∞n=1 being its subsequences also converge to p = Au. That is

(2.10) limn→∞

Sx2n−2 = limn→∞

Tx2n−1 = limn→∞

Ax2n = p = Au.


Write x = y = x2n in (1.4). Then applying the limit as n → ∞, s = limn→∞

Tx2n

and using (2.10), we find that

d(p, s) ≤ ϕ

(max

0, 0, d(s, p),

d(s, p) + 0

2

),

so that d(p, s) = 0 or p = s. This proves that (T,A) satisfies the propertyE.A. While taking x = y = x2n−1 in (1.4), applying the limit as n → ∞, using(2.10), t = lim

n→∞Sx2n−1 and proceeding as above, it follows that (S,A) satisfies

the property E.A. If S(X) is orbitally complete, then p ∈ S(X) and (1.3) gives,p ∈ A(X) and the conclusion follows from the earlier case. Similarly the case thatT (X) is orbitally complete can be handled. The conclusion follows from Theorem2.1. Thus Theorem 1.1 is a particular case of Theorem 2.1.

Remark 2.2 When T = S in (2.1), we get an (S,A)-orbit [6] at x0 ∈ X given by

(2.11) Sxn−1 = Axn for n = 1, 2, 3, ...

and hence the (S,A)-orbital completeness of X at x0. Therefore, taking T = S inTheorem 2.1, we get

Corollary 2.1 (Theorem 3.1, [4]) Let S and A be self-maps on X satisfyingthe property E.A. and the inequality

(2.12) ψ(d(Sx, Sy), d(Ax,Ay), d(Sx,Ax), d(Sy,Ay), d(Sx,Ay), d(Sy,Ax)) ≤ 0

for all x, y ∈ X,

If A(X) is complete and (S,A) is a weakly compatible pair, then S and A have aunique common fixed point.

Let Y an arbitrary nonempty subset ofX. Imitating the proof of Theorem 2.1,we can establish the following result for nonself-maps:

Theorem 2.2 Let S, T and A : Y → X satisfy (2.3) for all x, y ∈ Y. Supposethat either (S,A) or (T,A) satisfies the property E.A. on Y, and any one of thefollowing conditions holds good:

(1) A(Y ) is closed subspace of Y,

(2) S(Y ) ⊂ A(Y ),

(3) T (Y ) ⊂ A(Y ).

Then there is a coincidence point u common to S, T and A in Y. Further if thepoint of common coincidence of S, T and A with respect to u lies in Y, it willbe their unique common fixed point, provided either (S,A) or (T,A) is a weaklycompatible pair.


Remark 2.3 Write

ψ(l1, l2, l3, l4, l5, l6) = l1 −max

l2, αl3, αl4,

l5 + l62

in Theorem 2.2, where 0 < α < 1. Then (C1)-(C3) hold good and (1.5) is aparticular case of (2.3). Interestingly, Theorem 1.2 required both the containments(2) and (3) and weak compatibility of both the pairs (S,A) and (T,A). Also acommon fixed point was obtained under the condition that Y = X. However, theproof of our second results suggests that weak compatibility and property E.A.of either pair is sufficient to obtain a common fixed point under a generalizedinequality, even without the condition that Y = X.

Finally taking

ψ(l1, l2, l3, l4, l5, l6) = l1 −max

l2, βl3 + αl4,

l5 + l62

,

where β ≥ 0 and 0 < α < 1 and T = S in Theorem 2.2, we get

Corollary 2.2 (Theorem 3.1, [5]) Let S,A : Y → X satisfying

(2.13)

d(Sx, Sy)

< max

d(Ax,Ay), βd(Sx,Ax) + αd(Sy,Ay),

d(Sy,Ax) + d(Sx,Ay)

2

for all x, y ∈ Y with x = y,

Suppose that either A(Y ) is a complete subspace of Y or S(Y ) is a completesubspace of Y with S(Y ) ⊂ A(Y ). Then (S,A) have a coincidence point u in Y.Further if the point of coincidence of S and A with respect to u is in Y, then Sand A will have a unique unique common fixed point in Y, provided S and A areweakly compatible.

Acknowledgements. The author wishes to express sincere thanks to the refereefor his/her valuable suggestions in improving the paper.

References

[1] Aamri, M.A., Moutawaki, D. El., Some new common fixed point theo-rems under strict contractive conditions, J. Math. Anal. Appl., 270 (2002),181-188.

[2] Gerald Jungck, Compatible maps and common fixed points, Int. J. Math.& Math. Sci., 9 (1986), 771-779.


[3] Gerald Jungck, Rhoades, B.E., Fixed point for set valued functionswithout continuity, Indian J. Pure Appl. Math., 29 (1998), 227-238.

[4] Imdad, M., Javid Ali, Jungck’s common fixed point theorem and E.A.property, Acta Mathematica Sinica, English Ser., 24 (2008), 87-94.

[5] Khan, A.R., Domlo, A.A., Coincidence and fixed points of nonself con-tractive maps with applications, Ind. Jour. Math., 49 (2007), 17-13.

[6] Phaneendra, T., Coincidence Points of Two Weakly Compatible Self-Mapsand Common Fixed Point Theorem through Orbits, Ind. Jour. Math., 46 (2-3)(2003), 173-180.

[7] Phaneendra, T., Asymptotic regularity and common fixed point, Pure andAppl. Math. Sci., 59 (1-2) (2004), 45-49.

[8] Phaneendra, T., Orbital continuity and common fixed point, BuletiniShkencor, 3 (4) (2011), 375-380.

[9] Popa, V., Some fixed point theorems for compatible mappings satisfying animplicit relation, Demon. Math., 32 (1999), 157-163.

[10] Singh, S.L., Ashish Kumar, Common fixed point theorems for contractivemaps, Math. Vesnik., 58 (2006), 85-90.

[11] Singh, S.L., Mishra, S.N., Remarks on Jachymski’s fixed point theoremsfor compatible maps, Ind. J. Pure Appl. Math., 28 (1997), 611-615.



A CHARACTERIZATION OF PROJECTIVE SPECIAL LINEARGROUP L3(5) BY nse

Shitian Liu

School of ScienceSichuan University of Science and EngineeringZigong Sichuan, 643000Chinae-mail: [email protected]

Abstract. Let G be a group and ω(G) be the set of element orders of G. Let k ∈ ω(G)

and sk be the number of elements of order k in G. Let nse(G) = sk∣∣k ∈ ω(G). In

Khatami et al. and Liu’s works, the authors proved that the groups L3(2) and L3(4)

are unique determined by nse. In this paper, we prove that if G is a group such that

nse(G) =nse(L3(5)), then G ∼= L3(5).

Keywords and phrases: Element order, Projective special linear group, Thompson’s

problem, Number of elements of the same order.

AMS Subject Classification: 20D05, 20D06, 20D20.

1. Introduction

In 1987, Thompson posed a very interesting problem related to algebraic numberfields as follows (see [17]).

Thompson’s Problem. Let T (G) = (n, sn)∣∣n ∈ ω(G) and sn ∈nse(G), where

sn is the number of elements with order n. Suppose that T (G) = T (H). If G is afinite solvable group, is it true that H is also necessarily solvable?

It was proved that: Let G be a group and M some simple Ki-group, i = 3, 4,then G ∼= M if and only if |G| = |M | and nse(G)=nse(M) (see [13], [14]). Thegroup A12 is characterized by nse and order (see [8]). Recently, all sporadic simplegroups are characterizable by nse and these orders (see [1]).

Comparing the sizes of elements of same order but disregarding the actualorders of elements in T (G) of the Thompson’s Problem, in other words, it remainsonly nse(G), whether can it characterize finite simple groups? Up to now, somegroups especial for L2(q), where q = 7, 8, 9, 11, 13, can be characterized by onlythe set nse(G) (see [7], [16]). The author has proved that the groups L3(4), L5(2)

204 shitian liu

and U3(5) are characterizable by nse (see [9], [10] and [11], respectively). In thispaper, it is shown that the group L3(5) also can be characterized by nse(L3(5)).

We introduce some unfamiliar notations which will be used in this note. Leta · b denote the products of an integer a by an integer b. If n is an integer, thenwe denote by π(n) the set of all prime divisors of n. Let G be a group and r aprime. Then we denote the number of Sylow r-subgroups Pr of G by nr or nr(G).The set of element orders of G is denoted by ω(G). Let k ∈ ω(G) and sk be thenumber of elements of order k in G. Let nse(G) = sk

∣∣k ∈ ω(G). Let π(G)denote the set of prime p such that G contains an element of order p. The othernotations are standard (see [2]).

2. Some lemmas

Lemma 2.1 [4] Let G be a finite group and m be a positive integer dividing |G|.If Lm(G) = g ∈ G

∣∣gm = 1, then m∣∣|Lm(G)|.

Lemma 2.2 [12] Let G be a finite group and p ∈ π(G) be odd. Suppose that Pis a Sylow p-subgroup of G and n = psm with (p,m) = 1. If P is not cyclic ands > 1, then the number of elements of order n is always a multiple of ps.

Lemma 2.3 [16] Let G be a group containing more than two elements. If themaximal number s of elements of the same order in G is finite, then G is finiteand |G| ≤ s(s2 − 1).

Lemma 2.4 [5, Theorem 9.3.1] Let G be a finite solvable group and |G| = mn,where m = pα1

1 · · · pαrr , (m,n) = 1. Let π = p1, · · · , pr and hm be the number of

Hall π-subgroups of G. Then hm = qβ1

1 · · · qβss satisfies the following conditions for

all i ∈ 1, 2, · · · , s:

(1) qβi

i ≡ 1 (mod pj) for some pj.

(2) The order of some chief factor of G is divided by qβi

i .

Definition 2.5 A finite group G is called a simple Kn-group, if G is a simplegroup with |π(G)| = n.

Remark 2.6 If G is a simple K1-group, then G is a cyclic group of order p.

Remark 2.7 If G is a simple K2-group, then by Thompson’ paqb Theorem, G issoluble. Therefore there is no simple K2-group.

Lemma 2.8 [18] Let G be a simple K4-group. Then G is isomorphic to one ofthe following groups:

(1) A7, A8, A9 or A10.

(2) M11, M12 or J2.

a characterization of projective special linear group L3(5) ... 205

(3) One of the following:

(a) L2(r), where r is a prime and r2 − 1 = 2a · 3b · vc with a ≥ 1, b ≥ 1,c ≥ 1, and v is a prime greater than 3.

(b) L2(2m), where 2m − 1 = u, 2m + 1 = 3tb with m ≥ 2, u, t are primes,

t > 3, b ≥ 1.

(c) L2(3m), where 3m+1 = 4t, 3m− 1 = 2uc or 3m+1 = 4tb, 3m− 1 = 2u,

with m ≥ 2, u, t are odd primes, b ≥ 1, c ≥ 1.

(4) One of the following 28 simple groups: L2(16), L2(25), L2(49), L2(81),L3(4), L3(5), L3(7), L3(8), L3(17), L4(3), S4(4), S4(5), S4(7), S4(9), S6(2),O+

8 (2), G2(3), U3(4), U3(5), U3(7), U3(8), U3(9), U4(3), U5(2), Sz(8), Sz(32),2D4(2) or

2F4(2)′.

Lemma 2.9 Let G be a simple K4-group and 31 | |G| | 25.3.53.31. Then G ∼= L3(5)or L2(31).

Proof. Since G is a K4-group, then by Lemma 2.8(1)(2), order considerationrules out these cases.

So, by Lemma 2.8(3), the following cases are considered.

Case 1. G ∼= L2(r), where r ∈ 3, 5, 31.

• Let r = 3, then |π(r2 − 1)| = 1, which contradicts |π(r2 − 1)| = 3.

• Let r = 5, then |π(r2 − 1)| = 2, which contradicts |π(r2 − 1)| = 3.

• Let r = 31, then |π(312−1)| = 3. So G ∼= L2(31) since |G| = 25.3.5.31.

Case 2. G ∼= L2(2m), where u ∈ 3, 5, 31.

• Let u = 3, then m = 2 and so 5 = 3tb. But the equation has no solutionin N, a contradiction.

• Let u = 5, then the equation 2m − 1 = 5 has no solution in N, acontradiction..

• Let u = 31, then 32 = 2m and so m = 5. So 33 = 3tb, and so t = 11,which is a contradiction since 11 | |G|.

Case 3. G ∼= L2(3m).

The following two cases need to be considered.

• Subcase 3.1. 3m + 1 = 4t and 3m − 1 = 2uc.

Let t ∈ 3, 5, 31.If t = 3, 5, 31, the equation 3m + 1 = 4t has no solution. So the casecan be ruled out.

206 shitian liu

• Subcase 3.2. 3m + 1 = 4tb and 3m − 1 = 2u.

Let u ∈ 3, 5, 31.If u = 3, 5, 31, then the equation 3m − 1 = 2u has no solution in N, acontradiction.

By Lemma 2.8(4), order consideration rules out the groups except for L3(5).Therefore, G is isomorphic to L2(31) or L3(5).This completes the proof.

3. Main theorem and its proof

Let G be a group such that nse(G) =nse(L3(5)), and sn be the number of elementsof order n. By Lemma 2.3 we have that G is finite. We note that sn = kϕ(n),where k is the number of cyclic subgroups of order n. Also, we note that if n > 2,then ϕ(n) is even. If m ∈ ω(G), then by Lemma 2.1 and the above discussion, wehave

(3.1)

ϕ(m) | sm

m |∑

d |m sd

Theorem 3.1 Let G be a group with nse(G)=nse(L3(5)) = 1, 775, 15500, 15624,18600, 24800, 31000, 37200, 62000, 120000. Then G ∼= L3(5).

Proof. We first prove that π(G) ⊆ 2, 3, 5, 31, then show that |G| = |L3(5)|and so by [14], G ∼= L3(5).

By (3.1), π(G) ⊆ 2, 3, 5, 7, 11, 31, 37201.If k > 2, then ϕ(k) is even, s2=775, and so 2 ∈ π(G).If 5, 31, 3 ∈ π(G), then s5=15624, s31=120000, s3=15500, 24800, 62000.In the following, we prove 37201 ∈ π(G).If 37201 ∈ π(G), then by (3.1), s37201=37200.If 2.37201 ∈ ω(G), then s74402 ∈nse(G).Therefore 2.37201 ∈ ω(G).It follows that the Sylow 37201-subgroup of G acts fixed point freely on the

set of elements of order 2 and so |P37201| | s2, a contradiction. Thus, 37201 ∈ π(G).If 2.7 ∈ ω(G), then by Lemma 2.3 of [15], s2.7 = s7.t for some integer t. Then

s2.7 = s7. But by Lemma 2.1, 2.7 | 1 + s2 + s7 + s2.7(240776), a contradiction.It follows that the Sylow 7-subgroup of G acts fixed point freely on the set ofelements of order 2 and 7 | s2, a contradiction. Hence 7 ∈ π(G).

If 2a ∈ ω(G), then ϕ(2a) = 2a−1|s2a and so 0 ≤ a ≤ 7.If 3a ∈ ω(G), then 1 ≤ a ≤ 3.If 5a ∈ ω(G), then 1 ≤ a ≤ 5. If 55 ∈ ω(G), then s3125 ∈nse(G) and so

1 ≤ a ≤ 4.If 11a ∈ ω(G), then a = 1.


If 31a ∈ ω(G), then a = 1 or 2. If a = 2, then s961 ∈nse(G) and so 961 ∈ ω(G).Hence a = 1.

If 2a.3b ∈ ω(G), then 1 ≤ a ≤ 6 and 1 ≤ b ≤ 3.

If 3a.5b ∈ ω(G), then 1 ≤ a ≤ 3 and 1 ≤ b ≤ 4.

If 2a.3b.5c ∈ ω(G), then 1 ≤ a ≤ 4, 1 ≤ b ≤ 3 and 1 ≤ c ≤ 4.

To remove the prime 11, the fact that the prime 31 which belong to π(G) willbe proved.

Suppose that 31 ∈ π(G).

• If 3, 5, 11 ∈ π(G), then G is a 2-group and so 372000+ 15500k1 +15624k2 +18600k3 + 24800k4 + 31000k5 + 37200k6 + 62000k7 + 120000k8 = 2m whereki, i = 1, 2, ..., 8 and m are non-negative integers. Since |ω(G)| = 8, theequation has no solution in N since the number of elements of nse(G) isten. So, the following cases are considered: 3, 3, 5, 3, 11, 3, 5, 11,5, 11, 5, 7, 11 and 11,

• Let 11∈ π(G), then as exp(P11)=11, |P11||1 + s11 and so |P11|=11. Sincen11 = s11/ϕ(11)=1860, then 31∈ π(G), a contradiction.

Similarly, the set which contains the prime 11 can be excluded as the theset 11.

• Let 5 ∈ π(G). The exponent of P5 is equal to 5, 25, 125, and 625.

* If exp(P5)=5, then |P5| | 1 + s5 and so |P5| | 56.If |P5| = 5, then n5 = s5/ϕ(5)=3906 and so 31∈ π(G), a contradiction.

If |P5| = 52, then since 31, 11 ∈ π(G), 372000 + 15500k1 + 15624k2 +18600k3+24800k4+31000k5+37200k6+62000k7+120000k8 = 2m.3n.52

where ki, i = 1, 2, ..., 8 and m,n are non-negative integers, and

0 ≤8∑

i=1

ki ≤ 77. Since 372000 ≤ |G| = 2m.3n.52 ≤ 37200 + 77.120000,

then if n = 0, m = 12, ..., 18; if n = 1, then m = 13, ..., 16; if n = 2,then m = 11, ..., 15; if n = 3, then m = 10, ..., 13. But m is at mostseven, and so these cases can be excluded. If |P5| = 53, then similarlyif n = 0, then m = 12, ..., 16; if n = 1, then m = 10, ..., 14; if n = 2,then m = 9, ..., 13; if n = 3, then m = 7, ..., 11. Therefore only m = 7is considered. In this case, 372000 + 15500k1 + 15624k2 + 18600k3 +24800k4 + 31000k5 + 37200k6 + +6200k7 + 120000k8 = |G| = 27.33.53

where k1, ..., k8 are non-negative integers. By computer computation,the equation has no solution in N.Similarly, the other cases |P5| = 54, 55, 56 can be ruled out as the abovemethods.

* If exp(P5) = 25, then |P5| | 1 + s5 + s25 and so |P5| | 125. If |P5| = 25,then by (1), n5 = s25/π(25) if s25 ∈ 15500, 18600, 24800, 31000, 37200and so 31∈ π(G), a contradiction.

208 shitian liu

If s25 = 120000, then n5 = 6000. By Sylow’s theorem, n5 = 5k + 1 forsome integer k, but the equation has no solution in N.If |P5| = 125, then similarly as the case “exp(P5)=5 and |P5| = 25”,we can rule out this case.

* If exp(P5) = 125, then by Lemma 2.1, |P5| | 1 + s5 + s25 + s125 and so|P5| | 54.If |P5| = 53, then by (3.1), s125=15500, 24800, 31000, 37200, 120000. Ifs125=15500, 18600, 24800, 31000, 37200, then n5 = s125/ϕ(125)=155,186, 248, 310, 372 and so 31 belongs to π(G), a contradiction. Hences125=120000 and n5=1200. But by Sylow’s theorem, n5 = 5k + 1 forsome integer k, it is easy to see that the equation has no solution in N.If |P5| = 54, then similarly as the case “exp(P5)=5 and |P5| = 25”, wecan rule out this case.

* If exp(P5) = 625, then by Lemma 2.1, |P5| | 1 + s5 + s25 + s125 + s625(s625 = 15500, 31000, 120000) and so |P5| = 625. If s625=15500, 31000,n5 = s625/ϕ(625)=31, 62 and so 31 belongs to π(G), a contradiction.Therefore s625 = 120000 and t=240. But by Sylow’s theorem t = 5k+1for some integer k, so the equation has no solution in N.

Similarly, the set which contains the prime 5 can be excluded as the theset 5.

• Let 3 ∈ π(G). The exponent of P3 equal to 3, 9, 27.

* Let exp(P3)=3. Then by Lemma 2.1, |P3| | 1 + s3 and so |P3| | 9.If |P3| = 3, then n3 = s3/ϕ(3) and so 31 belongs to π(G), a contradic-tion.

If |P3| = 9, then s3 = 3t for some non-negative integer t. So s3=15624,18600, 37200, 120000 and t=5208, 6200, 12400, 40000, respectively. Ifs3=15624, 18600, 37200, then 31 ∈ π(G). So s3 = 120000. In thiscase, since 2∈ π(G) and if 31 ∈ π(G), the primes 11, 5 ∈ π(G), thenπ(G) = 2, 3 and so 372000+15500k1+15624k2+18600k3+24800k4+31000k5 +37200k6 +62000k7 +120000k8 = 2m.9 where ki, i = 1, 2, ..., 8

and m are non-negative integers, and 0 ≤8∑

i=1

ki ≤ 7. Since 372000 ≤

|G| = 2m.32 ≤ 37200+ 7.120000, then m = 16, a contradiction since mis at most 7.

* Let exp(P3)=9. Then by Lemma 2.1, |P3| | 1+ s3 + s9 and so |P3| | 81.If |P3| = 9, then 5 or 31 belongs to π(G), a contradiction.

If |P3| = 27, then similarly m = 16, a contradiction. If |P3| = 81, thenm > 7, a contradiction.

* Let exp(P3)=27. Then by Lemma 2.1, |P3| | 1 + s3 + s9 + s27 and so|P3| = 27. So n3 = s27/ϕ(27) and 31 belongs to π(G), a contradiction.


Similarly, the set which contains the prime 3 can be excluded as thethe set 3.

Therefore, 31 ∈ π(G).By Lemma 2.1, |P31| | 1 + s31 and so |P31| = 31.In the following, that the prime 11 do not belong to π(G) are proved.Let 11 ∈ π(G). If 11.31 ∈ ω(G), then by Lemma 2.3 of [15], s11.31 = 10.s31.t

for some integer t. But the equation has no solution since s11.31 ∈nse(G). Hence11.31 ∈ ω(G). It follows that the Sylow 11-subgroup of G acts fixed freely on theset of elements of order 31 and 11 | s31, a contradiction. So11 ∈ π(G).

Therefore, π(G) ⊆ 2, 3, 5, 31.From the above arguments, 2, 31 ∈ π(G), so the following cases will be con-

sidered: 2, 31, 2, 3, 31, 2, 5, 31 and 2, 3, 5, 31.

Case a. π(G) = 2, 31.Since exp(P31)=31, then by Lemma 2.1, |P31| | 1+ s31 and so |P31|=31. Since

n31 = s31/ϕ(31) = 4000, then 5 belongs to π(G), a contradiction.

Case b. π(G) = 2, 3, 31.The proof is the same as Case a.

Case c. π(G) = 2, 5, 31.Since exp(P31)=31, then by Lemma 2.1, |P31| | 1 + s31 and so |P31| = 31.If 2.31 ∈ ω(G), set P and Q are Sylow 31-subgroups of G, then P and Q

are conjugate in G and so CG(P ) and CG(Q) are conjugate in G. Then s2.31 =ϕ(2.31).n31.k, where k is the number of cyclic subgroups of order 2 in CG(P31),and so n31 | s2.31. So s2.31 = n31.t for some integer t. Since n31 = s31/ϕ(31),s2.31 = 4000t for some integer t and so s2.31 = s31. On the other hand, 2.31 |1 + s2 + s31 + s2.31(=240776), a contradiction. Thus 2.31 ∈ ω(G). It follows thatthe Sylow 2-subgroup of G acts fixed point freely on the set of elements of order31 and so |P2| | s31. So |P2| | 26. Also by (3.1), 5.31 ∈ ω(G) and |P5| | 54.

We know that exp(P5) = 5, 52, 53, 54.Let exp(P5) = 5.

• If |P5| = 5, then n5 = s5/ϕ(5) = 3906 and so 3 ∈ π(G), a contradiction.

• If |P5| = 52, then 372000+15500k1+15624k2+18600k3+24800k4+31000k5+37200k6 + 62000k7 + 120000k8 = 2m.52.31 where ki, i = 1, 2, ..., 8 and m are

non-negative integers, and 0 ≤8∑

i=1

ki ≤ 3. Since 372000 ≤ |G| = 2m.52.31 ≤

37200 + 3.120000, the equation has no solution since m is at most 6.

• If |P5| = 53, then similarly, the equation has no solution.

• If |P5| = 54, then similarly m = 6, 5. Let G be a group such that |G| =26.54.31 or |G| = 25.54.31 and nse(G)=nse(L3(5)). Then by programmeof [7], there is no such group.

210 shitian liu

Let exp(P5) = 25. Then by (3.1), s52 ∈ 15500, 18600, 24800, 31000, 37200,62000, 120000

• If |P5| = 52 and s52 ∈ 15500, 18600, 37200, 120000, then n5 = s52/ϕ(52)

and 3 ∈ π(G), a contradiction.

If |P5| = 52 and s52 ∈ 24800, 31000, then n5 = 1240, 1550. But by Sylow’stheorem, n5 = 5k+ 1 for some integer k, the equation has no solution in N.

• If |P5| ≥ 53, then similarly as the proof of “exp(P5) = 5”, we can rule outthis case.

Let exp(P5) = 53. Then by (3.1), s53 ∈ 15500, 24800, 31000, 37200, 120000.

• If s53 ∈ 37200, 120000 and |P5| = 53, then n5 = s53/ϕ(53) and 3 ∈ π(G),

a contradiction.

If s53 ∈ 15500, 24800, 31000 and |P5| = 53, then n5 = 155, 248, 310. Onthe other hand, by Sylow’s theorem, n5 = 5k + 1 for some integer k, theequation has no solution in N.

• If |P5| = 54, then similarly as the proof of “exp(P5) = 5”, we can rule outthis case.

Let exp(P5) = 54. Then similarly as the proof of “exp(P5) = 5”, we can ruleout this case.

Case d. π(G) = 2, 3, 5, 31.From the above, |P31|=31.If 3.31 ∈ ω(G), set P and Q are Sylow 31-subgroups of G, then P and Q

are conjugate in G and so CG(P ) and CG(Q) are also conjugate in G. Thereforewe have s3.31 = ϕ(3.31).n31.k, where k is the number of cyclic subgroups of order3 in CG(P31). As n31 = s31/ϕ(31) = 120000/30 = 40000, 240000 | s3.31 and sos3.31 = 240000t for some integer t, but the equation has no solution in N. So3.31 ∈ ω(G). It follows that the Sylow 3-subgroup P3 of G acts fixed point freelyon the set of elements of order 31 and so |P3| | s31. Therefore |P3| | 3. Similarlywe can prove that 3.5 ∈ ω(G) and |P5| | 53.

Therefore, |G| = 2m.3.5p.31. Since 372000 = 25.3.53.31 ≤ |G| = 2m.3.5p.31,then (m, p) = (5, 3), (6, 3).

In the following, first show that here is no group such that |G| = 26.3.53.31and nse(G)=nse(L3(5)), second get the desired result.

Since |π(G)| ≥ 3 and 3.31, 5.31, 3.5 ∩ ω(G) = Φ, then by Lemma 2.5 of [3],G is insoluble.

Therefore, we can suppose that G has a normal series 1 K L G suchthat L/K is isomorphic to a simple Ki-group with i = 3, 4 as 9 and 961 do notdivide order of G.

If L/K is isomorphic to a K3-simple group, then from [6], L/K ∼= A5, U4(2).From [2], n5(L/K) = n5(A5) = 6, and so n5(G) = 6t and 5 - t. Hence thenumber of elements of order 5 in G is: s5 = 6t · 4 = 24t for some integer t. Since


s5 ∈nse(G), then s5=15624 and t = 651. Hence 3.7.31 | |K| | 23.53 · 31, which is acontradiction. Similarly the other group U4(2) can be ruled out as the group “A5”.

If L/K is isomorphic to a K4-group, then by Lemma 2.9, L/K ∼= L2(31) orL/K ∼= L3(5).

Let L/K ∼= L2(31).Let G = G/K and L = L/K. Then

L2(31) ≤ L ∼= LCG(L)/CG(L) ≤ G/CG(L) = NG(L)/CG(L) ≤ Aut(L)

Set M = xK | xK ∈ CG(L), then G/M ∼= G/CG(L) and so L2(31) ≤G/M ≤Aut(L2(31)). Therefore G/M ∼= L2(31), or G/M ∼= 2.L2(31).

If G/M ∼= L2(31), then order consideration |M | = 2.52. Then there exist agroup M such that M is a Frobenius subgroup with a complement of order 2 anda Frobenius kernel of order 52. So there exists an element of order 3.52, which isa contradiction.

If G/M ∼= 2.L2(31), then |M | = 25 and M is a normal subgroups generatedby 5-central elements or with exponent of 52. If the former, then there is anelement of order 5.31, a contradiction. If the latter, also we have a contradiction.

If L/K ∼= L3(5), then similarly we have that G/M ∼= L3(5) or G/M ∼=2.L3(5).

If G/M ∼= L3(5), then |M | = 2. It follows that M is a normal subgroupgenerated by a 2-central element and so there exists an element of order 2.31, acontradiction.

G/M ∼= 2.L3(5), then M = 1. Since nse(2.L3(5)) =nse(G), then we rule outthis case.

Hence |G| = 25.3.53.31 and by assumption, nse(G)=nse(L3(5)), so by [14],G ∼= L3(5).

This completes the proof of the theorem.

Acknowledgments. The object is supported by the Department of Educationof Sichuan Province (Grant No: 12ZB085, 12ZB291 and 13ZA0119). The authoris very grateful for the helpful suggestions of the referee.

References

[1] Asboei, A.K.R., Amiri, S.S.S, Iranmanesh, A., Tehranian, A.,A characterization of sporadic simple groups by NSE and order, J. Algebraand its Applications, 12 (2) (2013). DOI: 10.1142/S0219498812501587.

[2] Conway, J.H., Curtis, C.R., Norton, S.P., Parker, R.A., Wilson,R.A., Atlas of finite groups: Maximal subgroups and ordinary characters forsimple groups, Clarendon Press, Oxford, 1985.

[3] Darafsheh, M.R., and Moghaddamfar, A.R., Characterization of thegroups PSL5(2), PSL6(2), and PSL7(2), Comm. Algebra, 29 (1) (2001),465–475.

212 shitian liu

[4] Frobenius G., Verallgemeinerung des Sylowschen Satze, Berliner Sitz, 1895,981-993.

[5] Hall, M., The theory of groups, Macmilan, New York, 1959.

[6] Herzog, M., Finite simple groups divisible by only three primes, J. Alg., 10(1968), 383-388.

[7] Khatami, M. Khosravi, B., Akhlaghi, Z., A new characterization forsome linear groups, Monatsh Math., 163 (2011), 39-50.

[8] Liu, S., Zhang, R., A new characterization of A12, Math. Sci., 6 (2012),30.

[9] Liu, S., A characterization of L3(4), Science Asia. In press.

[10] Liu, S., NSE characterization of projective special linear group L5(2), Rend.Semin. Mat. Univ. Padova. In press.

[11] Liu, S., A characterization of projective special unitary group U3(5) by nse,Arab J. Math. Sci. In press.

[12] Miller, G., Addition to a theorem due to Frobenius, Bull. Amer. Math.Soc., 11 (1) (1904), 6-7.

[13] Shao, C.G., Shi, W.J., Jiang, Q.H., A new characterization of simpleK3-groups, Adv. Math., 38 (2009), 327-330 (in Chinese).

[14] Shao, C., Shi, W., Jiang, Q., A new characterization of simple K4-groups,Front. Math. China., 3 (3) (2008), 355-370.

[15] Shao, C., Jiang, Q., A new characterization of some linear groups by NSE,J. Algebra Appl. DOI: 10.1142/S0219498813500941.

[16] Shen, R., Shao, C., Jiang, Q., Shi, W., Mazurov, V., A new charac-terization of A5, Monatsh Math., 160 (2010), 337-341.

[17] Shi, W.J., A new characterization of the sporadic simple groups, GroupTheory. In: Proc. of the 1987 Singapore Conf. Walter de Gruyter, pp. 531-540. Berlin, New York, 1989.

[18] Shi, W., On simple K4-group, Chin. Sci. Bul., 36 (17) (1991), 1281-1283 (inChinese).



NEW INEQUALITIES OF HERMITE-HADAMARD TYPEFOR FUNCTIONS WHOSE FIRST DERIVATIVES ABSOLUTEVALUES ARE s-CONVEX

Feixiang Chen

Yuming Feng

School of Mathematics and StatisticsChongqing Three Gorges UniversityWanzhou, Chongqing, 404000P.R. Chinae-mails: [email protected]

[email protected]

Abstract. In this paper, some new inequalities of the left-hand side of Hermite-

Hadamard-type are obtained for functions whose first derivatives absolute values are

s-convex.

Keywords: Hermite-Hadamard’s inequality; s-convex functions; Holder inequality;

power mean inequality.

AMS Subject Classifications: 26D15; 26D10.

1. Introduction

If f : I ⊂ R+ → R+ where R+ = [0,∞) is said to be s-convex on I if the inequality

(1.1) f(αx+ (1− α)y) ≤ αsf(x) + (1− α)sf(y)

holds for all x, y ∈ I and α ∈ [0, 1]. It can be easily seen that for s = 1, s-convexityreduces to ordinary convexity of functions defined on [0,∞).

One of the most famous inequality for the class of convex functions is so calledHermite-Hadamard inequality, which states that: Let f : I ⊂ R → R be a convexfunction on the interval I, then for any a, b ∈ I with a = b we have the followingdouble inequality

(1.2) f(a+ b

2

)≤ 1

b− a

∫ b

a

f(t)dt ≤ f(a) + f(b)

2.

Since then, some refinements of the Hermite-Hadamard inequality on convex func-tions have been extensively investigated by a number of authors (e.g., [1], [3], [4],[7], [8], [9] and [10]).

214 f. chen, y. feng

In [6], Dragomir and Fitzpatrick established a variant of Hermite-Hadamardinequality which holds for the s-convex functions.

Theorem 1.1 Suppose that f : I ⊂ [0,∞) → [0,∞) is an s-convex function inthe second sense, where s ∈ (0, 1] and let a, b ∈ [0,∞), a < b. If f ∈ L[0, 1], thenthe following inequality holds

(1.3) 2s−1f(a+ b

2

)≤ 1

b− a

∫ b

a

f(t)dt ≤ f(a) + f(b)

s+ 1.

Along this paper, we consider a real interval I ⊂ R, and we denote that I0 isthe interior of I.

In [5], Dragomir and Agarwal obtained the following Hermite-Hadamard typeintegral inequality.

Theorem 1.2 Let f : I ⊂ R → R be differentiable mapping on I0, where a, b ∈ Iwith a < b. If |f ′| is convex on [a, b], then the following inequality holds

(1.4)∣∣∣f(a) + f(b)

2− 1

b− a

∫ b

a

f(t)dt∣∣∣ ≤ b− a

8[|f ′(a)|+ |f ′(b)|].

In [2], Alomari, Darus and Kirmaci proved the following inequalities of Hermite-Hadamard type for differentiable convex mappings.

Theorem 1.3 Let f : I ⊂ [0,∞) → R be a differentiable mapping on I0, suchthat f ′ ∈ L[a, b], where a, b ∈ I with a < b. If |f ′| is s-convex on [a, b], for somefixed s ∈ (0, 1], then the following inequality holds:∣∣∣f(a+ b

2

)− 1

b− a

∫ b

a

f(t)dt∣∣∣

≤ b− a

4(s+ 1)(s+ 2)

[|f ′(a)|+ 2(s+ 1)

∣∣∣f ′(a+ b

2

) ∣∣∣+ |f ′(b)|]

≤ (22−s + 1)(b− a)

4(s+ 1)(s+ 2)

[|f ′(a)|+ |f ′(b)|

].

In [12], Pearce and Pecaric proved the following theorem.

Theorem 1.4 Let f : I ⊂ R → R be a differentiable mapping on I0, such thatf ′ ∈ L[a, b], where a, b ∈ I with a < b. If |f ′|q is convex on [a, b], for some q ≥ 1,then the following inequality holds:

(1.5)∣∣∣f(a+ b

2

)− 1

b− a

∫ b

a

f(t)dt∣∣∣ ≤ b− a

4

[ |f ′(a)|q + |f ′(b)|q

2

]1/q.

If |f ′|q is concave on [a, b], for some q ≥ 1, then

(1.6)∣∣∣f(a+ b

2

)− 1

b− a

∫ b

a

f(t)dt∣∣∣ ≤ b− a

4

∣∣∣f ′(a+ b

2

)∣∣∣.

new inequalities of hermite-hadamard type for functions ... 215

For recent results and generalizations concerning Hermite-Hadamard’s in-equality, see [6]-[12] and the references given therein.

In this paper, we establish some new inequalities of Hadamard’s type for theclass of s-convex functions in the second sense.

2. Lemmas

To prove our main results, we consider the following lemma:

Lemma 2.1 Let f : I ⊂ R → R be differentiable mapping on I0, where a, b ∈ Iwith a < b. If f ′ ∈ L[a, b], then the following inequality holds

f(λa+ (1− λ)b)− 1

b− a

∫ b

a

f(t)dt

= (b− a)(1− λ)2∫ 1

0

tf′(t(λa+ (1− λ)b) + (1− t)a

)dt

+(b− a)λ2

∫ 1

0

(t− 1)f′(tb+ (1− t)(λa+ (1− λ)b)

)dt.

for each λ ∈ [0, 1].

Proof. We note that

I1 =

∫ 1

0

tf ′(t(λa+ (1− λ)b) + (1− t)a

)dt

=1

(b− a)(1− λ)tf(t(λa+ (1− λ)b) + (1− t)a

)∣∣∣10

− 1

(b− a)(1− λ)

∫ 1

0

f(t(λa+ (1− λ)b) + (1− t)a

)dt

=1

(b− a)(1− λ)f(λa+ (1− λ)b)

− 1

(b− a)(1− λ)

∫ 1

0

f(t(λa+ (1− λ)b) + (1− t)a

)dt.

Setting x = t(λa+ (1− λ)b) + (1− t)a, and dx = (b− a)(1− λ)dt, which gives

I1 =1

(b− a)(1− λ)f(λa+ (1− λ)b)− 1

(b− a)2(1− λ)2

∫ λa+(1−λ)b

a

f(x)dx.

Similarly, we can show that

I2 =

∫ 1

0

(t− 1)f ′(tb+ (1− t)(λa+ (1− λ)b)

)dt

=1

(b− a)λf(λa+ (1− λ)b)− 1

(b− a)2λ2

∫ b

λa+(1−λ)b

f(x)dx,


and therefore,

I = (b− a)(1− λ)2I1 + (b− a)λ2I2

= f(λa+ (1− λ)b)− 1

b− a

∫ b

a

f(x)dx,

which completes the proof.

Remark 1. Applying Lemma 2.1 for λ =1

2, we get the Lemma 2.1 in [2].

3. The new Hermite-Hadamard type inequalities

Theorem 3.1 Let f : I ⊂ R → R be a differentiable mapping on I0, such thatf ′ ∈ L[a, b], where a, b ∈ I with a < b. If |f ′| is s-convex on [a, b], for some fixeds ∈ (0, 1], then the following inequality holds∣∣∣f(λa+ (1− λ)b)− 1

b− a

∫ b

a

f(t)dt∣∣∣

≤ (b− a)(1− λ)2( 1

(s+ 1)(s+ 2)|f ′(a)|+ 1

s+ 2|f ′(λa+ (1− λ)b)|

)+(b− a)λ2

( 1

(s+ 1)(s+ 2)|f ′(b)|+ 1

s+ 2|f ′(λa+ (1− λ)b)|

),


Proof. From Lemma 2.1, we have∣∣∣f(λa+ (1− λ)b)− 1

b− a

∫ b

a

f(t)dt∣∣∣

≤ (b− a)(1− λ)2∫ 1

0

t|f ′(t(λa+ (1− λ)b) + (1− t)a

)|dt

+(b− a)λ2

∫ 1

0

(1− t)|f ′(tb+ (1− t)(λa+ (1− λ)b)

)|dt

Because |f ′| is s-convex, we have∣∣∣f(λa+ (1− λ)b)− 1

b− a

∫ b

a

f(t)dt∣∣∣

≤ (b− a)(1− λ)2∫ 1

0

t(ts|f ′(λa+ (1− λ)b)|+ (1− t)s|f ′(a)|

)dt

+(b− a)λ2

∫ 1

0

(1− t)(ts|f ′(b)|+ (1− t)s|f ′(λa+ (1− λ)b)|

)dt

= (b− a)(1− λ)2( 1

(s+ 1)(s+ 2)|f ′(a)|+ 1

s+ 2|f ′(λa+ (1− λ)b)|

)+(b− a)λ2

( 1

(s+ 1)(s+ 2)|f ′(b)|+ 1

s+ 2|f ′(λa+ (1− λ)b)|

),

which completes the proof.


Remark 2. Applying Theorem 3.1 for λ =1

2, we get the result in Theorem 1.3.

Theorem 3.2 Let f : I ⊂ R → R be a differentiable mapping on I0, such thatf ′ ∈ L[a, b], where a, b ∈ I with a < b. If |f ′|p/(p−1) is s-convex on [a, b], for somefixed s ∈ (0, 1], p > 1, then the following inequality holds∣∣∣f(λa+ (1− λ)b)− 1

b− a

∫ b

a

f(t)dt∣∣∣

≤ (b− a)( 1

p+ 1

)1/p( 1

s+ 1

)1/q((1− λ)2

[(λs + 1)|f ′(a)|q + (1− λ)s|f ′(b)|q

]1/q+λ2

[λs|f ′(a)|q + ((1− λ)s + 1)|f ′(b)|q

]1/q),

for each λ ∈ [0, 1] and p is the conjugate of q, q = p/(p− 1).

Proof. From Lemma 2.1 and using the Holder inequality, we have∣∣∣f(λa+ (1− λ)b)− 1

b− a

∫ b

a

f(t)dt∣∣∣

≤ (b− a)(1− λ)2∫ 1

0

t|f ′(t(λa+ (1− λ)b) + (1− t)a

)|dt

+(b− a)λ2∫ 1

0(1− t)|f ′

(tb+ (1− t)(λa+ (1− λ)b)

)|dt

≤ (b− a)(1− λ)2(∫ 1

0

tpdt)1/p(∫ 1

0

∣∣∣f ′(t(λa+ (1− λ)b) + (1− t)a

)∣∣∣qdt)1/q+(b− a)λ2

(∫ 1

0

(1− t)pdt)1/p(∫ 1

0

∣∣∣f ′(tb+ (1− t)(λa+ (1− λ)b)

)∣∣∣qdt)1/q.Because |f ′|q is s-convex, we have∫ 1

0

∣∣∣f ′(t(λa+ (1− λ)b) + (1− t)a

)∣∣∣qdt ≤ |f ′(λa+ (1− λ)b)|q + |f ′(a)|q

s+ 1,

and ∫ 1

0

∣∣∣f ′(tb+ (1− t)(λa+ (1− λ)b)

)∣∣∣qdt ≤ |f ′(b)|q + |f ′(λa+ (1− λ)b)|q

s+ 1.

Therefore, we have∣∣∣f(λa+ (1− λ)b)− 1

b− a

∫ b

a

f(t)dt∣∣∣

≤ (b− a)( 1

p+ 1

)1/p( 1

s+ 1

)1/q((1− λ)2

[|f ′(λa+ (1− λ)b)|q + |f ′(a)|q

]1/q+λ2

[|f ′(λa+ (1− λ)b)|q + |f ′(b)|q

]1/q).


Now, since |f ′|q is s-convex on [a,b], for any λ ∈ [0, 1], then by (1) we have

(3.1) |f ′(λa+ (1− λ)b)|q ≤ λs|f ′(a)|q + (1− λ)s|f ′(b)|q.

Combining all the above inequalities, we obtain

∣∣∣f(λa+ (1− λ)b)− 1

b− a

∫ b

a

f(t)dt∣∣∣

≤ (b−a)( 1

p+1

)1/p( 1

s+1

)1/q((1−λ)2

[λs|f ′(a)|q+(1−λ)s|f ′(b)|q+|f ′(a)|q

]1/q+λ2

[λs|f ′(a)|q + (1− λ)s|f ′(b)|q + |f ′(b)|q

]1/q).

This proves the theorem.

Theorem 3.3 Let f : I ⊂ R → R be a differentiable mapping on I0, such thatf ′ ∈ L[a, b], where a, b ∈ I with a < b. If |f ′|q is s-convex on [a, b], for some fixeds ∈ (0, 1], q ≥ 1, then the following inequality holds

∣∣∣f(λa+ (1− λ)b)− 1

b− a

∫ b

a

f(t)dt∣∣∣

≤ (b− a)( 1

(s+ 1)(s+ 2)

)1/q(12

)1− 1q((1− λ)2

[((s+ 1)λs + 1)|f ′(a)|q

+(s+ 1)(1− λ)s|f ′(b)|q]1/q

+λ2[(s+ 1)λs|f ′(a)|q + ((s+ 1)(1− λ)s + 1)|f ′(b)|q

]1/q),


Proof. From Lemma 2.1 and using the using the well-known power-mean inequa-lity, we have

∣∣∣f(λa+ (1− λ)b)− 1

b− a

∫ b

a

f(t)dt∣∣∣

≤ (b− a)(1− λ)2∫ 1

0

t|f ′(t(λa+ (1− λ)b) + (1− t)a)|dt

+(b− a)λ2

∫ 1

0

(1− t)|f ′(tb+ (1− t)(λa+ (1− λ)b))|dt

≤ (b− a)(1− λ)2(∫ 1

0

tdt)1− 1

q(∫ 1

0

t∣∣∣f ′(t(λa+ (1− λ)b) + (1− t)a)

∣∣∣qdt)1/q+(b− a)λ2

(∫ 1

0

(1− t)dt)1− 1

q(∫ 1

0

(1− t)|f ′(tb+ (1− t)(λa+ (1− λ)b))|qdt)1/q

.


Because |f ′|q is s-convex, by (1) we have∫ 1

0

t∣∣∣f ′(t(λa+ (1− λ)b) + (1− t)a

)∣∣∣qdt≤∫ 1

0

(ts+1|f ′(λa+ (1− λ)b)|q + t(1− t)s|f ′(a)|q)dt

=1

(s+ 1)(s+ 2)|f ′(a)|q + 1

s+ 2|f ′(λa+ (1− λ)b)|q,

and ∫ 1

0

(1− t)∣∣∣f ′(tb+ (1− t)(λa+ (1− λ)b)

)∣∣∣qdt≤∫ 1

0

((1− t)ts|f ′(b)|q + (1− t)s+1|f ′(λa+ (1− λ)b)|q

)dt

=1

(s+ 1)(s+ 2)|f ′(b)|q + 1

s+ 2|f ′(λa+ (1− λ)b)|q.

Therefore, we have∣∣∣f(λa+ (1− λ)b)− 1

b− a

∫ b

a

f(t)dt∣∣∣

≤ (b−a)( 1

(s+1)(s+2)

)1/q(12

)1− 1q((1−λ)2[(s+1)|f ′(λa+(1−λ)b)|q+|f ′(a)|q]1/q

+λ2[(s+ 1)|f ′(λa+ (1− λ)b)|q + |f ′(b)|q]1/q).

Now, since |f ′|q is s-convex on [a,b], for any λ ∈ [0, 1], then by (1) we have

(3.2) |f ′(λa+ (1− λ)b)|q ≤ λs|f ′(a)|q + (1− λ)s|f ′(b)|q.Combining all the above inequalities, we have∣∣∣f(λa+ (1− λ)b)− 1

b− a

∫ b

a

f(t)dt∣∣∣

≤ (b− a)( 1

(s+ 1)(s+ 2)

)1/q(12

)1− 1q((1− λ)2

[((s+ 1)λs + 1)|f ′(a)|q

+(s+ 1)(1− λ)s|f ′(b)|q]1/q

+λ2[(s+ 1)λs|f ′(a)|q + ((s+ 1)(1− λ)s + 1)|f ′(b)|q

]1/q),

we get the desired result.

Theorem 3.4 Let f : I ⊂ R → R be a differentiable mapping on I0, such thatf ′ ∈ L[a, b], where a, b ∈ I with a < b. If |f ′|q is s-concave on [a, b], for some fixeds ∈ (0, 1], q > 1, then the following inequality holds∣∣∣f(λa+ (1− λ)b)− 1

b− a

∫ b

a

f(t)dt∣∣∣

≤ (b−a)( q−1

2q−1

)1− 1q2

s−1q

((1−λ)2

∣∣∣f ′((1+λ)a+(1−λ)b

2

)∣∣∣+λ2∣∣∣f ′(

λa+(2−λ)b

2)∣∣∣),



Proof. From Lemma 2.1 and using the Holder inequality, we have∣∣∣f(λa+ (1− λ)b)− 1

b− a

∫ b

a

f(t)dt∣∣∣

≤ (b− a)(1− λ)2∫ 1

0

t|f ′(t(λa+ (1− λ)b) + (1− t)a

)|dt

+(b− a)λ2∫ 1

0(1− t)|f ′

(tb+ (1− t)(λa+ (1− λ)b)

)|dt

≤ (b− a)(1− λ)2(∫ 1

0

tpdt)1/p(∫ 1

0

∣∣∣f ′(t(λa+ (1− λ)b) + (1− t)a

)∣∣∣qdt)1/q+(b− a)λ2

(∫ 1

0

(1− t)pdt)1/p(∫ 1

0

∣∣∣f ′(tb+ (1− t)(λa+ (1− λ)b)

)∣∣∣qdt)1/q.Because |f ′|q is s-concave, by the reversed direction of (3) we have∫ 1

0

∣∣∣f ′(t(λa+ (1− λ)b) + (1− t)a

)∣∣∣qdt ≤ 2s−1∣∣∣f ′((1 + λ)a+ (1− λ)b

2

)∣∣∣q,and ∫ 1

0

∣∣∣f ′(tb+ (1− t)(λa+ (1− λ)b)

)∣∣∣qdt ≤ 2s−1∣∣∣f ′(λa+ (2− λ)b

2

)∣∣∣q,so∣∣∣f(λa+ (1− λ)b)− 1

b− a

∫ b

a

f(t)dt∣∣∣

≤ (b−a)( 1

p+1

)1/p2

s−1q

((1−λ)2

∣∣∣f ′((1+λ)a+(1−λ)b

2

)∣∣∣+λ2∣∣∣f ′(λa+(2−λ)b

2

)∣∣∣)= (b−a)

( q−1

2q−1

)1− 1q2

s−1q

((1−λ)2

∣∣∣f ′((1+λ)a+(1−λ)b

2

)∣∣∣+λ2∣∣∣f ′(

λa+(2−λ)b

2)∣∣∣),

which yields the desired result.

Remark 3. Applying Theorem 3.4 for λ =1

2, we get

∣∣∣f(a+ b

2

)− 1

b− a

∫ b

a

f(t)dt∣∣∣

≤ b− a

4

( q − 1

2q − 1

)1− 1q2

s−1q

(∣∣∣f ′(3a+ b

4

)∣∣∣+ ∣∣∣f ′(a+ 3b

4

)∣∣∣)≤ b− a

4

( q − 1

2q − 1

)1− 1q(∣∣∣f ′

(3a+ b

4

)∣∣∣+ ∣∣∣f ′(a+ 3b

4

)∣∣∣),which is an improved result comparing with Theorem 2.5 in [2].


4. Applications to special means

Now, using the results of Section 3, we give some applications to special means ofreal numbers.

We shall consider the means for arbitrary real numbers α, β (α = β).

(1) Weighted mean

W (α, β) = λα+ (1− λ)β, α, β ∈ R, λ ∈ [0, 1].

(2) Generalized log-mean:

Ln(α, β) =

[βn+1 − αn+1

(n+ 1)(β − α)

] 1n

, n ∈ Z\0, 1, α, β ∈ R, (α = β).

Therefore, by applying the s-convex mapping f : [0, 1] → [0, 1], f(x) = xs, thefollowing inequalities hold:

Proposition 4.1 Let a, b ∈ I0, a < b and 0 < s < 1. Then, we have

|W s(a, b)− Lss(a, b)|

≤ (b− a)(1− λ)2( s

(s+ 1)(s+ 2)|a|s−1 +

s

s+ 2|λa+ (1− λ)b|s−1

)+(b− a)λ2

( s

(s+ 1)(s+ 2)|b|s−1 +

s

s+ 2|λa+ (1− λ)b|s−1

),


Proposition 4.2 Let a, b ∈ I0, a < b and 0 < s < 1. Then, for all q > 1, wehave

|W s(a, b)− Lss(a, b)|

≤ s(b− a)(

1p+1

)1/p(1

s+1

)1/q((1− λ)2

[(λs + 1)|a|q(s−1) + (1− λ)s|b|q(s−1)

]1/q+λ2

[λs|a|q(s−1) + ((1− λ)s + 1)|b|q(s−1)

]1/q),


Acknowledgments. This work is supported by the Youth Project of CTGU(Grant No.13QN11) and the Scientific and Technological Research Program ofCMEC (Grant Nos. KJ1401006, KJ1401019).


References

[1] Abramovich, S., Farid, G., Pecaric, J., More about Hermite-Hadamardinequalities, Cauchy’s Means, and superquadracity, J. Inequal. Appl., 2010,2010:102467.

[2] Alomari, M.W., Darus, M., Kirmaci, U.S., Some inequalities ofHermite-Hadamard type for s-convex functions, Acta Math. Sci., 4 (31)(2011), 1643-1652.

[3] Barani, A., Barani, S., Dragomir, S.S., Refinements of Hermite-Hadamard inequalities for functions when a power of the absolute value ofthe second derivative is P-convex, J. Appl. Math., 2012, 2012:615737.

[4] Bessenyei, M., Pales, Z., Hadamard-type inequalities for generalized con-vex functions, Math. Inequal. Appl., 6 (3) (2003), 379-392.

[5] Dragomir, S.S., Agarwal, R.P., Two inequalities for differentiable map-pings and applications to special means of real numbers and to trapezoidalformula, Appl. Math. Lett., 11 (5) (1998), 91-95.

[6] Dragomir, S.S., Fitzpatrick, S., The Hadamard inequalities for s-convexfunctions in the second sense, Demonstration Math., 32 (4) (1999), 687-696.

[7] Dragomir, S.S., Pearce, C.E.M., Selected Topics on Hermite-HadamardInequalities and Applications, RGMIA Monographs, Victoria University,2000.

[8] Farissi, A.E., Simple proof and refinement of Hermite-Hadamard inequality,J. Math. Inequal., 4 (3) (2010), 365-369.

[9] Gao, X., A note on the Hermite-Hadamard inequality, J. Math. Inequal., 4(4) (2010), 587-591.

[10] Kavurmaci, H., Avci, M., Ozdemir, M.E., New inequalities of Hermite-Hadamard type for convex functions with applications, J. Inequal. Appl., 2011,2011:86.

[11] Kirmaci, U.S., Bakula, M.K., Ozdemir, M.E., Pecaric, J.,Hadamard-type inequalities for s-convex functions, Appl. Math. Comput.,1 (193) (2007), 26-35.

[12] Pearce, C.E.M., Pecaric, J., Inequalities for differentiable mappingswith application to special means and quadrature formula, Appl. Math. Lett.,2 (13) (2000), 51-55.



ON THOMPSON’S CONJECTURE FOR Aut(J2) AND Aut(McL)

Yanheng Chen

Yuming Feng

School of Mathematics and StatisticsChongqing Three Gorges UniversityChongqing 404100P.R. Chinae-mails: math [email protected]

[email protected]

Guiyun Chen

School of Mathematics and StatisticsSouthwest UniversityChongqing 400715P.R. Chinae-mail: [email protected]

Abstract. Let G be a finite group and N(G) be the set of conjugacy class sizes of G.

In 1980s J. G. Thompson conjectured: If G is a finite group with trivial center and S is

a non-abelian finite simple group satisfying that N(G) = N(S), then G ∼= S. Here, we

generalize the conjecture to the automorphism groups of J2 and McL. As a corollary

this result extends the conjecture to all almost sporadic simple groups.

Keywords: finite group, almost sporadic simple groups, conjugacy class sizes, Thomp-

son’s conjecture.

AMS Mathematics Subject Classification (2010): 20D08, 20D60.

1. Introduction

All groups considered in this paper are finite and simple groups mentioned arenon-abelian.

Let G be a group. For an element x ∈ G, we denote by xG the conju-gacy class of x in G and N(G) the set of conjugacy class sizes of G, that is,N(G) = |xG|

∣∣ x ∈ G. In 1987, J.G. Thompson posed the following conjecture(ref. to [9], Problem 12.38):

Thompson’s conjecture. Let G be a group with Z(G) = 1 and S is a simplegroup satisfying that N(G) = N(S), then G ∼= S.

224 y. chen, y. feng, g. chen

The prime graph of a group G is defined as follows, whose vertices are theprime divisors of |G| and two distinct primes p and q are joined by an edge if andonly if G contains an element of order pq. J.S. Williams in [17] and A.S. Kon-dratiev in [18] obtained the classification of simple groups with disconnected primegraph. Based on these results, in 1994 G.Y. Chen proved that Thompson’s conjec-ture is true for all finite simple groups with disconnected prime graph in [1], partsof the proof were given in [2], [3], [4]. The method used by G.Y. Chen requires agroup with non-connected prime graph and several authors have worked on suchgroups in the same way as Chen.

For the simple groups with connected prime graph, there is not any progresson Thompson’s conjecture for a long time. Until 2009, A.V. Vasil’ev first dealtwith the groups with connected prime graph and proved that Thompson’s conjec-ture holds for A10 and L4(4) (see [5]). Jiang Qinhui et al in [7] and Gerald Pientkain [6] also independently gave a positive answer to Thompson’s conjecture for A10

in the year of 2011 and 2012, respectively. Note that Gerald Pientka’s work doesnot use the classification theorem of simple groups. In 2011, N. Ahanjideh in[8] proved that Thompson’s conjecture is true for some projective special lineargroups PSLn(q) which have connected prime graphs. Recently, I.B. Gorshkovin [9] established the validity of Thompson’s conjecture for U4(4), U4(5), S6(4),O+

8 (4), and A16.

It is worth to mention that some authors generalize Thompson’s conjectureto some almost simple groups, that is, a group M is said to be an almost simplerelated to S if and only if S ≤ M ≤Aut(S) for some simple group S. Forexample, in 2002 A. Khosravi and B. Khosravi in [11] generalized Thompson’sconjecture to almost sporadic simple groups except Aut(J2) and Aut(McL). In2005, S.H. Alavi and A. Daneshkhah in [16] generalized Thompson’s conjecture tosymmetric groups Sn, where n = p, p+1, and p is an odd prime number. In 2011,B. Khosravi and M. Khatami in [12], [13] established the validity of Thompson’sconjecture for the general projective groups PGL(2, q). It is pity that they stillneed to assume that the groups discussed have the non-connected prime graphs.Surely, it is an interesting topic to check Thompson’s conjecture for some almostsimple groups which have the connected prime graphs. In this paper, we give apositive answer to Thompson’s conjecture for Aut(J2) and Aut(McL), both ofwhich have the connected prime graphs. Further more, we get that Thompson’sconjecture can be generalized to all the almost sporadic simple groups.

Our main contribution is the following theorem and we will give the proofsin Sections 3 and 4.

Main Theorem. Let G be a group with Z(G) = 1 and M one of Aut(J2) andAut(McL) satisfying that N(G) = N(M), then G ∼= M .

Let M be an almost simple group related to sporadic simple group S. Then,by [14], we have that |Aut(S) : S| ≤ 2 such that M = S or Aut(S). G.Y. Chenproved Thompson’s conjecture for all sporadic simple groups in [2]. A. Khosraviand B. Khosravi in [11] proved Thompson’s conjecture for other almost sporadic

on thompson’s conjecture for Aut(J2) and Aut(McL) 225

simple groups except Aut(J2) and Aut(McL). Therefore, by our main conclusionthe following corollary holds.

Corollary. Thompson’s conjecture holds for all the almost sporadic simple groups.

For a group G, we denote by π(G) the set of prime divisors of |G| and denoteby Soc(G) the socle of G which is a subgroup generated by all minimal normalsubgroups of G. In addition, we also denote by π(n) the set of all primes dividingn where n is a positive integer, and then nπ to denote π-part of n for π ⊆ π(n).The other notation and terminologies in this paper are standard and the readeris referred to [14] and [19] if necessary.

2. Preliminaries

First, we cite here some known results which are useful in the sequel.

Lemma 2.1. ([5], Lemma 5, and [9], Lemma 1.4]) Let K be a normal subgroupof G and G = G/K. Then

(1) If x is the image of an element x of G in the group G, then |xK |∣∣|xG| and

|xG|∣∣|xG|.

(2) If x ∈ G and (|x|, |K|) = 1, then CG(x) = CG(x)K/K.

(3) If x, y ∈ G, (|x|, |y|) = 1, and xy = yx, then CG(xy) = CG(x) ∩ CG(y).

Lemma 2.2. ([5], Lemma 4) Let G be a group with trivial center, p ∈ π(G) andp2 not divide n for any n ∈ N(G). Then a Sylow p-subgroup of G is elementaryabelian.

Lemma 2.3. ([9], Lemma 1.10) Let a Sylow p-subgroup of G be of order p, xbe an element of order p, and |xG| be a number that is maximal with respect todivisibility in N(G). Then CG(x) is an abelian group.

Lemma 2.4. ([9], Lemma 1.9) Let G be a group, and p and q be numbers inπ(G). If G satisfies the following conditions:

(a) N(G) contains no number divisible by p2 or q2;

(b) N(G) contains no number except 1 co-prime to pq;

(c) N(G) contains a number hq such that any n in N(G) not divisible by q doesnot divide hq and N(G) contains no number divisible by hq and n;

(d) N(G) contains a number hp such that any l in N(G) not divisible by p doesnot divide hp and N(G) contains no number divisible by hp and l.


Then, the Sylow p-subgroups and q-subgroups of G are cyclic groups of primeorder. In addition, G has no element of order pq.

Lemma 2.5. ([9], Lemma 1.11) Let G be a group such that π(G) has no numbergreater than 17, and there are numbers p and q in π(G) such that:

(a) p and q are nonadjacent in the prime graph of G;

(b) Sylow p-subgroups and q-subgroups of G have prime orders;

(c) q − 1 is not divisible by p, while p− 1 is not divisible by q ;

(d) the centralizer of any element of order p or q is abelian;

(e) p and q are greater than 5.

Then G is insoluble and possesses a unique non-abelian composition factor S,whose order is divisible by pq. Moreover, If K is the soluble radical of G, thenS ≤ G/K ≤Aut(S).

Lemma 2.6. ([9], Lemma 1.12) Let G be a group, K the soluble radical of G,and G/K = S a simple non-abelian group. Suppose that there exists a prime psuch that p ∈ π(G) \ π(K). Assume that an element g of order p of G satisfiesthe following conditions:

(a) |gG| = |gS|, where g is the image of an element g in the group S;

(b) the number |gG| is maximal with respect to divisibility in N(G);

(c) the subgroup CG(g) is abelian.

Then K ≤ Z(G).

By [14], we know that |Out(J2)| = |Out(McL)| = 2 such thatM is isomorphicto Aut(J2) = J2 : 2 or Aut(McL) = McL : 2. Information on the set N(M) andthe order of M are given in the next two lemmas is obtained via [14].

Lemma 2.7. Let M ∼= J2 : 2. Then

(1) |M | = 28 · 33 · 52 · 7;

(2) N(M) = n1 = 1, n2 = 32 · 5 · 7, n3 = 23 · 32 · 5 · 7, n4 = 24 · 5 · 7,n5 = 25 · 3 · 52 · 7, n6 = 22 · 32 · 52 · 7, n7 = 26 · 32 · 7, n8 = 27 · 33 · 7,n9 = 24 · 32 · 52 · 7, n10 = 25 · 32 · 52 · 7, n11 = 27 · 33 · 52, n12 = 24 · 33 · 52 · 7,n13 = 26 · 33 · 5 · 7, n14 = 27 · 33 · 5 · 7, n15 = 28 · 32 · 5 · 7, n16 = 23 · 32 · 52,n17 = 23 · 32 · 52 · 7, n18 = 26 · 32 · 52 · 7, n19 = 23 · 33 · 52 · 7.

Especially,

(3) N(M) contains no number other than n11 and n16 not divisible by 7;


(4) N(M) contains no number divided by 72;

(5) n4 is only one in N(M) not divisible by 3.

Lemma 2.8. Let M ∼= McL : 2. Then

(1) |M | = 28 · 36 · 53 · 7 · 11;

(2) N(M) = n1 = 1, n2 = 34 · 52 · 11, n3 = 24 · 52 · 7 · 11, n4 = 25 · 3 · 53 · 7 · 11,n5 = 22·35·53·7·11, n6 = 26·35·7·11, n7 = 27·36·5·7·11, n8 = 24·34·52·7·11,n9 = 25 · 34 · 53 · 7 · 11, n10 = 27 · 36 · 53 · 11, n11 = 24 · 36 · 53 · 7 · 11,n12 = 28 · 33 · 53 · 7 · 11, n13 = 26 · 35 · 52 · 7 · 11, n14 = 27 · 36 · 53 · 7,n15 = 25 · 35 · 53 · 7 · 11, n16 = 27 · 35 · 52 · 7 · 11, n17 = 23 · 34 · 52 · 7,n18 = 23 · 34 · 52 · 7 · 11, n19 = 26 · 34 · 53 · 7 · 11, n20 = 23 · 35 · 53 · 7 · 11,n21 = 23 · 36 · 53 · 7 · 11, n22 = 27 · 36 · 52 · 7 · 11, n23 = 26 · 36 · 52 · 7 · 11.

In particular,



(5) For any n ∈ N(M) and p ∈ 7, 11, it follows that p2 | n;

(6) For any n ∈ N(M), either 7 or 11 divides n.

Lemma 2.9. Let M be one of J2 : 2 and McL : 2, and G be a group with trivialcenter. If N(G) = N(M), then |M |

∣∣|G| and π(M) = π(G).

Proof. Since the number in N(G) divides |G|, under the hypothesis we see that|M |

∣∣|G| by Lemmas 2.7 and 2.8. π(M) = π(G) is the result of Lemma 1.2.1 in [1]or Lemma 3 in [5].

Let S be a simple group with π(S) ⊆ 2, 3, 5, 7, 11. Up to isomorphism, thereare finitely many finite non-abelian simple groups S with π(S) ⊆ 2, 3, 5, 7, 11.Using the classification of finite simple groups one can list them all. For conve-nience, we list all the cases of S as well as the orders of S, the orders of the outerautomorphism of S in Table 1. Especially, we have that π(Out(S)) ⊆ 2, 3.

3. Proof of Main Theorem for J2 : 2

In this section, we prove the Main Theorem for J2 : 2 according to the informationin Lemma 2.7.

Theorem 3.1. Let M ∼= J2 : 2 and G be a group with trivial center. IfN(G) = N(M), then G ∼= M .

Proof. We divide the proof of this theorem into eight steps.


Table 1. Non-abelian simple groups S with π(S) ⊆ 2, 3, 5, 7, 11

S Order of S |Out(S)| S Order of S |Out(S)|A5 22 · 3 · 5 2 A9 26 · 34 · 5 · 7 2

L2(7) 23 · 3 · 7 2 J2 27 · 33 · 52 · 7 2A6 23 · 32 · 5 22 U3(5) 24 · 32 · 53 · 7 |S3|

L2(8) 23 · 32 · 7 3 S6(2) 29 · 34 · 5 · 7 1A7 23 · 32 · 5 · 7 2 U4(3) 27 · 36 · 5 · 7 |D8|

U3(3) 25 · 33 · 7 2 S4(7) 28 · 32 · 52 · 74 2A8 26 · 32 · 5 · 7 2 A10 27 · 34 · 52 · 7 2

L3(4) 26 · 32 · 5 · 7 |D12| O+8 (2) 212 · 35 · 52 · 7 |S3|

U4(2) 26 · 34 · 5 2 L2(49) 24 · 3 · 52 · 72 22

M12 26 · 33 · 5 · 11 2 M22 27 · 32 · 5 · 7 · 11 2M11 24 · 32 · 5 · 11 1 A11 27 · 34 · 52 · 7 · 11 2

L2(11) 22 · 3 · 5 · 11 2 HS 29 · 35 · 53 · 7 · 11 2U5(2) 210 · 35 · 5 · 11 2 A12 29 · 35 · 52 · 7 · 11 2McL 27 · 36 · 53 · 7 · 11 2 U6(2) 215 · 36 · 5 · 7 · 11 |S3|

Step 1. The Sylow 7-subgroup P of G is order of 7.Using Lemma 2.2 and (2) of Lemma 2.7, we derive that P is elementary

abelian. Assume that 72 divides G. Since N(G) = N(M), the centralizer of everyelement of G contains an element of order 7 by (4) of Lemma 2.7. Consider anelement y of G such that |yG| = n8 = 27 · 33 · 7.

Suppose that 7 does not divide |y|. Let x be an element of order 7 in CG(y).Then by (3) of Lemma 2.1, CG(xy) = CG(x)∩CG(y), and so lcm(|xG|, |yG|) divides|(xy)G|. Since P is abelian, CG(x) includes P up to conjugacy. Hence 7 does notdivide |xG|. It follows that |xG| is equal to n11 = 27 · 33 · 52 or n16 = 23 · 32 · 52. Inboth cases, 27 · 33 · 52 · 7 divides |(xy)G|, which is impossible by (2) of Lemma 2.7.

Suppose that 7 divides |y|. Let |y| = 7t. Since P is elementary abelian,one has that gcd(7, t) = 1. Put u = y7 and v = yt. Then y = uv, and soCG(uv) = CG(u)∩CG(v) by (3) of Lemma 2.1. Therefore |vG| divides |yG| = n8 =27 · 33 · 7. On the other hand, the element v is order of 7, and thus |vG| is equalto n11 = 27 · 33 · 52 or n16 = 23 · 32 · 52, a contradiction.

Hence P has order of 7.

Step 2. For an element x ∈ G of order 7, it follows that |xG| = n11 = 27 · 33 · 52and CG(x) is abelian.

For any 1 = y ∈ CG(x), since 7 ∥ |G| by Step 1, one has that 7 ∣∣ |yG|, and

hence |yG| = n11 or n16 by (2) of Lemma 2.7. Assume that |xG| = n16 = 23 · 32 · 52and let H be a Sylow 3-subgroup of CG(x). Then H is a nontrivial group of order|G|3/32 by Lemma 2.9. Hence there exists a 3-subgroup K of G such that H isa normal subgroup of K and |K/H| = 3. Then 1 = H ∩ Z(K) ≤ CG(x). Take1 = h ∈ H ∩ Z(K), we have that K ≤ CG(h), and so |hG|3 ≤ 3. But |hG| = n11

or n16, a contradiction. It follows that |xG| = n11 = 27 · 33 · 52.Since n11 is maximal with respect to divisibility in N(G), Lemma 2.3 implies

that the group CG(x) is abelian.


Step 3. Suppose that q ∈ 2, 3, 5, Q is a Sylow q-subgroup of G, and Z(Q) isits center. Let x ∈ Z(Q), then 7

∣∣ |CG(x)|.Let x ∈ Z(Q). Then q does not divide |xG|, and so by Lemma 2.7 we have

that |xG| = n2 = 32 · 5 · 7 while q = 2, |xG| = n4 = 24 · 5 · 7 while q = 3, and |xG|is equal to n7 = 26 · 32 · 7 or n8 = 27 · 33 · 7 while q = 5. The Step 3 follows.

Step 4. G is non-soluble and O2, 2′(G) = O2(G).Let K = O2(G), G = G/K, and denote by x the images of an element x of G

in G. If the statement is false, then there exists r ∈ 3, 5, 7 such that Or(G) = 1.If O7(G) = 1, then |O7(G)| = 7 by Step 1. Let y be an element of the center

Z(Q) of a Sylow 5-subgroup Q. Obviously, the subgroup O7(G)⟨y⟩ is cyclic. Hence7 divides |CG(y)|. Since (5, |K|) = 1, Lemma 2.1(2) implies that 7 divides |CG(y)|,which is impossible by Step 3. Thus, O7(G) = 1.

Let q ∈ 3, 5, and Q be a Sylow q-subgroup of G. If Oq(G) = 1, thenV = Z(Oq(G) is a nontrivial normal subgroup of G. If x is an element of order 7in G, then V = CV (x) × [V, x] such that x acts fixed-point freely on [x, V ], and

then |[x, V ]| − 1 is divisible by 7. Lemma 2.1 (1) implies that |xG| is a divisorof 27 · 33 · 52, and hence |V : CV (x)| divides 52 or 33 so that |[V, x]| ≤ q3. Butn = 6 is the least number such that 7 divides qn − 1, we have that [V, x] = 1 andV = CV (x). On the other hand, the center Z(Q) of a Sylow subgroup Q has anontrivial intersection with V . Let z be of order q from this intersection. Since(|K|, q) = 1, there exists a pre-image z of z in G such that z lies in the centerof a Sylow q-subgroup of G by Lemma 2.1 (2). Further, the centralizer of z alsocontains an element of order 7, which contradicts Step 3. Therefore, Oq(G) = 1.The Step 4 follows.

Step 5. Let K = O2(G), G = G/K. Then, every minimal normal subgroup of Gis non-soluble. Especially, Soc(G)EG .Aut(Soc(G)).

Let N be any minimal normal subgroup of G and assume that N is sol-vable. Then N is an elementary abelian p-group for some p ∈ 3, 5, 7, andso N ≤ Op(G). It follows that Op(G) is nontrivial, contradicts Step 4. Hencea very minimal normal subgroup of G is non-soluble. Let N1, N2, ..., Ns be allminimal normal subgroups of G, where s is a positive integer. Then Soc(G) =N1 × N2 × · · · × Ns. We assert that CG(Soc(G)) = 1. Otherwise, there exists aminimal normal subgroup N of G so that N ≤ CG(Soc(G)) ∩ Soc(G). Thus Nis an abelian group, a contradiction. By N/C theorem, we have Soc(G) E G =G/CG(Soc(G)) .Aut(Soc(G)), as desired.

Step 6. Let L = Soc(G). Then L is a non-abelian simple group and 7∣∣|L|.

By Step 5, we can have that L = S1 × S2 × · · · × Sk is a direct product ofnon-abelian simple groups of S1, S2, . . . , and Sk. Let g be an element of order7 of G and suppose that 7 /∈ π(L). Then g is of order 7 in G and inducesa nontrivial outer automorphism of the group L. Suppose that there exists isuch that Sg

i = Si. Without loss of generality, we assume that i = 1. Let

H = ⟨s |s = s1sg1s

g2

1 · · · sg6

1 , s1 ∈ S1⟩. Then H lies in the centralizer of the


element g and is isomorphic to S1, but the centralizer of g is abelian by Step 2,a contradiction. Hence g induces a nontrivial outer automorphism of the groupSi such that 7

∣∣|Out(Si)|. In view of π(Si) ⊆ π(G) = 2, 3, 5, 7 and by Table 1,the prime divisors of |Out(Si)| are not greater than 5, a contradiction. Therefore,7∣∣|L|.

If n > 1 and g ∈ Sj, then Si ≤ CG(g) for any 1 ≤ j ≤ n, j = i. On the otherhand, CG(g) is abelian by Step 2, a contradiction. Therefore, n = 1, and so L isa non-ableian simple group and 7

∣∣|L|.Step 7. L ∼= J2.

By Step 6 and Step 1, we have that L is a non-ableian simple group such that7 ∥ |M |. Then, by Table 1, L may be isomorphic to one of the following groups:

L2(7), L2(8), U3(3), A7, A8, L3(4), U3(5), A9, A10, U4(3), J2, S6(2), O+8 (2).

Hence, by Table 1 and Step 5, we see that π(Out(L)) ⊆ 2, 3 and LEG .Aut(L).Since K = O2(G) and |M |

∣∣|G|, we have that 52 divides |L|, which implies that Lcan be only isomorphic to one of the following groups:

U3(5), A10, J2, O+8 (2).

If L ∼= U3(5) or O+8 (2), then let x be an element of order 7 in G and x be its

image in G, of course x ∈ L, |xG| is a multiple of 53 or 35, so is |xG| by (1) ofLemma 2.1. This contradicts (2) of Lemma 2.7.

Assume that L ∼= A10. Then, there exists an element w of order 5 in L suchthat |wL| is a multiple of 34, so is |wG| by (2) of Lemma 2.1, which implies thatN(G) has one number divided by 34. This is impossible by (2) of Lemma 2.7.Hence, L must be isomorphic to J2.

Step 8. G ∼= M .Let x be an element of order 7 in G and x be its image in G. It is clear that

x ∈ L. By Lemma 2.1 and [14], we have that |xL| = |xG| = |xG| = 27 · 33 · 52such that K ≤ CG(x). If K = 1, then x centralizes an element from the center ofa Sylow 2-subgroup of G, which is impossible by Step 3. Hence G = G. Recallthat L E G .Aut(L). Then G ∼= J2 or J2 : 2 by Table 1. If G ∼= J2, then|N(G)| = |N(J2)|, a contradiction by [14]. Therefore G ∼= J2 : 2 ∼= M , as claimed.

4. Proof of Main Theorem for McL : 2

In this section, we prove the following theorem according to Lemma 2.8.

Theorem 4.1. Let M ∼= McL : 2 and G be a group with trivial center. IfN(G) = N(M), then G ∼= M .

Proof. We divide the proof of this theorem into seven steps.


Step 1. Sylow 7-subgroups and Sylow 11-subgroups of G are cyclic groups ofprime order and there are no elements of order 77 in G.

In view of N(G) = N(M) and Lemma 2.9, we can choose p = 7 and q = 11,and take h7 = n14 and h11 = n10 such that G meets the hypotheses of Lemma2.4. Hence Sylow 7-subgroups and Sylow 11-subgroups of G are cyclic groups ofprime order and there are no elements of order 7 · 11 in G.

Step 2. Let g, h ∈ G be elements of orders 7 and 11, respectively. Then |gG| =n10 = 27 · 36 · 53 · 11 and |hG| = n14 = 27 · 36 · 53 · 7, and CG(g) and CG(h) areabelian.

We only state the case of p = 7, and the case of q = 11 is similar.

Since the Sylow 7-subgroup of G is order of 7 by Step 1, one has that 7∣∣ |xG|

for any 1 = x ∈ CG(g). Hence |xG| = n2 or n10. Assume that |gG| = n2 = 34·52·11.Let H be a Sylow 5-subgroup of CG(g). Then H is a nontrivial group of order|G|5/52 by Lemma 2.9. Hence there exists a 5-subgroup K of G such that His a normal subgroup of K and |K/H| = 5. Further, 1 = H ∩ Z(K) ≤ CG(x).Taking 1 = h ∈ H ∩ Z(K), one has that 52 | |hG|5 and 7 | |hG|, which implies|hG| = 1 by Lemma 2.8, and so y ∈ Z(G), a contradiction. It follows that|gG| = n10 = 27 · 36 · 53 · 11.

Since n10 is maximal with respect to divisibility in N(G), Lemma 2.3 impliesthat the group CG(g) is abelian.

In the following discussion, we assume that K is the soluble radical of a groupG, and G = G/K.

Step 3. G is non-soluble and has a unique non-abelian composition factor S suchthat 7 · 11

∣∣|S| and S ≤ G ≤Aut(S). Moreover, S may be isomorphic to one of thefollowing groups:

M22, A11, HS,A12,McL, U6(2).

In view of Step 1 and Step 2, we can choose p = 7 and q = 11 such that Gsatisfies the hypotheses of Lemma 2.5. Hence G is non-soluble and has a uniquenon-abelian composition factor S such that 7 · 11

∣∣|S| and S ≤ G ≤Aut(S). Since7, 11 ⊆ π(S) ⊆ 2, 3, 5, 7, 11, we can easily get that S can be isomorphic toone of the groups: M22, A11, HS,A12,McL, U6(2) by Table 1.

Step 4. 53∣∣|S|.

Since N(G) contains an integer divisible by 53, |G| is divisible by 53. Assume|S| is not divisible by 53. By Table 1, |Out(S)| is not divisible by 5. Consequently,|K| is divisible by 5. For |K| is not divisible by 7, there is an element g in G suchthat |g| = 7 and the image g of g in the group G is nontrivial. By virtue of Step2, |gG|5 = |n10|5 = 53, the subgroup K contains an element h of order 5 such thatg does not centralize h. Let R1 R2 · · · Rn K be an un-refinable series ofcharacteristic subgroups in K and l a number such that h ∈ Rl+1 \Rl. The image

h of h in the group K/Rl is nontrivial and lies in H, which is a normal abelian

5-subgroup of G/Rl. Let G = G/Rl and g be the image of g in the group G. Then


H = CH(g)× [g, H] and g acts fixed-point freely on [g, H] such that |[g, H]| − 1 isdivisible by 7. Thus |[g, H]| ≥ 56, which implies that

56∣∣∣∣ |H||CH(g)|

∣∣∣∣ |K|5|CK(g)|5

.

By Lemma 2.1,

CG(g) = CG(g)K/K = CG(g)/(CG(g) ∩ K) = CG(g)/CK(g),

and then|gG|5 = |G|5|K|5/(|CG(g)|5|CK(g)|5) ≥ 56,

so is |gG|5. It follows a contradiction by (2) of Lemma 2.8, which implies thatN(G) has no number divisible by 56. Hence 53

∣∣|S|.Step 5. S ∼= McL.

By Step 3 and Step 4, we can get that S may be isomorphic to one of groupsHS and McL by Table 1. If M ∼= HS, then let x be an element of order 7 in Gand x be its image in G, of course x ∈ S, |xS| is a multiple of 29, so are |xG| and|xG| by Lemma 2.1. This contradicts (2) of Lemma 2.8. Hence S ∼= McL.

Step 6. G/K ∼= McL : 2.By virtue of S ≤ G/K ≤Aut(S) and [14], we have G/K ∼= McL or McL : 2

and K = 1. Assume that G/K ∼= McL. Then, let g ∈ G, |g| = 7, andg ∈ G/K = G be the image of the element g. In view of Lemma 2.8 (2), Step 2,and [14], the number n10 is maximal in N(G) and the number |gG| = |gS| = n10.Thus G satisfies the hypothesis of Lemma 2.6, and so K ≤ Z(G), a contradictionto Z(G) = 1. Therefore it is impossible that G/K ∼= S ∼= McL, and so G/Kmust be isomorphic to McL : 2.

Step 7. K is a trivial group such that G ∼= M ∼= McL : 2.Also let g ∈ G, |g| = 7, and g ∈ G/K = G ∼= M be the image of the

element g. By Lemma 2.8 (2), Step 2, and [14], the number n10 is maximal in

N(G) and the number |gG| = |gG| = |gS| = n10. Thus K ≤ CG(g). If K = 1 andlet p

∣∣|K|, then g centralizes an element from the center of a Sylow p-subgroupof G, which is impossible by Lemma 2.8. Hence K is a trivial group such thatG ∼= M ∼= McL : 2.

Proof of Main Theorem. It follows directly from Theorems 3.1 and 4.1.

Acknowledgement. This work was supported by National Natural ScienceFoundation of China (Grants Nos. 11271301, 11001226), the Scientific and Tech-nological Research Program of Chongqing Municipal Education Commission(Grant Nos. KJ1401006, KJ1401019), the Scientific Research Foundation ofChongqing Municipal Science and Technology Commission (Grant No.cstc2014jcyjA00009), the Fundamental Research Funds for the Central Univer-sities, and the Youth Foundation of Chongqing Three Gorges University (GrantNo. 12QN-23).


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ON SOME GROWTH PROPERTIES OF DIFFERENTIALPOLYNOMIALS IN THE LIGHT OF RELATIVE ORDER

Sanjib Kumar Datta

Department of MathematicsUniversity of KalyaniP.O. Kalyani, Dist-NadiaPIN–741235, West BengalIndiaemail: sanjib kr [email protected]

Tanmay Biswas

Rajbari, RabindrapalliR.N. Tagore RoadP.O. Krishnagar, Dist-NadiaPIN–741101, West BengalIndiaemails: tanmaybiswas [email protected]

tanmaybiswas [email protected]

M.D. Azizul Hoque

Gobargara High Madrasah (H.S.)P.O.+P.S. Hariharpara, Dist-MurshidabadPIN–742166, West BengalIndiaemail: [email protected]

Abstract. In the paper we establish some newly developed results based on the growth

properties of relative order (relative lower order), relative type (relative lower type) and

relative weak type of differential polynomials generated by entire and meromorphic

functions.

Keywords: entire function, meromorphic function, relative order (relative lower order),

relative type (relative lower type), relative weak type, polynomials.

2010 Mathematics Subject Classification: 30D35, 30D30, 30D20.

1. Introduction, definitions and notations

We denote by C the set of all finite complex numbers. Let f be a non-constant me-romorphic function defined in the open complex plane C. Also, let n0j, n1j, ..., nkj

(k ≥ 1) be non-negative integers such that for each j,k∑

i=0

nij ≥ 1.

236 s.k. datta, t. biswas, m.d. azizul hoque

We call Mj [f ] = Aj (f)n0j(f (1))n1j · · ·

(f (k))nkj , where T (r, Aj) = S (r, f)

to be a differential monomial generated by f . The numbers γMj=

k∑i=0

nij and

ΓMj=

k∑i=0

(i+ 1)nij are called, respectively, the degree and weight of Mj [f ]

([2], [8]). The expression P [f ] =s∑

j=1

Mj [f ] is called a differential polynomial

generated by f . The numbers γP = max1<j<s

γMjand ΓP = max

1<j<sΓMj

are called,

respectively, the degree and weight of P [f ] ([2], [8]). Also, we call the numbersγP−

= min1<j<s

γMjand k (the order of the highest derivative of f) the lower degree

and the order of P [f ], respectively. If γP−

= γP , P [f ] is called a homogeneous

differential polynomial. Throughout the paper we consider only the non-constantdifferential polynomials and we denote by P0 [f ] a differential polynomial notcontaining f , i.e., for which n0j = 0 for j = 1, 2, ..., s. We consider only thoseP [f ] , P0 [f ] singularities of whose individual terms do not cancel each other.

The following definitions are well known.

Definition 1. The quantity Θ (a; f) of a meromorphic function f is defined asfollows

Θ (a; f) = 1− lim supr→∞

−N (r, a; f)

T (r, f).

Definition 2. [7] For a ∈ C ∪ ∞,let np (r, a; f) denotes the number of zerosof f − a in |z| ≤ r, where a zero of multiplicity < p is counted according toits multiplicity and a zero of multiplicity > p is counted exactly p times; andNp (r, a; f) is defined in terms of np (r, a; f) in the usual way. We define

δp (a; f) = 1− lim supr→∞

Np (r, a; f)

T (r, f).

Definition 3. [1] P [f ] is said to be admissible if

(i) P [f ] is homogeneous, or

(ii) P [f ] is non homogeneous and m (r, f) = S (r, f) .

The following definitions are also well known.

Definition 4. The order ρf and lower order λf of a meromorphic function f aredefined as

ρf = lim supr→∞

log Tf (r)

log rand λf = lim inf

r→∞

log Tf (r)

log r.

Definition 5. The type σf and lower type−σf of a meromorphic function f are

defined as

σf = lim supr→∞

Tf (r)

rρfand

−σf = lim inf

r→∞

Tf (r)

rρf, 0 < ρf < ∞.

on some growth properties of differential polynomials ... 237

In this connection, Datta and Jha [3] gave the definition of weak type of ameromorphic function of finite positive lower order in the following way:

Definition 6. [3] The weak type τ f of a meromorphic function f of finite positivelower order λf is defined by

τ f = lim infr→∞

Tf (r)

rλf.

Similarly, one can define the growth indicator−τ f of a meromorphic function f of

finite positive lower order λf as

−τ f = lim sup

r→∞

Tf (r)

rλf.

For an entire function g, the Nevanlinna’s characteristic function Tg (r) is de-fined as

Tg (r) =1

2π

2π∫0

log+∣∣g(reiθ)∣∣ dθ,

where log+ x = max (0, log x) for x > 0.If g is non-constant then Tg (r) is strictly increasing and continuous and its

inverse T−1g : (Tg (0) ,∞) → (0,∞) exists and is such that lim

s→∞T−1g (s) = ∞.

Lahiri and Banerjee [6] introduced the definition of relative order of a mero-morphic function with respect to an entire function which is as follows:

Definition 7. [6] Let f be meromorphic and g be entire. The relative order of fwith respect to g denoted by ρg (f) is defined as

ρg (f) = inf

µ > 0 : Tf (r) < Tg (r

µ)for all sufficiently large r

= lim sup

r→∞

log T−1g Tf (r)

log r.

The definition coincides with the classical one [6] if g (z) = exp z.Similarly, one can define the relative lower order of a meromorphic function f

with respect to an entire g denoted by λg (f) in the following manner:

λg (f) = lim infr→∞

log T−1g Tf (r)

log r.

Datta and Biswas [4] gave the definition of relative type and relative weaktype of a meromorphic function with respect to an entire function g which are asfollows:

Definition 8. [4] The relative type σg (f) of a meromorphic function f withrespect to an entire function g are defined as

σg (f) = lim supr→∞

T−1g Tf (r)

rρg(f), where 0 < ρg (f) < ∞.


Similarly, one can define the lower relative type−σg (f) in the following way

−σg (f) = lim inf

r→∞

T−1g Tf (r)

rρg(f), where 0 < ρg (f) < ∞.

Definition 9. [4] The relative weak type τ g (f) of a meromorphic function f withrespect to an entire function g with finite positive relative lower order λg (f) isdefined by

τ g (f) = lim infr→∞

T−1g Tf (r)

rλg(f).

Analogously, one can define the growth indicator−τ g (f) of a meromorphic function

f with respect to an entire function g with finite positive relative lower orderλg (f) as

−τ g (f) = lim sup

r→∞

T−1g Tf (r)

rλg(f).

In this paper we establish some newly developed results based on the growthproperties of relative order (relative lower order) relative type(relative lower type)and relative weak type of polynomials generated by entire and meromorphic func-tions. We do not explain the standard notations and definitions in the theory ofentire and meromorphic functions because those are available in [5] and [9].

2. Lemmas

In this section we present some lemmas which will be needed in the sequel.

Lemma 1. [1] Let f be either of finite order or of non-zero lower order such thatΘ(∞; f) =

∑a=∞

δp (a; f) = 1 or δ (∞; f) =∑a =∞

δ (a; f) = 1. Then, for homoge-

neous P0 [f ],

limr→∞

TP0[f ] (r)

Tf (r)= γP0[f ].

Lemma 2. [1] Let P0 [f ] be admissible. If f is of finite order or of non zero lowerorder and

∑a=∞

Θ(a; f) = 2, then

limr→∞

TP0[f ] (r)

Tf (r)= ΓP0[f ].

Lemma 3. [3] If f be a meromorphic function of regular growth, i.e., ρf = λf

thenσf =

−σf = τ f =

−τ f .

3. Theorems

In this section, we present the main results of the paper. It is needless to mentionthat in the paper, the admissibility and homogeneity of P0[f ] will be needed asper the requirements of the theorems.


Theorem 1. Let f be a meromorphic function either of finite order or of non-zerolower order such that Θ(∞; f) =

∑a =∞

δp (a; f) = 1 or δ (∞; f) =∑a =∞

δ (a; f) = 1

and g be an entire function with 0 < τ g ≤ −τ g < ∞ and 0 <

−σg ≤ σg < ∞. Also

let Θ(∞; g) =∑a=∞

δp (a; g) = 1 or δ (∞; g) =∑a =∞

δ (a; g) = 1. Then

max

(γP0[f ]

γP0[g]

) 1λg

.

(τ g−τ g

) 1λg

,

(γP0[f ]

γP0[g]

) 1ρg

.

(−σg

σg

) 1ρg

≤ lim inf

r→∞

T−1P0[g]

TP0[f ] (r)

T−1g Tf (r)

≤ lim supr→∞

T−1P0[g]

TP0[f ] (r)

T−1g Tf (r)

≤ min

(γP0[f ]

γP0[g]

) 1λg

.

( −τ gτ g

) 1λg

,

(γP0[f ]

γP0[g]

) 1ρg

.

(σg

−σg

) 1ρg

,

where P0 [f ] and P0 [g] are homogeneous.

Proof. For any ε(> 0), we get from Lemma 1, for all sufficiently large values of r,

(1) TP0[f ] (r) ≤γP0[f ] + ε

Tf (r)

and

(2) TP0[f ] (r) ≥γP0[f ] − ε

Tf (r) .

Also, from Lemma 1, we get for all sufficiently large values of r that

TP0[g] (r) ≥γP0[g] − ε

Tg (r)

i.e., r ≥ T−1P0[g]

[γP0[g] − ε

Tg (r)

](3) i.e., T−1

g

(r

γP0[g] − ε

)≥ T−1

P0[g](r) .

andTP0[g] (r) ≤

γP0[g] + ε

Tg (r)

i.e., r ≤ T−1P0[g]

[γP0[g] + ε

Tg (r)

](4) i.e., T−1

g

(r

γP0[g] + ε

)≤ T−1

P0[g](r) .

Now, from (1) and (3), it follows for all sufficiently large values of r,

T−1P0[g]

TP0[f ] (r) ≤ T−1P0[g]

[γP0[f ] + ε

Tf (r)

]i.e., T−1

P0[g]TP0[f ] (r) ≤ T−1

g

[(γP0[f ] + ε

γP0[g] − ε

)Tf (r)

].(5)


Again from (2) and (4) it follows for all sufficiently large values of r,

T−1P0[g]

TP0[f ] (r) ≥ T−1P0[g]

[γP0[f ] − ε

Tf (r)

]i.e., T−1

P0[g]TP0[f ] (r) ≥ T−1

g

[(γP0[f ] − ε

γP0[g] + ε

)Tf (r)

].(6)

Now, for the definition of type and lower type we get for all sufficiently largevalues of r that

Tg

(Tf (r)

(σg + ε)

1ρg

)≤ Tf (r)

(7) i.e., T−1g Tf (r) ≥

Tf (r)

(σg + ε)

1ρg

and

Tg

γP0[f ] + ε(

γP0[g] − ε) (−

σg − ε)Tf (r)

1ρg

≥

[(γP0[f ] + ε

γP0[g] − ε

)Tf (r)

]

i.e.,

γP0[f ] + ε(γP0[g] − ε

) (−σg − ε

)Tf (r)

1ρg

≥ T−1g

[(γP0[f ] + ε

γP0[g] − ε

)Tf (r)

].(8)

Therefore, from (5) and (8), it follows for all sufficiently large values of r that

(9) T−1P0[g]

TP0[f ] (r) ≤

γP0[f ] + ε(γP0[g] − ε

) (−σg − ε

)Tf (r)

1ρg

.

Therefore, from (7) and (9), it follows for all sufficiently large values of r that

T−1P0[g]

TP0[f ] (r)

T−1g Tf (r)

≤

[(γP0[f ]

+ε

(γP0[g]−ε)

(−σg−ε

))Tf (r)

] 1ρg

Tf (r)

(σg+ε)

1ρg

i.e.,T−1P0[g]

TP0[f ] (r)

T−1g Tf (r)

≤

(γP0[f ] + ε)(σg + ε)(

γP0[g] − ε) (−

σg − ε) 1

ρg

i.e., lim supr→∞

T−1P0[g]

TP0[f ] (r)

T−1g Tf (r)

≤

(γP0[f ]

γP0[g]

) 1ρg

.

(σg

−σg

) 1ρg

.(10)


Similarly, from (6), it can be shown for all sufficiently large values of r,

(11) lim infr→∞

T−1P0[g]

TP0[f ] (r)

T−1g Tf (r)

≥

(γP0[f ]

γP0[g]

) 1ρg

.

(−σg

σg

) 1ρg

.

Therefore, from (10) and (11), we obtain that

(12)

(γP0[f ]

γP0[g]

) 1ρg

.

(−σg

σg

) 1ρg

≤ lim infr→∞

T−1P0[g]

TP0[f ] (r)

T−1g Tf (r)

≤ lim supr→∞

T−1P0[g]

TP0[f ] (r)

T−1g Tf (r)

≤

(γP0[f ]

γP0[g]

) 1ρg

.

(σg

−σg

) 1ρg

.

Similarly, using the weak type, one can easily verify that

(13)

(γP0[f ]

γP0[g]

) 1λg

.

(τ g−τ g

) 1λg

≤ lim infr→∞

T−1P0[g]

TP0[f ] (r)

T−1g Tf (r)

≤ lim supr→∞

T−1P0[g]

TP0[f ] (r)

T−1g Tf (r)

≤

(γP0[f ]

γP0[g]

) 1λg

.

( −τ gτ g

) 1λg

.

Thus the theorem follows from (12) and (13).

Theorem 2. Let f be a meromorphic function either of finite order or of non-zero lower order such that

∑a =∞

Θ(a; f) = 2 and g be an entire function with

0 < τ g ≤−τ g < ∞ and 0 <

−σg ≤ σg < ∞. Also let

∑a =∞

Θ(a; g) = 2. Then

max

(ΓP0[f ]

ΓP0[g]

) 1λg

.

(τ g−τ g

) 1λg

,

(ΓP0[f ]

ΓP0[g]

) 1ρg

.

(−σg

σg

) 1ρg

≤ lim inf

r→∞

T−1P0[g]

TP0[f ] (r)

T−1g Tf (r)

≤ lim supr→∞

T−1P0[g]

TP0[f ] (r)

T−1g Tf (r)

≤ min

(ΓP0[f ]

ΓP0[g]

) 1λg

.

( −τ gτ g

) 1λg

,

(ΓP0[f ]

ΓP0[g]

) 1ρg

.

(σg

−σg

) 1ρg

,

where P0 [f ] and P0 [g] are admissible.


With the help of Lemma 2, Theorem 2 can be carried out in the line ofTheorem 1. So, the proof is omitted.

Corollary 1. Under the same conditions of Theorem 1, if g is of regular growth,then by Lemma 3 one can easily verify that

limr→∞

T−1P0[g]

TP0[f ] (r)

T−1g Tf (r)

=

(γP0[f ]

γP0[g]

) 1ρg

.

Corollary 2. Under the same conditions of Theorem 2, if g is of regular growth,then by Lemma 3 one can also verify that

limr→∞

T−1P0[g]

TP0[f ] (r)

T−1g Tf (r)

=

(ΓP0[f ]

ΓP0[g]

) 1ρg

.


∑a =∞

δp (a; f) = 1 or δ (∞; f) =∑a =∞

δ (a; f) = 1

and g be an entire function with 0 < λg ≤ ρg < ∞. Also let Θ(∞; g) =∑a=∞

δp (a; g) = 1 or δ (∞; g) =∑a =∞

δ (a; g) = 1. Then, for homogeneous P0 [f ]

and P0 [g],

λg

ρg≤ lim inf

r→∞

log T−1P0[g]

TP0[f ] (r)

log T−1g Tf (r)

≤ lim supr→∞

log T−1P0[g]

TP0[f ] (r)

log T−1g Tf (r)

≤ρgλg

.

Proof. From (5) and (6), we get for all sufficiently large values of r that

(14) log T−1P0[g]

TP0[f ] (r) ≤ log T−1g

[(γP0[f ] + ε

γP0[g] − ε

)Tf (r)

]and

(15) log T−1P0[g]

TP0[f ] (r) ≥ log T−1g

[(γP0[f ] − ε

γP0[g] + ε

)Tf (r)

].

Now for the definition of order and lower order we get for all sufficiently largevalues of r that

Tg

(Tf (r)

1ρg+ε

)≤ Tf (r)

i.e., log T−1g Tf (r) ≥ 1(

ρg + ε) log Tf (r) .(16)

and

Tg

γP0[f ] + ε(

γP0[g] − ε) (−

σg − ε)Tf (r)

1

λg−ε

≥

[(γP0[f ] + ε

γP0[g] − ε

)Tf (r)

]


i.e.,

γP0[f ] + ε(γP0[g] − ε

) (−σg − ε

)Tf (r)

1λg−ε

≥ T−1g

[(γP0[f ] + ε

γP0[g] − ε

)Tf (r)

]

(17) i.e.,1

(λg − ε)log Tf (r) +O(1) ≥ log T−1

g

[(γP0[f ] + ε

γP0[g] − ε

)Tf (r)

].

Therefore from (14) and (17) it follows for all sufficiently large values of r that

(18) log T−1P0[g]

TP0[f ] (r) ≤1

(λg − ε)log Tf (r) +O(1).

Therefore from (16) and (18) it follows for all sufficiently large values of r that

log T−1P0[g]

TP0[f ] (r)

log T−1g Tf (r)

≤(ρg + ε

λg − ε

).log Tf (r) +O(1)

log Tf (r)

i.e., lim supr→∞

log T−1P0[g]

TP0[f ] (r)

log T−1g Tf (r)

≤ρgλg

.(19)

Similarly from (15) it can be shown for all sufficiently large values of r that

(20) lim infr→∞

log T−1P0[g]

TP0[f ] (r)

log T−1g Tf (r)

≥ λg

ρg.

Therefore from (19) and (20) we obtain that

λg

ρg≤ lim inf

r→∞

log T−1P0[g]

TP0[f ] (r)

log T−1g Tf (r)

≤ lim supr→∞

log T−1P0[g]

TP0[f ] (r)

log T−1g Tf (r)

≤ρgλg

.

Thus the theorem follows from above.

Remark 1. The conclusion of Theorem 3 can also drawn under the hypothesis∑a=∞

Θ(a; f) = 2 and∑a =∞

Θ(a; g) = 2 instead of “Θ (∞; f) =∑a =∞

δp (a; f) = 1

or δ (∞; f) =∑a=∞

δ (a; f) = 1” and “Θ (∞; g) =∑a =∞

δp (a; g) = 1 or δ (∞; g) =∑a=∞

δ (a; g) = 1” where P0 [f ] and P0 [g] are admissible.

Corollary 3. Under the same conditions of Theorem 3 and Remark 1 if g is ofregular growth then one may get that

limr→∞

log T−1P0[g]

TP0[f ] (r)

log T−1g Tf (r)

= 1.


Theorem 4. If f be a meromorphic function either of finite order or of non-zerolower order such that Θ(∞; f) =

∑a =∞

δp (a; f) = 1 or δ (∞; f) =∑a =∞

δ (a; f) = 1

and g be an entire function of regular growth having non zero finite order andΘ(∞; g) =

∑a =∞

δp (a; g) = 1 or δ (∞; g) =∑a =∞

δ (a; g) = 1. Then the relative order

and relative lower order of P0 [f ] with respect to P0 [g] are same as those of f withrespect to g where P0 [f ] and P0 [g] are homogeneous.

Proof. By Corollary 3, we obtain that

ρP0[g] (P0 [f ]) = lim supr→∞

log T−1P0[g]

TP0[f ] (r)

log r

= lim supr→∞

log T−1g Tf (r)

log r· limr→∞

log T−1P0[g]

TP0[f ] (r)

log T−1g Tf (r)

= ρg (f) .1 = ρg (f) .

In a similar manner,λP0[g] (P0 [f ]) = λg (f) .

Thus, the theorem follows.

Theorem 5. If f be a meromorphic function either of finite order or of non-zero lower order such that

∑a =∞

Θ(a; f) = 2 and g be an entire function of regular

growth having non zero finite order and∑a=∞

Θ(a; g) = 2. Then the relative order

and relative lower order of P0 [f ] with respect to P0 [g] are same as those of f withrespect to g where P0 [f ] and P0 [g] are admissible.

We omit the proof of Theorem 5 because it can be carried out in the line ofTheorem 4 and with the help of Corollary 3.

Theorem 6. If f be a meromorphic function either of finite order or of non-zerolower order such that Θ(∞; f) =

∑a =∞

δp (a; f) = 1 or δ (∞; f) =∑a =∞

δ (a; f) = 1

and g be an entire function of regular growth having non zero finite type andΘ(∞; g) =

∑a =∞

δp (a; g) = 1 or δ (∞; g) =∑a =∞

δ (a; g) = 1. Then the relative type

and relative lower type of P0 [f ] with respect to P0 [g] are(

γP0[f ]

γP0[g]

) 1ρg times that of f

with respect to g if ρg (f) is positive finite and P0 [f ] and P0 [g] are homogeneous.

Proof. From Corollary 1 and Theorem 4, we get that

σP0[g] (P0 [f ]) = lim supr→∞

T−1P0[g]

TP0[f ] (r)

rρP0[g](P0[f ])

= limr→∞

T−1P0[g]

TP0[f ] (r)

T−1g Tf (r)

.lim supr→∞

T−1g Tf (r)

rρg(f)=

(γP0[f ]

γP0[g]

) 1ρg

σg (f) .


Similarly,

−σP0[g] (P0 [f ]) =

(γP0[f ]

γP0[g]

) 1ρg

.−σg (f) .



∑a =∞

δp (a; f) = 1 or δ (∞; f) =∑a =∞

δ (a; f) = 1

and g be an entire function of regular growth having non zero finite type andΘ(∞; g) =

∑a =∞

δp (a; g) = 1 or δ (∞; g) =∑a =∞

δ (a; g) = 1. Then τP0[g] (P0 [f ]) and

−τP0[g] (P0 [f ]) are

(γP0[f ]

γP0[g]

) 1ρg times that of f with respect to g, i.e., τP0[g] (P0 [f ]) =(

γP0[f ]

γP0[g]

) 1ρg .τ g (f) and

−τP0[g] (P0 [f ]) =

(γP0[f ]

γP0[g]

) 1ρg .

−τ g (f) when λg (f) is positive

finite and P0 [f ] and P0 [g] are homogeneous.

We omit the proof of Theorem 7 because it can be carried out in the line ofTheorem 6 and with the help of Theorem 5 and Corollary 2.

In a similar manner, we can state the following two theorem without proof:

Theorem 8. If f be a meromorphic function either of finite order or of non-zerolower order such that

∑a =∞

Θ(a; f) = 2 and g be an entire function of regular growth

having non zero finite type and∑a =∞

Θ(a; g) = 2, then the relative type and relative

lower type of P0 [f ] with respect to P0 [g] are(

ΓP0[f ]

ΓP0[g]

) 1ρg times that of f with respect

to g if ρg (f) is positive

finite and P0 [f ] and P0 [g] are admissible.

Theorem 9. Let f be a meromorphic function either of finite order or of non-zero lower order such that

∑a =∞

Θ(a; f) = 2 and g be an entire function of regular

growth having non zero finite type and∑a=∞

Θ(a; g) = 2. Then τP0[g] (P0 [f ]) and

−τP0[g] (P0 [f ]) are

(ΓP0[f ]

ΓP0[g]

) 1ρg times that of f with respect to g i.e., τP0[g] (P0 [f ]) =(

ΓP0[f ]

ΓP0[g]

) 1ρg .τ g (f) and

−τP0[g] (P0 [f ]) =

(ΓP0[f ]

ΓP0[g]

) 1ρg .

−τ g (f) when λg (f) is positive

finite and P0 [f ] and P0 [g] are homogeneous.

References

[1] Bhattacharjee, N., Lahiri, I., Growth and value distribution of differen-tial polynomials, Bull. Math. Soc. Sc. Math. Roumanie, vol. 39 (87), (1-4)(1996), 85-104.


[2] Doeringer, W., Exceptional values of differential polynomials, PacificJ. Math., 98 (1) (1982), 55-62.

[3] Datta, S.K., Jha, A., On the weak type of meromorphic functions, Int.Math. Forum, 4 (12) (2009), 569-579.

[4] Datta, S.K., Biswas, A., On relative type of entire and meromorphic func-tions, Advances in Applied Mathematical Analysis, 8 (2) (2013), 63-75.

[5] Hayman, W.K., Meromorphic Functions, The Clarendon Press, Oxford,1964.

[6] Lahiri, B.K., Banerjee, D., Relative order of entire and meromorphicfunctions, Proc. Nat. Acad. Sci. India, 69 (A) III (1999), 339-354.

[7] Lahiri, I., Deficiencies of differential polynomials, Indian J. Pure Appl.Math., 30 (5) (1999), 435-447.

[8] Sons, L.R., Deficiencies of monomials, Math. Z, 111 (1969), pp.53-68.

[9] Valiron, G., Lectures on the General Theory of Integral Functions, ChelseaPublishing Company, 1949.



JENSEN TYPE WEIGHTED INEQUALITIES FOR FUNCTIONSOF SELFADJOINT AND UNITARY OPERATORS

S.S. Dragomir

Mathematics, College of Engineering & ScienceVictoria University, PO Box 14428Melbourne City, MC 8001Australiae-mail: [email protected]: http://rgmia.org/dragomirandSchool of Computational & Applied MathematicsUniversity of the WitwatersrandPrivate Bag 3, Johannesburg 2050South Africa

Abstract. On making use of the spectral representations in terms of the Riemann-

Stieltjes integral for the selfadjoint and unitary operators in Hilbert spaces we establish

here some weighted inequalities of Jensen’s type for convex, square-convex and Arg-

square-convex functions. Some applications for simple functions of operators that be-

long to those classes are also provided.

Keywords: Jensen’s inequality, Holder’s inequality, Measurable functions, Lebesgue

integral, Selfadjoint operators, Unitary operators, Spectral family.

1991 Mathematics Subject Classification: Primary 26D15, 26D20; Secondary

47A63.

1. Introduction

Let A be a selfadjoint operator on the complex Hilbert space (H, ⟨·, ·⟩) with thespectrum Sp (A) included in the interval [m,M ] for some real numbersm < M andlet Eλλ be its spectral family. Then for any continuous function f : [m,M ] → R,it is well known that we have the following spectral representation in terms of theRiemann-Stieltjes integral (see, for instance, [19, p. 257]):

(1) ⟨f (A) x, y⟩ =∫ M

m−0

f (λ) d (⟨Eλx, y⟩) ,

248 s.s. dragomir

and

(2) ∥f (A)x∥2 =∫ M

m−0

|f (λ)|2 d ∥Eλx∥2 ,

for any x, y ∈ H.The function gx,y (λ) := ⟨Eλx, y⟩ is of bounded variation on the interval [m,M ]

and

gx,y (m− 0) = 0 while gx,y (M) = ⟨x, y⟩

for any x, y ∈ H. It is also well known that gx (λ) := ⟨Eλx, x⟩ is monotonicnondecreasing and right continuous on [m,M ] for any x ∈ H.

The following result that provides an operator version for the Jensen inequa-lity is due to Mond & Pecaric [23] (see also [18, p. 5]):

Theorem 1 (Mond-Pecaric, 1993, [23]) Let A be a selfadjoint operator onthe Hilbert space H and assume that Sp (A) ⊆ [m,M ] for some scalars m,Mwith m < M. If h is a convex function on [m,M ] , then

(MP) h (⟨Ax, x⟩) ≤ ⟨h (A)x, x⟩

for each x ∈ H with ∥x∥ = 1.

As a special case of Theorem 1 we have the following Holder-McCarthyinequality:

Theorem 2 (Holder-McCarthy, 1967, [21]) Let A be a selfadjoint positiveoperator on a Hilbert space H. Then, for all x ∈ H with ∥x∥ = 1,

(i) ⟨Arx, x⟩ ≥ ⟨Ax, x⟩r for all r > 1;

(ii) ⟨Arx, x⟩ ≤ ⟨Ax, x⟩r for all 0 < r < 1;

(iii) If A is invertible, then ⟨Arx, x⟩ ≥ ⟨Ax, x⟩r for all r < 0.

The following reverse for the Mond-Pecaric inequality that generalizes thescalar Lah-Ribaric inequality for convex functions is well known, see for instance[18, p. 57]:

Theorem 3 Let A be a selfadjoint operator on the Hilbert space H and assumethat Sp (A) ⊆ [m,M ] for some scalars m,M with m < M. If h is a convexfunction on [m,M ] , then

(LR) ⟨h (A) x, x⟩ ≤ M − ⟨Ax, x⟩M −m

· h (m) +⟨Ax, x⟩ −m

M −m· h (M)

for each x ∈ H with ∥x∥ = 1.

jensen type weighted inequalities for functions ... 249

We recall that the bounded linear operator U : H → H on the Hilbert spaceH is unitary iff U∗ = U−1.

It is well known that (see for instance [19, p. 275-p. 276]), if U is a unitaryoperator, then there exists a family of projections Eλλ∈[0,2π], called the spectralfamily of U with the following properties

a) Eλ ≤ Eµ for 0 ≤ λ ≤ µ ≤ 2π;

b) E0 = 0 and E2π = 1H (the identity operator on H);

c) Eλ+0 = Eλ for 0 ≤ λ < 2π;

d) U =∫ 2π

0eiλdEλ where the integral is of Riemann-Stieltjes type.

Moreover, if Fλλ∈[0,2π] is a family of projections satisfying the requirementsa)-d) above for the operator U, then Fλ = Eλ for all λ ∈ [0, 2π] .

Also, for every continuous complex valued function f : C (0, 1) → C on thecomplex unit circle C (0, 1), we have

(3) f (U) =

∫ 2π

0

f(eiλ)dEλ

where the integral is taken in the Riemann-Stieltjes sense.In particular, we have the equalities

(4) ⟨f (U)x, y⟩ =∫ 2π

0

f(eiλ)d ⟨Eλx, y⟩

and

(5) ∥f (U)x∥2 =∫ 2π

0

∣∣f (eiλ)∣∣2 d ∥Eλx∥2 =∫ 2π

0

∣∣f (eiλ)∣∣2 d ⟨Eλx, x⟩ ,

for any x, y ∈ H.From the above properties it follows that the function gx (λ) := ⟨Eλx, x⟩ is

monotonic nondecreasing and right continuous on [0, 2π] for any x ∈ H.For z ∈ Cr 0 we call the principal value of log (z) the complex number

Log (z) := ln |z|+ iArg (z)

where 0 ≤ Arg (z) < 2π.We observe that for t ∈ [0, 2π) we have

Log(eit)= it.

If we extend this equality by continuity in the point t = 2π, then we can definethe operator Log(U) : H → H as

(6) Log(U)x =

∫ 2π

0

Log(eiλ)dEλx =

∫ 2π

0

(iλ) dEλx, x ∈ H.

250 s.s. dragomir

Utilizing these spectral representations in terms of the Riemann-Stieltjes inte-gral for the selfadjoint and unitary operators we establish here some weightedinequalities of Jensen’s type for three classes of functions: convex, square-convexand Arg-square-convex functions. Some applications for simple functions of ope-rators that belong to those classes are also provided.

For classical and recent results concerning inequalities for continuos functionsof selfadjoint operators, see [23], [24], [25], [20], [18], [6], [9], [10], [12], [11], [16],[15], [14], [13], [7], and [8].

2. Weighted inequalities for the Riemann-Stieltjes integral

We can state the following result concerning the weighted Riemann-Stieltjes inte-gral of monotonic nondecreasing integrators:

Theorem 4 Let Φ : [γ,Γ] ⊂ R → R be a continuous convex function on theinterval [γ,Γ] , f : [a, b] ⊂ R → R be a continuous function on the interval [a, b]and with the property that

(7) γ ≤ f (t) ≤ Γ for any t ∈ [a, b]

and w : [a, b] → [0,∞) be continuous on [a, b]. Then, for each monotonic non-

decreasing function u : [a, b] → R such that∫ b

aw (t) du (t) > 0, we have the

inequalities

(8)

Φ

(∫ b

aw (t) f (t) du (t)∫ b

aw (t) du (t)

)≤∫ b

aw (t) (Φ f) (t) du (t)∫ b

aw (t) du (t)

≤Φ (γ)

(Γ−

∫ b


aw (t) du (t)

)+ Φ(Γ)

(∫ b


aw (t) du (t)

− γ

)Γ− γ

.

Proof. Utilizing the gradient inequality for the convex function Φ, namely,

Φ (ς)− Φ (τ) ≥ δΦ (τ) (ς − τ)

for any ς, τ ∈ [γ,Γ] where δΦ (τ) ∈[Φ′

− (τ) ,Φ′+ (τ)

], (for τ = γ we take δΦ (τ) =

Φ′+ (γ) and for τ = Γ we take δΦ (τ) = Φ′

− (Γ)), then we get

Φ (ς)− Φ

(∫ b


aw (t) du (t)

)(9)

≥ δΦ

(∫ b


aw (t) du (t)

)(ς −

∫ b


aw (t) du (t)

)for any ς ∈ [γ,Γ] , since obviously, by (7)∫ b


aw (t) du (t)

∈ [γ,Γ] .


Since f satisfies (7), then by (9) we get

(Φ f) (s)− Φ

(∫ b


aw (t) du (t)

)(10)

≥ δΦ

(∫ b


aw (t) du (t)

)(f (s)−

∫ b


aw (t) du (t)

)for any s ∈ [a, b] .

Now, if we multiply (10) by w (s) ≥ 0 and integrate the result over themonotonic nondecreasing integrator u on the interval [a, b] we obtain the firstinequality in (8).

By the convexity of Φ we also have the inequality

Φ (τ) ≤ (Γ− τ) Φ (γ) + (τ − γ) Φ (Γ)

Γ− γ

for any τ ∈ [γ,Γ] , which, by (9) implies that

(11) (Φ f) (s) ≤ (Γ− f (s)) Φ (γ) + (f (s)− γ) Φ (Γ)

Γ− γ

for any s ∈ [a, b] .Now, if we multiply (11) by w (s) ≥ 0 and integrate the result over the

monotonic nondecreasing integrator u on the interval [a, b] we obtain the secondinequality in (11).

The proof is complete.

Remark 1 The above inequality provides a generalization for the unweightedcase, namely w (t) = 1, t ∈ [a, b] , which can be stated as

(12)

Φ

(∫ b

af (t) du (t)

u (b)− u (a)

)≤∫ b

a(Φ f) (t) du (t)u (b)− u (a)

≤Φ (γ)

(Γ−

∫ b

af (t) du (t)

u (b)− u (a)

)+ Φ(Γ)

(∫ b

af (t) du (t)

u (b)− u (a)− γ

)Γ− γ

.

For inequalities related to the Jensen’s result, see [1], [2], [3], [17], [4], [26]and [27].

Corollary 1 Let h : [a, b] ⊂ R → R be a continuous function on the interval [a, b]and with the property that

(13) 0 ≤ γ ≤ h (t) ≤ Γ for any t ∈ [a, b]

and w : [a, b] → [0,∞) be continuous on [a, b]. Assume also that u : [a, b] → R is

a monotonic nondecreasing function such that∫ b

aw (t) du (t) > 0.

252 s.s. dragomir

(i) If p ≥ 1, then(∫ b

a

w (t)h (t) du (t)

)p

(14)

≤[∫ b

a

w (t) du (t)

]p−1 ∫ b

a

w (t)hp (t) du (t)

≤ 1

Γ− γ

[∫ b

a

w (t) du (t)

]p×

[γp

(Γ−

∫ b

aw (t)h (t) du (t)∫ b

aw (t) du (t)

)+ Φp

(∫ b


aw (t) du (t)

− γ

)].

(ii) If p ∈ (0, 1) , then the inequalities reverse in (14).

(iii) If p < 0 and γ > 0, then the inequality (14) also holds.

The proof follows by Theorem 4 applied for the convex (concave) functionf (t) = tp, p ∈ (−∞, 0) ∪ [1,∞) (p ∈ (0, 1)).

The following result is the well known version of the Holder inequality for theRiemann-Stieltjes integral with monotonic nondecreasing integrators u : [a, b] → R:

(15)

∫ b

a

|f (t) g (t)| du (t) ≤[∫ b

a

|f (t)|p du (t)]1/p [∫ b

a

|g (t)|q du (t)]1/q

,

where f, g : [a, b] ⊂ R → C are continuous and p, q > 1 with 1/p+ 1/q = 1.

Proposition 1 Let f, g : [a, b] ⊂ R → Cr 0 be continuous on [a, b] andu: [a, b]→R monotonic nondecreasing on [a, b]. Let p, q ∈ Rr0 with 1/p+1/q=1and assume that

(16) 0 ≤ γ ≤ |f (t)||g (t)|q−1 ≤ Γ for any t ∈ [a, b] .

(i) If p > 1, then∫ b

a

|f (t) g (t)| du (t)(17)

≤[∫ b

a

|g (t)|q du (t)]1/q [∫ b

a

|f (t)|p du (t)]1/p

≤ 1

(Γ− γ)1/p

∫ b

a

|g (t)|q du (t)

×

[γp

(Γ−

∫ b

a|f (t) g (t)| du (t)∫ b

a|g (t)|q du (t)

)+ Φp

(∫ b

a|f (t) g (t)| du (t)∫ b

a|g (t)|q du (t)

− γ

)]1/p.

(ii) If p ∈ (0, 1) , then the inequalities in (17) reverse.

(iii) If p < 0 and γ > 0 then the inequalities in (17) also reverse.


Proof. Follows by Corollary 1, by choosing

h =|f |

|g|q−1 and w = |g|q

and performing some simple calculation.The details are omitted.

Corollary 2 Let h : [a, b] ⊂ R → R be a continuous function on the interval [a, b]and with the property that

(18) 0 < γ ≤ h (t) ≤ Γ for any t ∈ [a, b]

and w : [a, b] → [0,∞) be continuos on [a, b]. Assume also that u : [a, b] → R is a

monotonic nondecreasing function such that∫ b

aw (t) du (t) > 0. Then

(19)

∫ b


aw (t) du (t)

≥ exp

[∫ b

aw (t) (ln h) (t) du (t)∫ b

aw (t) du (t)

]

≥ γ1

Γ−γ

(Γ−

∫ ba w(t)h(t)du(t)∫ b

a w(t)du(t)

)Γ

1Γ−γ

( ∫ ba w(t)h(t)du(t)∫ b

a w(t)du(t)−γ

).

The proof follows by Theorem 4 applied for the convex function Φ(t) = − ln t,t > 0.

3. Weighted inequalities for convex functions of selfadjoint operators

We can state the following result concerning the weighted Jensen’s inequality forcontinuous functions of selfadjoint operators:

Theorem 5 Let A be a selfadjoint operator on the Hilbert space H and assumethat Sp (A) ⊆ [m,M ] for some scalars m,M with m < M. If Φ : [γ,Γ] ⊂ R → Ris a continuous convex function on the interval [γ,Γ] , f : [m,M ] ⊂ R → R is acontinuous function on the interval [m,M ] and with the property that

(20) γ ≤ f (t) ≤ Γ for any t ∈ [m,M ]

and w : [m,M ] → [0,∞) is continuous on [m,M ], then

(21)

Φ

(⟨w (A) f (A)x, x⟩

⟨w (A) x, x⟩

)≤ ⟨w (A) (Φ f) (A) x, x⟩

⟨w (A)x, x⟩

≤Φ (γ)

(Γ− ⟨w (A) f (A)x, x⟩

⟨w (A) x, x⟩

)+ Φ(Γ)

(⟨w (A) f (A)x, x⟩

⟨w (A) x, x⟩− γ

)Γ− γ

,

for any x ∈ H with ∥x∥ = 1 and ⟨w (A) x, x⟩ = 0.

254 s.s. dragomir

Proof. Let Eλλ be the spectral family of the operator A. Let ε > 0 andwrite the inequality (8) on the interval [a, b] = [m− ε,M ] and for the monotonicnondecreasing function g (t) = ⟨Etx, x⟩ , x ∈ H with ∥x∥ = 1, to get

(22)

Φ

(∫M

m−εw (t) f (t) d ⟨Etx, x⟩∫M

m−εw (t) d ⟨Etx, x⟩

)≤∫M

m−εw (t) (Φ f) (t) d ⟨Etx, x⟩∫M


≤

(Γ−

∫M

m−εw (t) f(t)d ⟨Etx, x⟩∫M

m−εw(t)d ⟨Etx, x⟩

)Φ(γ)+

(∫M

m−εw(t)f(t)d ⟨Etx, x⟩∫M


− γ

)Φ(Γ)

Γ− γ.

Letting ε → 0+ and utilizing the spectral representation (1), we deduce from (22)the desired result (21).

Remark 2 If we choose w (t) = 1 and f (t) = t with t ∈ [m,M ] then we get from(21) the inequalities (MP) and (LR).

We have the following generalization and reverse for the Holder-McCarthyinequality:

Corollary 3 Let A be a selfadjoint positive operator on a Hilbert space H andassume that Sp (A) ⊆ [m,M ] for some scalars m,M with m < M . If the functionsf, w : [m,M ] → [0,∞) are continuous and f satisfies the condition (20) withγ ≥ 0, then for any p ≥ 1 we have

⟨w (A) f (A) x, x⟩p(23)

≤ ⟨w (A) f p (A) x, x⟩ ⟨w (A)x, x⟩p−1

≤ 1

Γ− γ⟨w (A) x, x⟩p−1

× [γp (⟨w (A) [Γ1H − f (A)] x, x⟩) + Γp (⟨w (A) [f (A)− γ1H ] x, x⟩)]

where x ∈ H with ∥x∥ = 1.If p ∈ (0, 1) then the inequalities reverse in (23).If γ > 0 and p < 0 the inequalities in (23) also hold.

Remark 3 If we choose w (t) = 1 and f (t) = t with t ∈ [m,M ] ⊂ [0,∞) thenwe get from (23)

⟨Ax, x⟩p ≤ ⟨Apx, x⟩(24)

≤ 1

M −m[mp (⟨(M1H − A) x, x⟩) +Mp (⟨(A−m1H) x, x⟩)]

for any p ≥ 1, where x ∈ H with ∥x∥ = 1.If p ∈ (0, 1) , then the inequalities reverse in (24).If m > 0 and p < 0 then the inequalities in (24) also hold.


Remark 4 If we choose w (t) = f (t) = t with t ∈ [m,M ] ⊂ [0,∞), then we getfrom (23)

(25)

⟨A2x, x⟩p ≤ ⟨Apx, x⟩ ⟨Ax, x⟩p−1

≤ 1

M−m⟨Ax, x⟩p−1 [mp (⟨A (M1H−A) x, x⟩) +Mp (⟨A (A−m1H)x, x⟩)]

for any p ≥ 1, where x ∈ H with ∥x∥ = 1.If p ∈ (0, 1) , then the inequalities reverse in (25).If m > 0 and p < 0 then the inequalities in (25) also hold.

Corollary 4 Let A be a selfadjoint positive operator on a Hilbert space H andassume that Sp (A) ⊆ [m,M ] for some scalars m,M with m < M . If the functionsf, w : [m,M ] → [0,∞) are continuous and f satisfies the condition (20) with γ > 0then

(26)

⟨w (A) f (A) x, x⟩⟨w (A)x, x⟩

≥ exp

[⟨w (A) (ln f) (A) x, x⟩

⟨w (A) x, x⟩

]≥ γ

1Γ−γ (Γ−

⟨w(A)f(A)x,x⟩⟨w(A)x,x⟩ )Γ

1Γ−γ (

⟨w(A)f(A)x,x⟩⟨w(A)x,x⟩ −γ)

for any x ∈ H with ∥x∥ = 1.

Remark 5 If we choose w (t) = 1 and f (t) = t with t ∈ [m,M ] ⊂ (0,∞) thenwe get from (26)

⟨Ax, x⟩ ≥ exp [⟨lnAx, x⟩](27)

≥ m1

M−m⟨(M1H−A)x,x⟩M

1M−m

⟨(A−1Hm)x,x⟩

for any x ∈ H with ∥x∥ = 1.Also, if we choose w (t) = f (t) = t with t ∈ [m,M ] ⊂ (0,∞) then we get

from (26) that

⟨A2x, x⟩⟨Ax, x⟩

≥ exp

[⟨A lnAx, x⟩⟨Ax, x⟩

](28)

≥ m1

M−m

(M−

⟨A2x,x⟩⟨Ax,x⟩

)M

1M−m

(⟨A2x,x⟩⟨Ax,x⟩ −m

)

for any x ∈ H with ∥x∥ = 1.

Remark 6 If we choose w (t) = tr and f (t) = tq with t ∈ [m,M ] ⊂ (0,∞) wherer, q > 0, then we get from (21) that

Φ

(⟨Ar+qx, x⟩⟨Arx, x⟩

)≤ ⟨ArΦ (Aq) x, x⟩

⟨Arx, x⟩(29)

≤Φ (γq)

(Γq − ⟨Ar+qx,x⟩

⟨Arx,x⟩

)+ Φ(Γq)

(⟨Ar+qx,x⟩⟨Arx,x⟩ − γq

)Γq − γq

,

for a continuous convex function Φ : [mq,M q] → R and for any x ∈ H with∥x∥ = 1.

256 s.s. dragomir

We have the following Holder type inequality for continuous functions ofselfadjoint operators:

Proposition 2 Let A be a selfadjoint positive operator on a Hilbert space Hand assume that Sp (A) ⊆ [m,M ] for some scalars m,M with m < M . Iff, g : [a, b] ⊂ R → Cr 0 are continuous on [a, b] and p, q ∈ Rr 0 with1/p+ 1/q = 1 are such that

(30) 0 ≤ γ ≤ |f (t)||g (t)|q−1 ≤ Γ for any t ∈ [a, b] ,

then we have the inequalities

⟨|f (A) g (A)|x, x⟩(31)

≤ [⟨|g (A)|q x, x⟩]1/q [⟨|f (A)|p x, x⟩]1/p

≤ 1

(Γ− γ)1/p⟨|g (A)|q x, x⟩

×[γp

(Γ− ⟨|f (A) g (A)| x, x⟩

⟨|g (A)|q x, x⟩

)+ Γp

(⟨|f (A) g (A)|x, x⟩

⟨|g (A)|q x, x⟩− γ

)]1/p,

for p > 1 and for any x ∈ H with ∥x∥ = 1 and ⟨|g (A)|q x, x⟩ = 0.If p ∈ (0, 1) , then the inequalities in (31) reverse;If p < 0 and γ > 0 then the inequalities in (31) also reverse.

4. Weighted inequalities for square-convex functions

We introduce the following class of complex valued functions:

Definition 1 A function Φ : [γ,Γ] ⊂ R → C is called square-convex on [γ,Γ]if the associated function φ : [γ,Γ] → [0,∞), φ (t) = |Φ (t)|2 is convex on [γ,Γ] .

A simple example of such a function is the concave power function Φ : [γ,Γ] ⊂[0,∞) → [0,∞), Φ (t) = tr with r ∈

[12, 1]. Also, if h : [γ,Γ] → [0,∞) is convex

then the complex valued function Φ : [γ,Γ] ⊂ R → C given by Φ (t) = h1/2 (t) eit

is square-convex on [γ,Γ] .Consider the function f (t) = ln (t+ 1) . We observe that it is concave and

positive on (0,∞) and if we define φ (t) = [ln (t+ 1)]2 , then we have that

φ′′ (t) =2

(t+ 1)2[1− ln (t+ 1)] , t > −1,

showing that f is square-convex on the interval [0, e− 1] .Another example for trigonometric functions is for instance f (t) = cos t,

t ∈[π4, π2

]. The function φ (t) = cos2 t has the second derivative φ′′ (t) = −2 cos (2t)

which is positive for t ∈[π4, π2

]. Therefore, f is square-convex on the interval[

π4, π2

].


Theorem 6 Let A be a selfadjoint operator on the Hilbert space H and assumethat Sp (A) ⊆ [m,M ] for some scalars m,M with m < M. If Φ : [γ,Γ] ⊂ R → Cis a continuous square-convex function on the interval [γ,Γ] , f : [m,M ] ⊂ R → Ris a continuous function on the interval [m,M ] and with the property that

(32) γ ≤ f (t) ≤ Γ for any t ∈ [m,M ]

and w : [m,M ] → [0,∞) is continuos on [m,M ], then

(33)

∣∣∣∣Φ(⟨w (A) f (A) x, x⟩⟨w (A)x, x⟩

)∣∣∣∣ ≤[⟨

w (A)(|Φ|2 f

)(A)x, x

⟩⟨w (A) x, x⟩

]1/2

≤

|Φ (γ)|2(Γ−⟨w (A) f (A)x, x⟩

⟨w (A) x, x⟩

)+ |Φ (Γ)|2

(⟨w (A) f (A) x, x⟩

⟨w (A)x, x⟩− γ

)Γ− γ

1/2

for any x ∈ H with ∥x∥ = 1 and ⟨w (A) x, x⟩ = 0.

The proof follows from Theorem 4 applied for the function φ : [γ,Γ] → [0,∞),φ (t) = |Φ (t)|2 that is continuous convex on [γ,Γ]. The details are omitted.

Remark 7 If w (t) = 1, then we get from (33) the following simpler result

(34)

|Φ (⟨f (A) x, x⟩)| ≤ ∥(Φ f) (A) x∥

≤

[|Φ (γ)|2 ⟨(Γ1H − f (A))x, x⟩+ |Φ (Γ)|2 ⟨(f (A)− 1Hγ) x, x⟩

Γ− γ

]1/2,

for any x ∈ H with ∥x∥ = 1.This is true since⟨(

|Φ|2 f)(A) x, x

⟩=

∫ M

m−0

|Φ (f (t))|2 d ⟨Etx, x⟩

= ∥Φ (f (A))x∥2

for any x ∈ H with ∥x∥ = 1 (for the second equality see for instance [19, p. 257]).

Corollary 5 With the assumptions of Theorem 6 for A, f, w and if γ > 0, thenwe have

(35)

(⟨w (A) f (A) x, x⟩

⟨w (A)x, x⟩

)q

≤[⟨w (A) f 2q (A) x, x⟩

⟨w (A)x, x⟩

] 12

≤

γ2q

(Γ− ⟨w (A) f (A)x, x⟩

⟨w (A) x, x⟩

)+ Γ2q

(⟨w (A) f (A)x, x⟩

⟨w (A) x, x⟩− γ

)Γ− γ

12

,

for any q ∈[12, 1]and any x ∈ H with ∥x∥ = 1 and ⟨w (A) x, x⟩ = 0.

258 s.s. dragomir

Remark 8 If we choose w (t) = 1 and f (t) = t with t ∈ [m,M ] ⊂ (0,∞) thenwe get from (35)

⟨Ax, x⟩q ≤ ∥Aqx∥(36)

≤[m2q ⟨(M1H − A) x, x⟩+M2q ⟨(A− 1Hm)x, x⟩

M −m

]1/2,

for any q ∈[12, 1]and any x ∈ H with ∥x∥ = 1.

Also, if we choose w (t) = f (t) = t with t ∈ [m,M ] ⊂ (0,∞) then we getfrom (35)(

⟨A2x, x⟩⟨Ax, x⟩

)q

≤[⟨A2q+1x, x⟩⟨Ax, x⟩

]1/2(37)

≤

m2q

(M − ⟨A2x,x⟩

⟨Ax,x⟩

)+M2q

(⟨A2x,x⟩⟨Ax,x⟩ −m

)M −m

1/2

,


Remark 9 If we choose w (t) = tr and f (t) = ts with t ∈ [m,M ] ⊂ (0,∞) wherer, s > 0, then we get from (35) that

(38)

(⟨Ar+sx, x⟩⟨Arx, x⟩

)q

≤[⟨Ar+2qsx, x⟩⟨Arx, x⟩

] 12

≤

m2qs

(M s − ⟨Ar+sx, x⟩

⟨Arx, x⟩

)+M2qs

(⟨Ar+sx, x⟩⟨Arx, x⟩

−ms

)M s −ms

12

,


5. Weighted inequalities for Arg-square-convex functions

The function Φ : C (0, 1) → C will be called Arg-square-convex if the compositefunction φ : [0, 2π] → [0,∞),

φ (t) :=

|Φ (eit)|2 , t ∈ [0, 2π)

lims→2π−

∣∣Φ (eis)∣∣2 , t = 2π

is continuous and convex on [0, 2π] .


To make the distinction between the value φ (0) = |Φ (ei0)|2 = |Φ (1)|2 and

the value φ (2π) = lims→2π−

∣∣Φ (eis)∣∣2, we denote by Φc (1) := lims→2π−

Φ(eis). With

this notation, we have φ (2π) = |Φc (1)|2 .The function Φn : C (0, 1) → C, Φn (z) = [Log (z)]n , where n is a positive

integer, is Arg-square-convex. We have

φn (t) =∣∣Φn

(eit)∣∣2 = ∣∣[Log (eit)]n∣∣2 = |it|2n = t2n, t ∈ [0, 2π),

andφn (2π) = lim

s→2π−

∣∣Φn

(eis)∣∣2 = |Φn,c (1)|2 = (2π)2n .

For q ≥ 12, define the function Φq : C (0, 1) → [0,∞) by Φq (z) = |Log (z)|q .

We have

φq (t) =∣∣Φq

(eit)∣∣2 = ∣∣Log (eit)∣∣2q = |it|2q = t2q, t ∈ [0, 2π)

andφq (2π) = lim

s→2π−

∣∣Φq

(eis)∣∣2 = |Φq,c (1)|2 = (2π)2q .

The function Φq for q ≥ 12is an Arg-square-convex function.

If g : [0, 2π] → [0,∞) is continuous and convex on [0, 2π] , then the compositefunction Φ : C (0, 1) → [0,∞) defined by

Φ (z) := [g (|Log (z)|)]1/2

is an Arg-square-convex function on C (0, 1) .

Theorem 7 Let U ∈ B (H) be a unitary operator on the Hilbert space Hand Φ : C (0, 1) → C a continuous and Arg-square-convex function on C (0, 1) .If w : C (0, 1) → [0,∞) is a continuous function, then we have

(39)

∣∣∣∣Φ(exp [⟨w (U)Log(U)x, x⟩⟨w (U) x, x⟩

])∣∣∣∣ ≤[⟨

w (U) |Φ (U)|2 x, x⟩

⟨w (U)x, x⟩

]1/2

≤

(2π − ⟨w (U) |Log(U)| x, x⟩

⟨w (U)x, x⟩

)|Φ(1)|2 + ⟨w (U) |Log(U)|x, x⟩

⟨w (U) x, x⟩|Φc(1)|2

2π

1/2

for any x ∈ H, ∥x∥ = 1, where Φc (1) := lims→2π−

Φ(eis).

Proof. We apply Theorem 4 to the function φ : [0, 2π] → [0,∞),

φ (t) =

|Φ (eit)|2 , t ∈ [0, 2π)

lims→2π−

∣∣Φ (eis)∣∣2 , t = 2π

that is continuous and convex on [0, 2π] .

260 s.s. dragomir

If Eλλ∈[0,2π] is the spectral family of the operator U, then we can writethe inequality (8) on the interval [a, b] = [0, 2π] for the monotonic nondecreasingintegrator u (t) = ⟨Etx, x⟩ and for the identity function f (t) = t, t ∈ [0, 2π] to get

(40)

∣∣∣∣∣Φ(exp

[i∫ 2π

0w (eit) td ⟨Etx, x⟩∫ 2π

0w (eit) d ⟨Etx, x⟩

])∣∣∣∣∣2

≤∫ 2π

0w (eit) |Φ (eit)|2 d ⟨Etx, x⟩∫ 2π


≤

(2π −

∫ 2π

0w (eit) td ⟨Etx, x⟩∫ 2π


)|Φ(1)|2+

(∫ 2π

0w (eit) Φ (t) d ⟨Etx, x⟩∫ 2π

0w (t) d ⟨Etx, x⟩

)|Φc(1)|2

2π

for any x ∈ H, ∥x∥ = 1.Since, by the spectral representation of functions of unitary operators (3),

we have

i

∫ 2π

0

w(eit)td ⟨Etx, x⟩ =

∫ 2π

0

w(eit)Log

(eit)d ⟨Etx, x⟩

= ⟨w (U)Log(U)x, x⟩∫ 2π

0

w(eit)d ⟨Etx, x⟩ = ⟨w (U)x, x⟩ ,∫ 2π

0

w(eit) ∣∣Φ (eit)∣∣2 d ⟨Etx, x⟩ =

⟨w (U) |Φ (U)|2 x, x

⟩and∫ 2π

0

w(eit)td ⟨Etx, x⟩ = ⟨w (U) |Log(U)|x, x⟩

for any x ∈ H, ∥x∥ = 1, then inequality (40) produces the desired result (39).

Remark 10 If w (t) = 1, then we get from (39) the following simpler result

(41)

|Φ (exp [⟨Log(U)x, x⟩])| ≤ ∥Φ (U) x∥

≤

[⟨(2π1H − |Log(U)|) x, x⟩ |Φ (1)|2 + ⟨|Log(U)|x, x⟩ |Φc (1)|2

2π

]1/2for any x ∈ H with ∥x∥ = 1.

This is true since⟨|Φ (U)|2 x, x

⟩=

∫ 2π

0

∣∣Φ (eit)∣∣2 d ⟨Etx, x⟩ = ∥Φ (U) x∥2

for any x ∈ H with ∥x∥ = 1 (for the second equality see (5)).

The interested reader may apply the inequality (39) for different examples ofArg-square-convex functions. We give here only one example, for instance if wechoose the function Φq (z) = |Log (z)|q , q ≥ 1/2 as introduced above, then we getfrom (39)


∣∣∣∣Log(exp [⟨w (U)Log(U)x, x⟩⟨w (U)x, x⟩

])∣∣∣∣q ≤[⟨

w (U) |Log (U)|2q x, x⟩

⟨w (U)x, x⟩

]1/2(42)

≤ ⟨w (U) |Log(U)|x, x⟩1/2

⟨w (U) x, x⟩1/2(2π)q−1/2

for any x ∈ H with ∥x∥ = 1 and w : C (0, 1) → [0,∞) a continuous function.In particular, we have

|Log (exp [⟨Log(U)x, x⟩])|q ≤ ∥|Log (U)|q x∥(43)

≤ (2π)q−1/2 ⟨|Log(U)|x, x⟩1/2

for any x ∈ H with ∥x∥ = 1.Finally, we notice that the following result providing Holder’s type inequalities

for continuous functions of unitary operators can be stated:

Proposition 3 Let U ∈ B (H) be a unitary operator on the Hilbert space H and.If f, g : C (0, 1) → Cr 0 are continuous on C (0, 1) and p, q ∈ Rr 0 with1/p+ 1/q = 1 are such that

(44) 0 ≤ γ ≤ |f (eit)||g (eit)|q−1 ≤ Γ for any t ∈ [0, 2π]

then we have the inequalities

⟨|f (U) g (U)|x, x⟩(45)

≤ [⟨|g (U)|q x, x⟩]1/q [⟨|f (U)|p x, x⟩]1/p

≤ 1

(Γ− γ)1/p⟨|g (U)|q x, x⟩

×[γp

(Γ− ⟨|f (U) g (U)| x, x⟩

⟨|g (U)|q x, x⟩

)+ Γp

(⟨|f (U) g (U)|x, x⟩

⟨|g (U)|q x, x⟩− γ

)]1/p,

for p > 1 and for any x ∈ H with ∥x∥ = 1 and ⟨|g (U)|q x, x⟩ = 0.If p ∈ (0, 1) , then the inequalities in (45) reverse;If p < 0 and γ > 0 then the inequalities in (45) also reverse.

The proof follows by Proposition 1 and the spectral representation for conti-nuous functions of unitary operators.

If g : [0, 2π] → [0,∞) is continuous and convex on [0, 2π] , then the compositefunction f : C (0, 1) → [0,∞) defined by

f (z) := [g (|Log (z)|)]1/2

is an Arg-square-convex function on C (0, 1) .

262 s.s. dragomir

As examples of such functions we have

fα (z) := exp (α |Log (z)|)

which are Arg-square-convex functions on C (0, 1) for any real number α = 0.We also notice that the family of functions

fm,n : C (0, 1) → C, fm,n (z) = zm [Log (z)]n ,

where m = 0 is an integer and n is a positive integer, are Arg-square-convexfunctions.

The reader may apply the above inequalities for these functions as well.However, the details are omitted.

References

[1] Dragomir, S.S., A converse result for Jensen’s discrete inequality viaGruss’ inequality and applications in information theory, An. Univ. OradeaFasc. Mat., 7 (1999/2000), 178–189.

[2] Dragomir, S.S., On a reverse of Jessen’s inequality for isotonic linear func-tionals, J. Ineq. Pure & Appl. Math., 2 (3) (2001), Article 36.

[3] Dragomir, S.S., A Gruss type inequality for isotonic linear functionals andapplications, Demonstratio Math, 36 (3) (2003), 551–562. Preprint RGMIARes. Rep. Coll., 5 (2002), Suplement, Art. 12. http://rgmia.org/v5(E).php

[4] Dragomir, S.S., Bounds for the normalized Jensen functional, Bull. Aus-tral. Math. Soc., 74 (3) (2006), 471-476.

[5] Dragomir, S.S., Bounds for the deviation of a function from the chordgenerated by its extremities, Bull. Aust. Math. Soc., 78 (2) (2008), 225–248.

[6] Dragomir, S.S., Gruss’ type inequalities for functions of selfadjoint opera-tors in Hilbert spaces, Preprint, RGMIA Res. Rep. Coll., 11 (e) (2008), Art.11. http://rgmia.org/v11(E).php

[7] Dragomir, S.S., Some inequalities for convex functions of selfadjoint ope-rators in Hilbert spaces, Filomat, 23 (3) (2009), 81–92. Preprint RGMIA Res.Rep. Coll., 11 (e) (2008), Art. 10.

[8] Dragomir, S.S., Some Jensen’s type inequalities for twice differentiablefunctions of selfadjoint operators in Hilbert spaces, Filomat, 23 (3) (2009),211-222. Preprint RGMIA Res. Rep. Coll., 11 (e) (2008), Art. 13.

[9] Dragomir, S.S., Some new Gruss’ type inequalities for functions ofselfadjoint operators in Hilbert spaces, Sarajevo J. Math., 6 (18) (1)(2010), 89-107. Preprint RGMIA Res. Rep. Coll., 11(e) (2008), Art. 12.http://rgmia.org/v11(E).php


[10] Dragomir, S.S., New bounds for the Cebysev functional of two functions ofselfadjoint operators in Hilbert spaces, Filomat, 24 (2) (2010), 27-39.

[11] Dragomir, S.S., Some Jensen’s type inequalities for log-convex functions ofselfadjoint operators in Hilbert spaces, Bull. Malays. Math. Sci. Soc., 34 (3)(2011). Preprint RGMIA Res. Rep. Coll., 13 (2010), Sup. Art. 2.

[12] Dragomir, S.S., Some reverses of the Jensen inequality for functions ofselfadjoint operators in Hilbert spaces, J. Ineq. & Appl., vol. 2010, Ar-ticle ID 496821. Preprint RGMIA Res. Rep. Coll., 11 (e) (2008), Art. 15.http://rgmia.org/v11(E).php

[13] Dragomir, S.S., Some Slater’s type inequalities for convex functions of self-adjoint operators in Hilbert spaces, Rev. Un. Mat. Argentina, 52 (1) (2011),109-120. Preprint RGMIA Res. Rep. Coll., 11 (e) (2008), Art. 7.

[14] Dragomir, S.S., Hermite-Hadamard’s type inequalities for operator convexfunctions, Appl. Math. Comp., 218 (2011), 766-772. Preprint RGMIA Res.Rep. Coll., 13 (1) (2010), Art. 7.

[15] Dragomir, S.S., Hermite-Hadamard’s type inequalities for convex functionsof selfadjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll.,13 (2) (2010), Art 1.

[16] Dragomir, S.S., New Jensen’s type inequalities for differentiable log-convexfunctions of selfadjoint operators in Hilbert spaces, Sarajevo J. Math., 19 (1)(2011), 67-80. Preprint RGMIA Res. Rep. Coll., 13 (2010), Sup. Art. 2.

[17] Dragomir, S.S., Ionescu, N.M., Some converse of Jensen’s inequalityand applications, Rev. Anal. Numer. Theor. Approx., 23 (1) (1994), 71–78.MR:1325895 (96c:26012).

[18] Furuta, T., Micic Hot, J., Pecaric, J., Seo, Y., Mond-PecaricMethod in Operator Inequalities. Inequalities for Bounded Selfadjoint Opera-tors on a Hilbert Space, Element, Zagreb, 2005.

[19] Helmberg, G., Introduction to Spectral Theory in Hilbert Space, John Wi-ley, New York, 1969.

[20] Matkovic, A., Pecaric, J., Peric, I., A variant of Jensen’s inequalityof Mercer’s type for operators with applications, Linear Algebra Appl., 418(2-3) (2006), 551–564.

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[22] Micic, J., Seo, Y., Takahasi, S.-E., Tominaga, M., Inequalities ofFuruta and Mond-Pecaric, Math. Ineq. Appl., 2 (1999), 83-111.

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264 s.s. dragomir

[24] Mond, B., Pecaric, J., On some operator inequalities, Indian J. Math.,35 (1993), 221-232.

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[26] Niculescu, C.P., An extension of Chebyshev’s inequality and its connectionwith Jensen’s inequality, J. Inequal. Appl., 6 (4) (2001), 451–462.

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φJ-MULTIPLIERS AND φJ-MULTIPLIERS QUADRATICON JORDAN BANACH ALGEBRAS

Abdelaziz Tajmouati

Sidi Mohamed Ben Abdellah UniversityFaculty of Sciences Dhar El MarhazFezMoroccoe-mail: [email protected]

Abstract. In this work, we define the notion of φJ-multipliers on Jordan-Banach

algebras without order and investigate some of their properties. We show that a φJ-

multiplier satisfies the condition T [Ux(y)] = Uφ(x)[T (y)]. This suggests that we define

a new concept which is the φJ-multiplier quadratic. We show several algebraic or topo-

logical properties for both concepts. In particular, we extend some known results for

φ-multipliers to φJ-multipliers and φJ-multipliers quadratic.

Keywords: Multiplier, J-Multiplier, J-multiplier quadratic, φ-Multiplier, φJ-Multiplier,

φJ-multiplier quadratic, Idempotent homomorphism, Quadratic operator, Spectrum of

an element, Product of Jordan, Jordan Banach Algebras, Algebra without order, Spe-

cial Jordan algebra, Full algebra. Strong topology.

2010 Mathematics Subject Classification: 47B48.

1. Introduction

The aim of this work is to study the φJ-Multipliers and φJ-Multipliers Quadraticin the case of Jordan-Banach algebras.

In the first part, we are going to resume the following definition of a φ-Multiplier T which is due to M. Adib and A. Riazi [1]:

T (x · y) = φ(x) · Ty = T (x) · φ(y).

We will show that some results that are true in the case of Banach algebras remaintrue in the case of Jordan-Banach. We will demonstrate that every φJ-multiplierof Jordan-Banach algebras verifies the relation:

T [Ux(y)] = Uφ(x)[T (y)], ∀x, y ∈ A,

266 a. tajmouati

and that this condition is not sufficient for a linear operator to be a φJ-multiplier.This suggests to us to define a new notion which is that of φJ-multiplier quadratic.

In the second part, we will define φJ-multipliers quadratic of Jordan-Banachalgebras. We will show that this generalized definition is the one given in thecase of Banach. We will compare the two notions just defined and show sometheorems of the type Wang. More particularly, we will establish some algebraicand topological relations between all φJ-multipliers quadraticMφJQ(A) of Jordan-Banach and the algebra of continuous linear operators of A, noted L(A). We willalso show some algebraic properties of MφJQ(A).

2. Preliminaries

Let K be a commutative field of characteristic zero.We call K-algebra every K-vector space A provided with a bilinear product

(x, y) → x.y of A×A in A. If the product is associative (resp. commutative), wesay that the algebra is associative (resp. commutative).

If A is a non-commutative algebra, we define the following operators: Rx(y) =xy and Lx(y) = yx. In this case, we note A+ the algebra of the same vector spacestructure as A, provided with product Defined by

x y =1

2(xy + yx),∀x, y ∈ A

is called Jordan product.If A is an associative algebra not commutative, the subalgebras of A+ are

called special Jordan algebras.Among the remarkable identities of these algebras, we note the two following

identities:x y = y x (C)

(x2 y) x = x2 (y x) (J)

More generally, an algebra (A, ·) that verifies the two previous identities is calledJordan algebra.

In a Jordan algebra A, we define the following applications:

Ux(y) = 2x(xy)− x2y

Ux,y =1

2(Ux+y − Ux − Uy)

The application x → Ux is quadratic, thus we have: Uax = a2Ux for every a ∈ Kand the application (x, y) → Ux,y is bilinear.

If A has a unity, then Rx = Ux,e = Ue,x.In the case of special Jordan algebra, we have

Ux(y) = xyx, (∀x, y ∈ A).

φj-multipliers and φj-multipliers quadratic ... 267

a ∈ A is invertible if and only if Ua is invertible in L(A) and we have

Ua−1 = (Ua)−1.

A normed Jordan algebra A is a Jordan algebra provided with a vector spacenorm ∥.∥ verifying

∥xy∥ ≤ ∥x∥∥y∥, (∀x, y ∈ A).

A Jordan Banach algebra (A, ∥.∥) is a normed Jordan algebra and complete forits norm.

The spectrum of an element x of Jordan algebra A is a set of scalars λ verifyingthat x− λe is not invertible in A.

The spectrum of an element of Jordan-Banach algebra is a nonempty com-pact.

3. The φJ-multipliers of Jordan Banach Algebra

In [2], Helgason has defined a multiplier T of Banach algebra by posing for x et yelements of A:

(Tx) · y = x · (Ty)

Then, he has showed that, if A is without order, then this definition is equivalent to

(3.1) T (x · y) = x · Ty

This remark has allowed Wang to establish a number of results on multipliers [9].As the relation (3.1) arises out quite simply of the associativity of A, we have

had the idea of taking it as a definition of a J-multiplier of Jordan Banach algebra[8], in order to find in the case Jordan some results of the type Wang.

In [1], the authors defined the concept of φ-multipliers and proved a someuseful results.

We propose in this section to extend the concept of φ-multiplier on JordanBanach algebras by asking the Definition 3.2.

Definition 3.1 Let A be a Jordan-Banach algebra. A is without order if for allx element of A, xA = 0 implies x = 0, or, for all x element of A, Ax = 0 impliesx = 0.

Obviously, if A has a unit or if A is semi-simple, then it is without order.

Definition 3.2 Let A be a Jordan-Banach algebra and φ an idempotent homo-morphism A. A left (resp.right) φJ-multiplier on A is a bounded linear mappingT of A in A such that T (x · y) = T (x)φ(y) (resp. T (x · y) = φ(x)T (y)) for allx, y ∈ A. We say T is a φJ-multiplier on A if it is both a left φJ-multiplier andright φJ-multiplier. We denote MφJ(A) the collection of all φJ-multipliers of A.

268 a. tajmouati

Remark 3.1 It is clear that the condition ”idempotent” is not necessary in thedefinition of a φJ-multiplier. But it plays an important role in the followingtheorems.

Remark 3.2 It is obvious that φ is a φJ-multiplier of A.

To illustrate this theory, we give two important examples of idempotent ho-momorphisms algebra. It is therefore an example of φ-multiplier and anotherφJ-multiplier:

Exemple 3.1 Let ω = (ωn)n≥0 be a sequence of real numbers such that ωn ≥ 1.Consider the space c0(ω) of all sequences (xn)n≥0 of complex number for whichlim

n−→∞xnωn = 0 (see [6]). We equip this space with the weighted supremum norm

∥ · ∥w given by

∥x∥w := supn≥0

|xnωn|, for all x = (xn)n≥0

Moreover, with respect to coordinatewise operations c0(ω) is a commutative Ba-nach algebra, consider the standard basis (en)n≥0 where en = (δnj)j≥0, then en isidempotent for all n ≥ 0.

Now, let the multiplication operator by the element ek

Lek : c0(ω) −→ c0(ω)

x −→ ekx

Since ek is idempotent then Lek is idempotent homomorphism of algebra c0(ω),indeed:

(Lek(xy) = ekxy = e2kxy = ekxeky = Lek(x)Lek(y) because c0(ω) is commutative )

Therefore, Lek is a φ-multiplier of A.

Exemple 3.2 LetA be a non-commutative Banach algebra, E=e ∈ A : e2 = edenote the set of idempotents in A and C(A) = x ∈ A : ax = xa, for all a in Athe centre of A.

The characterization of Banach algebras itempotents in C(A) has been ob-tained by Zemanek ([10], Theorem 5.2), he showed that an idempotent e ∈ Ebelongs to the centre C(A) if and only if e is an isolated point in the set E. Inother words, the centre of a Banach algebra meets the set of idempotents just inits isolated points.

Let e ∈ E belongs in C(A), and consider the multiplication operator by e.Then Le is an idempotent homomorphism algebra forA+ (special Jordan algebra).Therefore, Le is a φJ-multiplier of A+.

Proposition 3.1 Let A be a non-commutative Banach algebra. If T is a φ-multiplier of A, then T is a φJ-multiplier in special Jordan algebra A+.


Proof. Let A be a non-commutative Banach algebra and T a φ-multiplier of A.Then we have:

T (x y) = T

[1

2(x · y + y · x)

]=

1

2[T (x · y) + T (y · x)]

=1

2[φ(x)T (y) + T (y)φ(x)]

= φ(x) T (y).

T (x y) = T

[1

2(x · y + y.x)

]=

1

2[T (x · y) + T (y · x)]

=1

2[T (x)φ(y) + φ(y)T (x)]

= T (x) φ(y).

T (x y) = φ(x) T (y) = T (x) φ(y).

Therefore, T is a φJ-multiplier in special Jordan algebra A+.

Remark 3.3 The definition of a φJ-multiplier of Jordan Banach algebra gene-ralizes well that given in the case of Banach algebra.

For the adopted definition of a φJ-multiplier of Jordan Banach algebra, wehave the following theorems the demonstration of which is exactly the same inthe case of Banach algebras.

Theorem 3.1 Let A be a Jordan-Banach algebra without order and φ be anidempotent homomorphism. Then the set MφJ(A) is a closed and commutativesubalgebra of Banach algebra of continuous linear operators L(A), for the topo-logy of simple convergence. Moreover if A2 = A and φ commutes with any φJ-multiplier of A, then MφJ(A) is commutative and without order.

Theorem 3.2 Let A be a Jordan-Banach algebra without order. Then MφJ(A)is complete in the strong operator topology.

Proof. Suppose (Tλ) is a Cauchy net in the strong operator. Then, for each xin A, (Tλ(x)) is a Cauchy net in A and hence there exists T (x) in A such thatlim

λ→+∞∥Tλ(z)− T (z)∥ = 0.

If x,y in A, then:

∥T (xy)− φ(x)T (y)∥ ≤ ∥T (xy)− Tλ(xy)∥+ ∥Tλ(xy)− φ(x)T (y)∥∥T (xy)− φ(x)T (y)∥ ≤ ∥T (xy)− Tλ(xy)∥+ ∥φ(x)Tλ(y)− φ(x)T (y)∥∥T (xy)− φ(x)T (y)∥ ≤ ∥T − Tλ∥∥xy∥+ ∥φ(x)∥∥Tλ − T∥∥y∥

270 a. tajmouati

and so T (xy) = φ(x)T (y).The same way, we show that T (xy) = T (x)φ(y), for all x, y in A. Therefore,

T in MφJ(A) and MφJ(A) is complete in the strong operator topology.

Theorem 3.3 Let A be a Jordan-Banach algebra without order and φ be anisomorphism from A into A. If T is a φJ-multiplier of A, then the followingstatements are equivalent:

i) T is bijective.

ii) T−1 exists and T−1 ∈ Mφ−1J(A).

Corollary 3.1 Let A be a Jordan-Banach unitary algebra. Then, for every mul-tiplier T , we have:

SpMφJ (A)(A) = SpL(A)(T ).

MφJ(A) is a full subalgebra de L(A).

Remark 3.4 Like all Jordan Banach algebra are commutative, we have the fol-lowing theorem whose proof is exactly the same as in the case of Banach algebras.

Theorem 3.4 Let A be a Jordan Banach algebra without order and φ be a ho-momorphism from A to A with dense range. Then T is a φJ-multiplier if, andonly if, Tx2 = φ(x)T (x) for all x in A.

Corollary 3.2 Let A be a Jordan Banach Algebra without order and T a linearmap from A to A. Then T is a φJ-multiplier of A if, and only if, Tx2 = xT (x)for all x in A.

Proof. Just take φ(x) = x.

Proposition 3.2 Let A be a Banach algebra and T a φ-multiplier of A. Then:

i) We have

(3.2) T Ux T = Uφ(x) T 2, ∀x ∈ A.

ii) (3.2) remains true in special algebra A+.

Proof. Let A be a Banach algebra and T ∈ MφJ(A).

i) We have :Ux(T (y)) = x2 · T (y), ∀x, y ∈ A

Therefore:

T [Ux(T (y))] = T [x2 · T (y)]= φ(x2) · T [T (y)]= [φ(x)]2 · T [T (y)]= Uφ(x)[T

2(y)]

Therefore we have :

T [Ux(T (y))] = Uφ(x)[T2(y)], ∀x, y ∈ A


ii) In every special Jordan algebra, we have:

Ux(y) = x · y · x, ∀x, y ∈ A

Therefore, we have:Ux(T (y)) = x · T (y) · x.

Thus,

T [Ux(T (y))] = T [x · T (y) · x]= φ(x)T [T (y) · x]= φ(x) · T [T (y)] · φ(x)= Uφ(x)[T

2(y)]

whenceUφ(x)[T

2(y)] = T Ux T (y), ∀x, y ∈ A.

Taking this into account, it is normal to ask ourselves the following questions:Given a Jordan-Banach algebra A and a φJ-multiplier T , is the (3.2) relationverified? This is the purpose of the following proposition:

Proposition 3.3 Let A be a Jordan-Banach algebra. If T is φJ-multiplier of A,we have:

T Ux T = Uφ(x) T 2, ∀x ∈ A

Proof. Let T ∈ MφJ(A) and x, y ∈ A. Then we have:

TUxT (y) = T [Ux(T (y))]

= T [2x · (x · T (y))− x2 · T (y)]= 2φ(x) · T [x · T (y)]− φ(x2) · T [T (y)]= 2φ(x) · [φ(x) · T [T (y)]− [φ(x)]2 · T 2(y)

= 2φ(x) · [φ(x) · T 2(y)− [φ(x)]2 · T 2(y))

= Uφ(x) T 2(y), ∀x, y ∈ A

Consequently, Uφ(x) T 2 = T Ux T , for all element x in A.

Proposition 3.4 Let A be a Jordan-Banach algebra. If T is a φJ-multiplier ofA, then for x and y elements of A, we have:

(3.3) T [Ux(y)] = Uφ(x)[T (y)]

Proof. Let T be a φJ-multiplier of A, x and y elements of A. Then we have:

T [Ux(y)] = T [2x · (x · y)− x2 · y]= φ(2x) · T (x · y)− φ(x2) · T (y)= 2φ(x) · [φ(x) · T (y)]− φ(x)2 · T (y)= Uφ(x)[T (y)].

272 a. tajmouati

Remark 3.5 Let A be a Jordan-Banach algebra. If T is a φJ-multiplier of A,then for x and y elements of A, we have:

(3.4) T [Uφ(x)(y)] = Uφ(x)[T (y)]

Proof. Let T be a φJ-multiplier of A, x and y elements of A. Then we have:

T [Uφ(x)(y)] = T [2φ(x) · (φ(x) · y)− (φ(x))2 · y]= 2φ(φ(x))T (φ(x) · y)− φ((φ(x))2)T (y)

= 2φ(x)[φ(x)T (y)]− [φ(x)]2T (y)

= Uφ(x)[T (y)]

Remark 3.6 The previous Proposition 3.3 and Remark 3.4. suggest to define anew notion which is that of φJ-multiplier quadratic; this is what we are doing inthe next section.

4. φJ-Multipliers quadratic of Jordan-Banach algebra

Definition 4.3 Let A be Jordan-Banach algebra, φ an idempotent homomor-phism A and T : A −→ A a continuous linear application. We say that T is aφJ-multiplier quadratic of A if, we have:

T [Ux(y)] = Uφ(x)[T (y)], ∀x, y ∈ A

The set of a φJ-multiplier quadratic of A is noted MφJQ(A).

Remark 4.7 φ is an element of MφJQ(A).

Proposition 4.5 Let A be Banach algebra. If T is a φ-multiplier of A, then wehave:

T [Ux(y)] = Uφ(x)[T (y)], ∀x, y ∈ A

Proof. T is a φ-multiplier of A, then we have:

T (x.y) = T (x).φ(y) = φ(x).T (y),∀x, y ∈ A

Therefore,

T [Ux(y)] = T (x2 · y) = T [x · (x · y)] = φ(x) · T (x · y)= φ(x) · [φ(x) · T (y)] = φ(x)2 · T (y) = Uφ(x)[T (y)].

Proposition 4.6 Let A be Banach algebra. If T is a φ-multiplier of A, then Tis a φJ-multiplier quadratic in the special Jordan algebra A+.

Proof. As in A+, Ux(y) = x.y.x, ∀x, y ∈ A, we have:

T [Ux(y)] = T (x · y · x) = φ(x) · T (y · x) = φ(x) · T (y) · φ(x) = Uφ(x)[T (y)].


Remark 4.8 The definition of φJ-multiplier quadratic well generalizes that ofφ-multiplier given in the case of Banach.

Remark 4.9 Let A be Jordan-Banach algebra. Then we have

MφJ(A) ⊂ MφJQ(A).

In what follows, L(A) will denote the Banach algebra of all continuous linearoperators from A to A.

Theorem 4.5 Let A be Jordan-Banach algebra. Then, MφJQ(A) is a closedsubalgebra of L(A), for the topology of simple convergence, which contains φ.

Proof. Let T and S be two elements of MφJQ(A) and α ∈ C. For any twoelements x and y of A, we have:

Uφ(x)[(αT )(y)] = Uφ(x)[α(T (y))] = αUφ(x)(T (y)) = α[T (Ux(y)].

Therefore, αT ∈ MφJQ(A).

(TS)[Ux(y)] = T [S(Ux(y)] = T [Uφ(x)(S(y))] = Uφ(φ(x))[T (S(y))] = Uφ(x))[(TS)(y))]

Therefore, TS ∈ MφJQ(A).

Uφ(x)[(T + S)(y)] = Uφ(x)[T (y) + S(y)] = Uφ(x)[T (y)] + Uφ(x)[S(y)]

= T [Ux(y)] + S[Ux(y)]

= (T + S)[Ux(y)]

Therefore, (T + S) ∈ MφJQ(A). Therefore, MφJQ(A) is a subalgebra of L(A)which contains the φ. Let’s show that M(A) is closed in L(A) for the topology ofsimple convergence. (Tλ)λ∈Λ is a suite of elements of MφJQ(A) and T an elementde L(A) such that, for every element z de A, we have:

limλ→+∞

∥Tλ(z)− T (z)∥ = 0

As with any λ ∈ Λ and Tλ ∈ MφJQ(A), we have:

Uφ(x)[Tλ(y)] = Tλ[Ux(y)], ∀x, y ∈ A

Therefore:

∥T [Ux(y)]− Uφ(x)[T (y)]| ≤ ∥T [Ux(y)]− Tλ[Ux(y)]|+ ∥Uφ(x)[Tλ(y)− Uφ(x)[T (y)]|

Since the application z → Ux(z) is continuous on A and (Tλ(z))λ converges toT (z) for every z in A then: (∀x, y ∈ A), (∀ϵ > 0), (∃λ ∈ Λ) such that

∥T [Ux(y)]− Tλ[Ux(y)]| ≤ ϵ/2

∥Uφ(x)[Tλ(y)]− Uφ(x)[T (y)]| ≤ ϵ/2.

Therefore,∥T [Ux(y)]− Uφ(x)[T (y)]| ≤ ϵ.

Then, we have:T [Ux(y)] = Uφ(x)[T (y)],∀x, y ∈ A,

i.e., T ∈ MφJQ(A).

274 a. tajmouati

Remark 4.10 For φJ multipliers, we have shown that if A2 = A, then MφJ iswithout order. We will show that in the quadratic case, we only assume that Ais unitary. Hence we have the following proposition.

Proposition 4.7 Let A be Jordan-Banach algebra with identity e. Then, MφJQ(A)is without order.

Proof. Let T a element of MφJQ(A) such that ST = 0 for each T ∈ MφJQ(A).We have, for all x, y in A

T [Ux(y)] = Uφ(x)[T (y)]

= Uφ(φ(x))T (y)

= φ[Uφ(x)(T (y)]

= Uφ(x)[φ(T (y)]

As φ is a φJ-multiplier quadratic and the application z → Ux(z) is linear, thenφ(T (y) = 0. Therefore, Uφ(x)[φ(T (y)] = 0, where T [Ux(y)] = 0, ∀x, y ∈ A.

In particular, if we take x = e, we obtain T (y) = 0, for all y in A. Therefore,T = 0.

Theorem 4.6 Let A be Jordan-Banach algebra. Then, MφJQ(A) is complete inthe strong operator topology.

Proof. Suppose (Tλ)λ∈Λ is a Cauchy net in the strong operator topology. Thenfor each element z in A, (Tλ(z))λ∈Λ is a cauchy net in A that is complete andhence there exists T (z) in A such that:

limλ→+∞

∥Tλ(z)− T (z)∥ = 0.

It is conventional to show that T is linear continuous and that

limλ→+∞

∥Tλ − T∥ = 0.

For any two elements x and y of A, we have:

∥T [Ux(y)]− Uφ(x)[T (y)]∥ ≤ ∥T [Ux(y)]− Tλ[Ux(y)]∥+ ∥Uφ(x)[Tλ(y)− Uφ(x)[T (y)]∥∥T [Ux(y)]− Uφ(x)[T (y)]∥ ≤ ∥(T − Tλ)(Ux(y))∥+ ∥(Uφ(x))[Tλ(y)− T (y)]∥

Since Tλ, T and z → Ux(z) are linear and continuous applications on A, then:

∥(T − Tλ)(Ux(y))∥ ≤ ∥T − Tλ∥∥Ux(y)∥

and

∥(Uφ(x))[Tλ(y)− T (y)]∥ ≤ ∥Uφ(x)∥∥T − Tλ∥∥y∥.


We havelim

λ→+∞∥Tλ − T∥ = 0.

Then∥T [Ux(y)]− Uφ(x)[T (y)]∥ = 0,

and soT [Ux(y)] = Uφ(x)[T (y)],∀x, y ∈ A.

Therefore, T ∈ MφJQ(A) and MφJQ(A) is complete in the strong operator to-pology.

Theorem 4.7 Let A be a Jordan-Banach algebra without order and φ be anisomorphism from A into A. If T is a φJ-multiplier quadratic of A, then thefollowing statements are equivalent:

i) T is bijective.

ii) T−1 exists and T−1 ∈ Mφ−1JQ(A).

Proof.

ii) ⇒ i) Obvious

i) ⇒ ii) T−1is linear. It is continuous according to the theorem of open applica-tion. On the other hand, we have by hypothesis:

T [Ux(y)] = Uφ(x)[T (y)], ∀x, y ∈ A

Let x and z are two elements of A. We pose y = T−1(z).

Then, we have

T [Uφ−1(x)(T−1(z))] = Ux[T (T

−1(z))] = Ux(z).

Thus, we obtain

Uφ−1(x)(T−1(z)) = T−1[Ux(z)], ∀x, z ∈ A.

Hence T−1 ∈ Mφ−1JQ(A).

Corollary 4.3 Let A be Jordan-Banach algebra without order and T ∈ MφJQ(A).Then, we have

SpMφJQ(A)(T ) = SpL(A)(T )

i.e., MφJQ(A) is a full subalgebra of L(A).

Acknowledgement. The author would like to thank the referee for helpfulsuggestions and remarks.

276 a. tajmouati

References

[1] Adib, M., Riazi, A., φ-Multipliers on Banach Algebras without order. Int.Journal of Math. Analysis, 3 (3) (2009), 121-132.

[2] Helgason, S., Multipliers of Banach Algebras. Ann. Maths., 64 (1956),240-254.

[3] Husain, T., Multipliers of topological Algebras. Dissertationes MathematicaCCLXXXV, Warszawa (1989).

[4] Jacobson, N., Structure and representations of Jordan Algebras. Amer.Maths. Soc. Providence, 1968.

[5] Larsen, R., An Introduction to the theory of multipliers, Springer-verlagBerlin Heidelberg New-York, 1971.

[6] Laursen, K.B., Neumann, M.M.,, An introduction to Local SpectralTheory, London Mathematical Society Monograph, New series, vol. 20,Clarendon Press, Oxford, 2000.

[7] Martinez Moreno, J., Sobre algebras de Jordan normadas completas.Tesis doctoral, Universidad de Granada, 1977.

[8] Tajmouati, A., On J-Multipliers and J-Multipliers Quadratic in JordanBanach Algebras. Int. Journal of Math. Analysis, 8 (4) (2014), 195-207.

[9] Wang, J.K., Multipliers of commutative Banach algebras. Pacific J. Math.,1961, 1131-1149.

[10] Zemanek, J., Properties of the spectral radius in Banach algebras. Spectraltheory, Banach centre publication, vol. 8, Pwn-Polish Scientific PublicationWarsaw, 1982.

Accepted: 2.11.2013


ANOTE ON BOOLEAN SUBSETS OF ORTHOMODULAR POSETS1

Dietmar Dorninger

Helmut Langer

Vienna University of TechnologyFaculty of Mathematics and GeoinformationInstitute of Discrete Mathematics and GeometryWiedner Hauptstraße 8-101040 ViennaAustriae-mails: [email protected]

[email protected]

Abstract. Modelling quantum systems by orthomodular posets P = (P,≤,′ , 0, 1) gives

rise to the question, when a finite subset A of P lies within a Boolean subalgebra of P,

in which case A is called Boolean. Boolean subsets A specify the physical subsystem

represented by A to be classical. We give a characterization of a subset of P to be

Boolean by only taking into account terms of elements of this subset and in such a way

that an inductive algorithm can be derived.

Keywords: orthomodular poset, Boolean algebra.

AMS Subject Classification: 06C15, 06E99, 03G12, 81P10

1. Introduction

Orthomodular lattices, and more general, orthomodular posets P = (P,≤,′ , 0, 1)have been intensively studied as models for quantum logics (cf., e.g., [2], [5], [6], [7]and [8]). It is well known that these algebraic structures correspond to classicalmechanical systems if and only if they are Boolean algebras. However, if onedeals with only a finite subset A of P the question arises whether the physicalsubsystem represented by A is classical, which then means that A lies within aBoolean subalgebra of P . If this is the case, then A will be called Boolean. Forthis definition, cf. the pioneering paper [1], in which the question of Booleansubsets was settled for the special case of so-called algebras of numerical events.For the general case of arbitrary orthomodular posets, Boolean subsets have been

1Support of the research of the second author by OAD, Cooperation between Austria andCzech Republic in Science and Technology, grant No. CZ 03/2013, is gratefully acknowledged.

278 d. dorninger, h. langer

identified by means of a generalization of the compatibility relation of two elements(cf. [3], [4] and [8]).

In this paper, we will stick to the classical commutativity relation of only twoelements and will only take into account terms of elements of the given finiteset of elements of P to answer the question, whether this set is Boolean. Thecharacterization we derive will be inductive and hence will give rise to a step bystep procedure.

We begin by recollecting the definition of an orthomodular poset and some ofits properties.

An orthomodular poset is a ordered quintuple P = (P,≤,′ , 0, 1) such that(P,≤, 0, 1) is a bounded poset and ′ is a unary operation on P satisfying thefollowing conditions for all x, y ∈ P :

(i) x ≤ y implies x′ ≥ y′.

(ii) (x′)′ = x

(iii) x ∨ x′ = 1

(iv) If x ⊥ y, i. e. if x ≤ y′, in which case x and y are said to be orthogonal(to each other), then x ∨ y, the supremum of x and y, exists.

(v) If x ≤ y then the orthomodular law y = x ∨ (y ∧ x′) holds.

y∧x′ denotes the infimum of y and x′. It is easy to see that within an orthomodularposet the existence of x∨y implies the existence of x′∧y′ and that the de Morganlaws (x ∨ y)′ = x′ ∧ y′ and (x ∧ y)′ = x′ ∨ y′ hold, in the sense that if one side isdefined then so is the other and they are equal. Moreover, if x ≤ y then x ⊥ y′

and hence x∨ y′ exists which in turn shows the existence of x′∧ y = (x∨ y′)′. Theexistence of the term on the right-hand side of the orthomodular law is securedby the fact that x ⊥ (y ∧ x′).

Two elements a and b of an orthomodular poset P = (P,≤,′ , 0, 1) are saidto commute (with each other) – in short, a C b – if there exist three mutuallyorthogonal elements c, d, e of P such that a = c∨d and b = d∨ e. It is well known(cf. e. g. [8]) that the elements c, d, e are unique if they exist, namely c = a ∧ b′,d = a ∧ b and e = a′ ∧ b.

There are numerous characterizations and implications concerning the pro-perty a C b (cf. e. g. [2] and [8]). We list the very conditions we need for ourfurther considerations, and when this seems essential, we will refer to them.

Properties of C

For an orthomodular poset P = (P,≤,′ , 0, 1) the following conditions for a, b ∈ Pare equivalent:

(i) a C b

(ii) The subset a, b of P is Boolean, i. e. there exists a Boolean subalgebra ofP containing a and b.

a note on boolean subsets of orthomodular posets 279

(iii) (x ∨ y) ∧ z = (x ∧ z) ∨ (y ∧ z) for three arbitrary but fixed elements of thefour elements a, a′, b, b′

(iv) (x ∧ y) ∨ z = (x ∨ z) ∧ (y ∨ z) for three arbitrary but fixed elements of thefour elements a, a′, b, b′

Moreover, the following holds:

(v) If a C b then x C y for all x, y ∈ 0, a, a′, b, b′, 1.

(vi) If a ≤ b then a C b.

(vii) If a ⊥ b then a C b.

(viii) If a C b then a ∨ b and a ∧ b exist.

2. Characterizing Boolean subsets

As usual, we will write∨i∈I

si and∧i∈I

si for the supremum and infimum, respec-

tively, of the elements si, i ∈ I, and we will use the notation∨

S and∧

S ifS = si | i ∈ I. Assume that S is a finite subset of pairwise orthogonal elementsof an orthomodular poset P = (P,≤,′ , 0, 1). Then it can immediately be seenthat

∨S exists.

The following result is well-known (cf. [3], [4] and [8]):

Lemma 2.1. Let A be a finite subset of mutually orthogonal elements of an ortho-modular poset P = (P,≤,′ , 0, 1). Then the subset

∨D |D ⊆ A of P is Boolean.

Now, we can prove our main theorem:

Theorem 2.2. Let P = (P,≤,′ , 0, 1) be an orthomodular poset, 1 ≤ k < n andA an n-element subset of P . If any k-element subset of A is Boolean and (

∧B)

C (∧

D) for all k-element subsets B and D of A then also any (k + 1)-elementsubset of A is Boolean.

Proof. With regard to the properties of C the case k = 1 is obvious. Now assumek > 1 and let A be the set a1, ..., an.First, we show by induction on s that for all s = k, k − 1, ..., 1 we have

a1 ∧ ... ∧ as ∧ a′s+1 ∧ ... ∧ a′k C ak+1.

Since (a1 ∧ ... ∧ ak) C (a2 ∧ ... ∧ ak+1) the infimum

(a1 ∧ ... ∧ ak) ∧ (a2 ∧ ... ∧ ak+1) = a1 ∧ ... ∧ ak+1

exists, and due to property (iii) of C we obtain


a1 ∧ ... ∧ ak = (a1 ∧ ... ∧ ak) ∧ ((a2 ∧ ... ∧ ak+1) ∨ (a2 ∧ ... ∧ ak+1)′)

= ((a1 ∧ ... ∧ ak) ∧ (a2 ∧ ... ∧ ak+1)) ∨ ((a1 ∧ ... ∧ ak) ∧ (a′2 ∨ ... ∨ a′k+1))

= (a1 ∧ ... ∧ ak+1) ∨ (a1 ∧ ... ∧ ak ∧ (a′2 ∨ ... ∨ a′k+1))

= (a1 ∧ ... ∧ ak+1) ∨ (a1 ∧ ((a2 ∧ ... ∧ ak) ∧ ((a2 ∧ ... ∧ ak)′ ∨ a′k+1)))

= (a1 ∧ ... ∧ ak+1) ∨ (a1 ∧ ((a2 ∧ ... ∧ ak) ∧ a′k+1))

= (a1 ∧ ... ∧ ak+1) ∨ (a1 ∧ ... ∧ ak ∧ a′k+1)

= ((a1 ∧ ... ∧ ak) ∧ ak+1) ∨ ((a1 ∧ ... ∧ ak) ∧ a′k+1).

Hence, again by property (iii) of C, a1∧ ...∧ak C ak+1, which proves our assertionfor s = k.

Now, assume 1 < s ≤ k and

a1 ∧ ... ∧ ai ∧ a′i+1 ∧ ... ∧ a′k C ak+1

for all i = s, ..., k. Our goal is to show that

a1 ∧ ... ∧ as−1 ∧ a′s ∧ ... ∧ a′k C ak+1.

According to the properties of C,

a1 ∧ ... ∧ as ∧ a′s+1 ∧ ... ∧ a′k C a′k+1

and hence

a1 ∧ ... ∧ as ∧ a′s+1 ∧ ... ∧ a′k ∧ a′k+1 = (a1 ∧ ... ∧ as ∧ a′s+1 ∧ ... ∧ a′k) ∧ a′k+1

exist. In the following, let us denote a by a1 and a′ by a−1. For reasons ofsymmetry also ai11 ∧ ... ∧ a

ik+1

k+1 exists whenever

|j ∈ 1, ..., k + 1 | ij = 1| ≥ s.

Now, we compute by relying on the properties of C

(a1 ∧ ... ∧ as−1 ∧ a′s ∧ ... ∧ a′k) ∧ (a′k+1 ∨ (a1 ∧ ... ∧ as−1 ∧ a′s ∧ ... ∧ a′k)′)

= (a1 ∧ ... ∧ as−1 ∧ a′s ∧ ... ∧ a′k) ∧ (a′k+1 ∨ a′1 ∨ ... ∨ a′s−1 ∨ as ∨ ... ∨ ak)

= (a1 ∧ ... ∧ as−1 ∧ a′s ∧ ... ∧ a′k−1) ∧∧(a′k ∧ (ak ∨ (a′1 ∨ ... ∨ a′s−1 ∨ as ∨ ... ∨ ak−1 ∨ a′k+1)))

= (a1 ∧ ... ∧ as−1 ∧ a′s ∧ ... ∧ a′k−1) ∧∧(a′k ∧ (a′1 ∨ ... ∨ a′s−1 ∨ as ∨ ... ∨ ak−1 ∨ a′k+1))

= a′k ∧ ((a1 ∧ ... ∧ as−1 ∧ a′s ∧ ... ∧ a′k−1) ∧∧((a1 ∧ ... ∧ as−1 ∧ a′s ∧ ... ∧ a′k−1)

′ ∨ a′k+1))

= a′k ∧ ((a1 ∧ ... ∧ as−1 ∧ a′s ∧ ... ∧ a′k−1) ∧ a′k+1)

= a1 ∧ ... ∧ as−1 ∧ a′s ∧ ... ∧ a′k−1 ∧ a′k ∧ a′k+1

= (a1 ∧ ... ∧ as−1 ∧ a′s ∧ ... ∧ a′k−1 ∧ a′k) ∧ a′k+1

a note on boolean subsets of orthomodular posets 281

showinga1 ∧ ... ∧ as−1 ∧ a′s ∧ ... ∧ a′k C ak+1

which completes the proof by induction.We have shown that

a1 ∧ ai22 ∧ ... ∧ aikk C ak+1.

for all i2, ..., ik ∈ −1, 1 and we point out that the elements

ai11 ∧ ... ∧ aik+1

k+1 = (ai11 ∧ ... ∧ aikk ) ∧ (ai22 ∧ ... ∧ aik+1

k+1 )

(i1, ...ik+1 ∈ −1, 1) exist and are mutually orthogonal. Further, we obtain

a1 =∨

i2,...,ik∈−1,1

(a1 ∧ ai22 ∧ ... ∧ aikk )

=∨

i2,...,ik∈−1,1

(((a1 ∧ ai22 ∧ ... ∧ aikk ) ∧ ak+1) ∨ ((a1 ∧ ai22 ∧ ... ∧ aikk ) ∧ a′k+1))

=∨

i2,...,ik∈−1,1

((a1 ∧ ai22 ∧ ... ∧ aikk ∧ ak+1) ∨ (a1 ∧ ai22 ∧ ... ∧ aikk ∧ a′k+1))

=∨

i2,...,ik+1∈−1,1

(a1 ∧ ai22 ∧ ... ∧ aik+1

k+1 ).

For reasons of symmetry, we also have

aj =∨

i1,...,ij−1,ij+1,...,ik+1∈−1,1

(ai11 ∧ ... ∧ aij−1

j−1 ∧ aj ∧ aij+1

j+1 ∧ ... ∧ aik+1

k+1 )

for all j = 1, ..., k + 1 which, according to Lemma 2.1, shows that a1, ..., ak+1is Boolean. By symmetry, it follows that every (k + 1)-element subset of A isBoolean.

Corollary 2.3. Let P = (P,≤,′ , 0, 1) be an orthomodular poset, n > 1 and A ann-element subset of P . Then A is Boolean if and only if (

∧B) C (

∧D) for every

k ∈ 1, ..., n− 1 and every k-element subsets B and D of A.

Remark 2.4. According to Theorem 2.2,∧

B exists for all k-element subsets ofA, k ∈ 1, 2, ..., n− 1.

3. Remarks about algorithmic aspects

If one wants to check whether a subset A = a1, ..., an of an orthomodular posetP is Boolean, one can proceed as follows:

(1) Check whether ai C aj for i, j ∈ 1, 2, ..., n, i = j, e. g. by looking for theexistence of ai ∧ aj, ai ∧ a′j and whether (ai ∧ aj) ∨ (ai ∧ a′j) = ai. If theseelements exist and the equation holds (which is equivalent to ai C aj) and ifP will be a lattice (as in the case of a Hilbert space logic) then one is done;A is Boolean. If these elements exist and the equation holds and P is not alattice then


(2) for r ∈ 1, ...,(n2

), denote the elements ai ∧ aj, i, j ∈ 1, ..., n, i = j, by br

and check, whether br C bs for r, s ∈ 1, ...,(n2

), r = s. Assuming, this is

the case then

(3) replenish the elements br to triples ai ∧ aj ∧ ak, i, j, k ∈ 1, ...,(n3

), i = j =

k = i, and so on,

until one comes across a pair (∧

B,∧D) of infima of subsets B and D of A with

|B| = |D| ≤ n − 1 which do not commute, in which case A is not Boolean;otherwise, the subset A is Boolean.

References

[1] Beltrametti, E.G., Maczynski, M.J., On the characterization of pro-babilities: A generalization of Bell’s inequalities, J. Math. Phys., 34 (1993),4919–4929.

[2] Beran, L., Orthomodular Lattices, Algebraic Approach, Academia, Prague,1984.

[3] Brabec, J., Compatibility in orthomodular posets, Casopis Pest. Mat., 104(1979), 149–153.

[4] Brabec, J., Ptak, P., On compatibility in quantum logics, Found. Phys.,12 (1982), 207–212.

[5] Dvurecenskij, A., Pulmannova, S., New Trends in Quantum Structures,Kluwer, Dordrecht, 2000.

[6] Kalmbach, G., Orthomodular Lattices, Academic Press, London, 1983.

[7] Maczynski, M.J., Traczyk, T., A characterization of orthomodular par-tially ordered sets admitting a full set of states, Bull. Polish Acad. Sci. Math.,21 (1973), 3–8.

[8] Ptak, P., Pulmannova, S., Orthomodular Structures as Quantum Logics,Kluwer, Dordrecht, 1991.



SOME DERIVATIONS ON THE BOUNDSFOR THE ZEROS OF ENTIRE FUNCTIONSBASED ON SLOWLY CHANGING FUNCTIONS

Sanjib Kumar Datta

Department of MathematicsUniversity of KalyaniKalyani, Dist-Nadia, PIN–741235West BengalIndiaemail: sanjib kr [email protected]

Dilip Chandra Pramanik

Department of MathematicsUniversity of North BengalRaja Rammohunpur, Dist-Darjeeling, PIN–734013West BengalIndiaemail: [email protected]

Abstract. A single valued function of one complex variable which is analytic in the fi-

nite complex plane is called an entire function. The purpose of this paper is to establish

the bounds for the moduli of zeros of entire functions in the light of slowly changing

functions.

Keywords: zeros of entire functions, proper ring shaped region.

2000 Mathematics Subject Classification: Primary 30C15, 30C10, Secondary

26C10.

1. Introduction, definitions and notations

Let

P (z) = a0 + a1z + a2z2 + a3z

3 + ........+ an−1zn−1 + anz

n; |an| = 0

be a polynomial of degree n. Datt and Govil [2], Govil and Rahaman [4], Marden[8], Mohammad [9], Chattopadhyay, Das, Jain and Konwer [1], Joyal, Labelle andRahaman [5], Jain [6],[7], Sun and Hsieh [12], Zilovic, Roytman, Combettes andSwamy [14], Das and Datta [3] etc. worked in the theory of the distribution ofthe zeros of polynomials and obtained some newly developed results.

284 s.k. datta, d.c. pramanik

In this paper we intend to establish some of sharper results concerning thetheory of distribution of zeros of entire functions in the light of slowly changingfunctions.

The following definitions are well known :

Definition 1. The order ρ and lower order λ of an entire function f are defined as

ρ = lim supr→∞

log[2]M(r, f)

log rand λ = lim inf

r→∞

log[2] M(r, f)

log r,

where log[k] x = log(log[k−1] x) for k = 1, 2, 3, ... and log[0] x = x.

Let L ≡ L (r) be a positive continuous function increasing slowly, i.e.,L (ar) ∼ L (r) as r −→ ∞ for every positive constant a. Singh and Barker[10]defined it in the following way:

Definition 2. [10] A positive continuous function L(r) is called a slowly changingfunction if, for ε (> 0),

1

kε≤ L (kr)

L (r)≤ kε for r > r (ε)

and uniformly for k(≥ 1).

If, further, L(r) is differentiable, the above condition is equivalent to

limr→∞

rL′(r)

L(r)= 0.

Somasundaram and Thamizharasi [11] introduced the notions of L-order and L-lower order for entire functions defined in the open complex plane C as follows:

Definition 3. [11] The L-order ρL and the L-lower order λL of an entire functionf are defined as

ρL = lim supr→∞

log[2] M(r, f)

log[rL(r)]and λL = lim inf

r→∞

log[2] M(r, f)

log[rL(r)].

The more generalized concept for L-order and L-lower order are L∗-order andL∗-lower order respectively. Their definitions are as follows:

Definition 4. The L∗-order ρL∗and the L∗-lower order λL∗

of an entire functionf are defined as

ρL∗= lim sup

r→∞

log[2] M(r, f)

log[reL(r)]and λL∗

= lim infr→∞

log[2] M(r, f)

log[reL(r)].

2. Lemmas

In this section, we present some lemmas which will be needed in the sequel.

some derivations on the bounds for the zeros of entire functions...285

Lemma 1. If f (z) is an entire function of L-order ρL, then for every ε > 0 theinequality

N (r) ≤ [rL(r)]ρL+ε

holds for all sufficiently large r where N (r) is the number of zeros of f (z) in|z| ≤ [rL(r)].

Proof. Let us suppose that f (0) = 1. This supposition can be made withoutloss of generality because if f (z) has a zero of order ′m′ at the origin then we

may consider g (z) = c · f(z)zm

where c is so chosen that g (0) = 1. Since the func-tion g (z) and f (z) have the same order therefore it will be unimportant for ourinvestigations that the number of zeros of g (z) and f (z) differ by m.

We further assume that f (z) has no zeros on |z| = 2[rL(r)] and the zeroszi’s of f (z) in |z| < [rL(r)] are in non decreasing order of their moduli so that|zi| ≤ |zi+1|. Also let ρL suppose to be finite.

Now, we shall make use of Jenson’s formula as state below

log |f (0)| = −n∑

i=1

logR

|zi|+

1

2π

∫ 2π

0

log∣∣f (R eiϕ

)∣∣ dϕ.(1)

Let us replace R by 2r and n by N (2r) in (1)

log |f (0)| = −N(2r)∑i=1

log2r

|zi|+

1

2π

∫ 2π

0

log∣∣f (2r eiϕ

)∣∣ dϕ.Since f (0) = 1, log |f (0)| = log 1 = 0.

∴N(2r)∑i=1

log2r

|zi|=

1

2π

∫ 2π

0

log∣∣f (2r eiϕ

)∣∣ dϕ.(2)

L.H.S. =

N(2r)∑i=1

log2r

|zi|≥

N(r)∑i=1

log2r

|zi|≥ N (r) log 2(3)

because for large values of r,

log2r

|zi|≥ log 2.

R.H.S =1

2π

∫ 2π

0

log∣∣f (2r eiϕ

)∣∣ dϕ≤ 1

2π

∫ 2π

0

logM (2r) dϕ = logM (2r) .

(4)

Again, by definition of order ρL of f (z), we have for every ε > 0, and asL (2r) ∼ L (r),

logM (2r) ≤ [2rL(2r)]ρL+ε/2

logM (2r) ≤ [2rL(r)]ρL+ε/2.

(5)


Hence, from (2) by the help of (3) , (4) and (5), we have

N (r) log 2 ≤ [2rL(r)]ρL+ε/2

i.e., N (r) ≤ 2ρL+ε/2

log 2· (rL(r))

ρL+ε

(rL(r))ε/2≤ [rL(r)]ρ

L+ε.

This proves the lemma.

In the line of Lemma 1, we may state the following lemma:

Lemma 2. If f (z) is an entire function of L∗-order ρL∗, then for every ε > 0 the

inequality

N (r) ≤ [reL(r)]ρL∗

+ε

holds for all sufficiently large r where N (r) is the number of zeros of f (z) in|z| ≤ [reL(r)].

Proof. With the initial assumptions as laid down in Lemma 1, let us supposethat f (z) has no zeros on |z| = 2[reL(r)] and the zeros zi’s of f (z) in |z| < [reL(r)]are in non decreasing order of their moduli so that |zi| ≤ |zi+1|. Also, let ρL

∗

supposed to be finite.In view of (1), (2), (3) and (4), by definition of ρL

∗and as L (2r) ∼ L (r) , we

get for every ε > 0 that

logM (2r) ≤ [2reL(2r)]ρL∗

+ε/2

i.e., logM(2r) ≤ [2reL(r)]ρL∗+ε/2

.(6)

Hence, by the help of (3) , (4) and (6), we obtain from (2) that

N (r) log 2 ≤ [2reL(r)]ρL∗

+ε/2

N (r) ≤ 2ρL∗

+ε/2

log 2· [reL(r)]ρ

L∗+ε

[rL(r)]ε/2≤ [reL(r)]ρ

L∗+ε.

Thus, the lemma is established.

3. Theorems

In this section, we present the main results of the paper.

Theorem 1. Let P (z) be an entire function having L-order ρL in the disc|z| ≤ [rL(r)] for sufficiently large r. Also, let the Taylor’s series expansion ofP (z) be given by

P (z) = a0 + ap1zp1 + · · ·+ apmz

pm + aN(r)zN(r), a0 = 0, aN(r) = 0

with 1 ≤ p1 < p2 < · · · < pm ≤ N(r)− 1, pi’s are integers such that for ρL > 0,

|a0| (ρL)N(r) ≥ |ap1 | (ρL)N(r)−p1 ≥ · · · ≥ |apm| (ρL)N(r)−pm ≥∣∣aN(r)

∣∣ .


Then, all the zeros of P (z) lie in the ring shaped region

1

ρL(1 +

|ap1 ||a0|(ρL)p1

) < |z| < 1

ρL

(1 +

|a0|∣∣aN(r)

∣∣(ρL)N(r)

).

Proof. Given that

P (z) = a0 + ap1zp1 + · · ·+ apmz

pm + aN(r)zN(r)

where pi’s are integers and 1 ≤ p1 < p2 < · · · < pm ≤ N(r)− 1. Then for ρL > 0 ,


∣∣ .Let us consider

Q(z) = (ρL)N(r)P

(z

ρL

)= (ρL)N(r)

a0 + ap1

zp1(ρL)p1

+ · · ·+ apmzpm

(ρL)pm+ aN(r)

zN(r)

(ρL)N(r)

= a0(ρ

L)N(r) + ap1(ρL)N(r)−p1zp1 + · · ·+ apm(ρ

L)N(r)−pmzpm + aN(r)zN(r).

Therefore,

|Q(z)| ≥∣∣aN(r)z

N(r)∣∣−∣∣a0(ρL)N(r) + ap1(ρ

L)N(r)−p1zp1 + · · ·+ apm(ρL)N(r)−pmzpm

∣∣ .(7)

Now, using the given condition of Theorem 1 we obtain that∣∣a0(ρL)N(r) + ap1(ρL)N(r)−p1zp1 + · · ·+ apm(ρ

L)N(r)−pmzpm∣∣

≤ |a0| (ρL)N(r) + |ap1 | (ρL)N(r)−p1 |z|p1 + · · ·+ |apm | (ρL)N(r)−pm |z|pm

≤ |a0| (ρL)N(r) |z|N(r)

(1

|z|N(r)−pm+ · · ·+ 1

|z|N(r)

)for |z| = 0.

Using (7), we get for |z| = 0 that

|Q(z)| ≥∣∣aN(r)

∣∣ |z|N(r) − |a0| (ρL)N(r) |z|N(r)

(1

|z|N(r)−pm+ · · ·+ 1

|z|N(r)

)

>∣∣aN(r)

∣∣ |z|N(r) − |a0| (ρL)N(r) |z|N(r)

(1

|z|N(r)−pm+ · · ·+ 1

|z|N(r)+ · · ·

)

=∣∣aN(r)

∣∣ |z|N(r) − |a0| (ρL)N(r) |z|N(r)

(∞∑k=1

1

|z|k

).

(8)


The geometric series∞∑k=1

1

|z|kis convergent for

1

|z|< 1

i.e., for |z| > 1

and converges to1

|z|1

1− 1|z|

=1

|z| − 1.

Therefore,∞∑k=1

1

|z|k=

1

|z| − 1for |z| > 1.

Using (8), we get from above that for |z| > 1

|Q(z)| >∣∣aN(r)

∣∣ |z|N(r) − |a0| (ρL)N(r) |z|N(r)

(1

|z| − 1

)= |z|N(r)

(∣∣aN(r)

∣∣− |a0| (ρL)N(r)

|z| − 1

).

Now, for |z| > 1,

|Q(z)| > 0 if∣∣aN(r)

∣∣− |a0| (ρL)N(r)

|z| − 1≥ 0

i.e., if∣∣aN(r)

∣∣ ≥ |a0| (ρL)N(r)

|z| − 1

i.e., if |z| − 1 ≥ |a0| (ρL)N(r)∣∣aN(r)

∣∣i.e., if |z| ≥ 1 +

|a0| (ρL)N(r)∣∣aN(r)

∣∣ > 1.

Therefore, |Q(z)| > 0 if

|z| ≥ 1 +|a0| (ρL)N(r)∣∣aN(r)

∣∣ .

Therefore, Q(z) does not vanish for

|z| ≥ 1 +|a0| (ρL)N(r)∣∣aN(r)

∣∣ .

So, all the zeros of Q(z) lie in

|z| < 1 +|a0| (ρL)N(r)∣∣aN(r)

∣∣ .


Let z = z0 be any zero of P (z). Therefore, P (z0) = 0. Clearly, z0 = 0 as a0 = 0.Putting z = ρLz0 in Q(z), we get that

Q(ρLz0) = (ρL)N(r).P (z0) = (ρL)N(r).0 = 0.

So z = ρLz0 is a zero of Q(z). Hence,∣∣ρLz0∣∣ < 1 +|a0| (ρL)N(r)∣∣aN(r)

∣∣i.e., |z0| <

1

ρL

(1 +


∣∣).

Since z0 is an arbitrary zero of P (z), therefore, all the zeros of Q(z) lie in

|z| < 1

ρL

(1 +


∣∣).(9)

Again, let us consider

R(z) = (ρL)N(r)zN(r)P

(1

ρLz

).

Therefore,

R(z)=(ρL)N(r)zN(r)

a0+ap1

1

(ρL)p1zp1+ · · ·+ apm

1

(ρL)pmzpm+aN(r)

1

(ρL)N(r)zN(r)

=a0(ρ

L)N(r)zN(r)+ap1(ρL)N(r)−p1zN(r)−p1 + · · ·+ apm(ρ

L)N(r)−pmzN(r)−pm+aN(r).

Now,

|R(z)| ≥∣∣a0(ρL)N(r)zN(r)

∣∣−∣∣ap1(ρL)N(r)−p1zN(r)−p1 + · · ·+ apm(ρ

L)N(r)−pmzN(r)−pm + aN(r)

∣∣ .(10)

Also∣∣ap1(ρL)N(r)−p1zN(r)−p1 + · · ·+ apm(ρL)N(r)−pmzN(r)−pm + aN(r)

∣∣≤∣∣ap1(ρL)N(r)−p1zN(r)−p1

∣∣+ · · ·+∣∣apm(ρL)N(r)−pmzN(r)−pm

∣∣+ ∣∣aN(r)

∣∣= |ap1 | (ρL)N(r)−p1 |z|N(r)−p1 + · · ·+ |apm | (ρL)N(r)−pm |z|N(r)−pm +

∣∣aN(r)

∣∣≤ |ap1 | (ρL)N(r)−p1

(|z|N(r)−p1 + · · ·+ |z|N(r)−pm + 1

).

(11)

Using (11), we get from (10) that for |z| = 0

|R(z)| ≥ |a0| (ρL)N(r) |z|N(r) − |ap1 | (ρL)N(r)−p1(|z|N(r)−p1 + · · ·+ |z|N(r)−pm + 1

)= |a0| (ρL)N(r) |z|N(r) − |ap1 | (ρL)N(r)−p1 |z|N(r)

(1

|z|p1+ · · ·+ 1

|z|pm+

1

|z|N(r)

)> |a0| (ρL)N(r) |z|N(r)

− |ap1 | (ρL)N(r)−p1 |z|N(r)(

1|z|p1 + · · ·+ 1

|z|pm + 1

|z|N(r) + · · ·).


Therefore, for |z| = 0,

|R(z)| > |a0| (ρL)N(r) |z|N(r) − |ap1 | (ρL)N(r)−p1 |z|N(r)

(∞∑k=1

1

|z|k

).(12)

Now, the geometric series∞∑k=1

1

|z|kis convergent for

1

|z|< 1

i.e., for |z| > 1

and converges to1

|z|1

1− 1|z|

=1

|z| − 1.

So∞∑k=1

1

|z|k=

1

|z| − 1for |z| > 1.

Therefore, for |z| > 1,

|R(z)| > |a0| (ρL)N(r) |z|N(r) − |ap1 | (ρL)N(r)−p1 |z|N(r)(

1|z|−1

)= |z|N(r) (ρL)N(r)−p1

(|a0| (ρL)p1 −

|ap1 ||z|−1

).

i.e., for |z| > 1

|R(z)| > |z|N(r) (ρL)N(r)−p1

(|a0| (ρL)p1 −

|ap1||z| − 1

).

Now,

|R(z)| > 0 if |a0| (ρL)p1 −|ap1 ||z| − 1

≥ 0

i.e., if |a0| (ρL)p1 ≥|ap1 ||z| − 1

i.e., if |z| − 1 ≥ |ap1 ||a0| (ρL)p1

i.e., if |z| ≥ 1 +|ap1 |

|a0| (ρL)p1> 1.

Therefore,

|R(z)| > 0 if |z| ≥ 1 +|ap1 |

|a0| (ρL)p1.

Since R(z) does not vanish in

|z| ≥ 1 +|ap1 |

|a0| (ρL)p1,


all the zeros of R(z) lie in

|z| < 1 +|ap1 |

|a0| (ρL)p1.

Let z = z0 be any zero of P (z). Therefore, P (z0) = 0. Clearly, z0 = 0 as a0 = 0.Putting z = 1/ρLz0 in R(z), we obtain that

R

(1

ρLz0

)= (ρL)N(r)

(1

ρLz0

)N(r)

· P (z0)

=

(1

z0

)N(r)

· 0 = 0.

So ∣∣∣∣ 1

ρLz0

∣∣∣∣ < 1 +|ap1 |

|a0| (ρL)p1

i.e.,

∣∣∣∣ 1z0∣∣∣∣ < ρL

(1 +

|ap1 ||a0| (ρL)p1

)i.e., |z0| >

1

ρL(1 +

|ap1 ||a0|(ρL)p1

) .

As z0 is an arbitrary zero of P (z), all the zeros of P (z) lie in

|z| > 1

ρL(1 +

|ap1 ||a0|(ρL)p1

) .(13)

So, from (9) and (13), we may conclude that all the zeros of P (z) lie in the properring shaped region

1

ρL(1 +

|ap1 ||a0|(ρL)p1

) < |z| < 1

ρL

(1 +

|a0|∣∣aN(r)

∣∣(ρL)N(r)

).


In the line of Theorem 1, we may state the following theorem in view ofLemma 2:

Theorem 2. Let P (z) be an entire function having L∗-order ρL∗in the disc

|z| ≤ [reL(r)] for sufficiently large r. Also, let the Taylor’s series expansion ofP (z) be given by

P (z) = a0 + ap1zp1 + · · ·+ apmz

pm + aN(r)zN(r), a0 = 0, aN(r) = 0


with 1 ≤ p1 < p2 < · · · < pm ≤ N(r)− 1, pi’s are integers such that for ρL∗> 0,

|a0| (ρL∗)N(r) ≥ |ap1 | (ρL

∗)N(r)−p1 ≥ · · · ≥ |apm| (ρL

∗)N(r)−pm ≥

∣∣aN(r)

∣∣ .Then, all the zeros of P (z) lie in the ring shaped region

1

ρL∗

(1 +

|ap1 ||a0|(ρL∗)

p1

) < |z| < 1

ρL∗

(1 +

|a0|∣∣aN(r)

∣∣(ρL∗)N(r)

).

The proof is omitted.

Corollary 1. In view of Theorem 1, we may conclude that all the zeros of

P (z) = a0 + ap1zp1 + · · ·+ apmz

pm + anzn

of degree n with 1 ≤ p1 < p2 < · · · < pm ≤ n − 1, pi’s are integers such that forρL > 0,

|a0| ≥ |ap1 | ≥ · · · ≥ |an|lie in the ring shaped region

1(1 +

|ap1 ||a0|

) < |z| <(1 +

|a0||an|

)

on putting ρL = 1 in Theorem 1.

Corollary 2. In view of Theorem 2, we may conclude that all the zeros of

P (z) = a0 + ap1zp1 + · · ·+ apmz

pm + anzn

of degree n with 1 ≤ p1 < p2 < · · · < pm ≤ n − 1, pi’s are integers such that forρL

∗> 0,

|a0| ≥ |ap1 | ≥ ...... ≥ |an|lie in the ring shaped region

1(1 +

|ap1 ||a0|

) < |z| <(1 +

|a0||an|

)

on putting ρL∗= 1 in Theorem 2.

Theorem 3. Let P (z) be an entire function having L-order ρL. For sufficientlylarge r in the disc |z| ≤ [rL(r)], the Taylor’s series expansion of P (z) be given byP (z) = a0 + a1z + · · ·+ aN(r)z

N(r), a0 = 0. Further, for ρL > 0,

|a0| (ρL)N(r) ≥ |a1| (ρL)N(r)−1 ≥ · · · ≥∣∣aN(r)

∣∣ .


Then, all the zeros of P (z) lie in the ring shaped region.

1

ρLt′0< |z| < 1

ρLt0,

where t0 and t′0 are the greatest roots of

g(t) ≡∣∣aN(r)

∣∣ tN(r)+1 −(∣∣aN(r)

∣∣+ (ρL)N(r) |a0|)tN(r) + (ρL)N(r) |a0| = 0

and

f(t) ≡ |a0| ρLtN(r)+1 −(|a0| ρL + |a1|

)tN(r) + |a1| = 0.

Proof. Let

P (z) = a0 + a1z + · · ·+ aN(r)zN(r)

by applying Lemma 1 and in view of Taylor’s series expansion of P (z). Also

|a0| (ρL)N(r) ≥ |a1| (ρL)N(r)−1 ≥ · · · ≥∣∣aN(r)

∣∣ .Let us consider

Q(z) = (ρL)N(r)P

(z

ρL

)=(ρL)N(r)

a0 + a1

z

ρL+ a2

z2

(ρL)2+ · · ·+ aN(r)

zN(r)

(ρL)N(r)

= a0(ρ

L)N(r) + a1(ρL)N(r)−1z + · · ·+ aN(r)z

N(r).

Now

|Q(z)| ≥∣∣aN(r)

∣∣ |z|N(r) −∣∣a0(ρL)N(r)+a1(ρ

L)N(r)−1z + · · ·+ aN(r)−1zN(r)−1

∣∣ .Also, applying the condition |a0| (ρL)N(r) ≥ |a1| (ρL)N(r)−1 ≥ · · · ≥

∣∣aN(r)

∣∣, we getfrom above that∣∣a0(ρL)N(r) + a1(ρ

L)N(r)−1z + · · ·+ aN(r)−1zN(r)−1

∣∣≤ |a0| (ρL)N(r) + |a1| (ρL)N(r)−1 |z|+ · · ·+

∣∣aN(r)−1

∣∣ |z|N(r)−1

≤ |a0| (ρL)N(r)(1 + |z|+ · · ·+ |z|N(r)−1

)= |a0| (ρL)N(r) |z|

N(r) − 1

|z| − 1for |z| = 1.

Therefore, it follows from above that

|Q(z)| ≥∣∣aN(r)

∣∣ |z|N(r) − |a0| (ρL)N(r) · |z|N(r) − 1

|z| − 1.


Now

|Q(z)| > 0 if∣∣aN(r)

∣∣ |z|N(r) − |a0| (ρL)N(r) · |z|N(r) − 1

|z| − 1> 0

i.e., if∣∣aN(r)

∣∣ |z|N(r) > |a0| (ρL)N(r).|z|N(r) − 1

|z| − 1

i.e., if∣∣aN(r)

∣∣ |z|N(r) (|z| − 1) > |a0| (ρL)N(r)(|z|N(r) − 1

)i.e., if

∣∣aN(r)

∣∣ |z|N(r)+1 −(∣∣aN(r)

∣∣+ |a0| (ρL)N(r))|z|N(r) + |a0| (ρL)N(r) > 0.

Let us consider

g(t) ≡∣∣aN(r)

∣∣ tN(r)+1 −(∣∣aN(r)

∣∣+ |a0| (ρL)N(r))tN(r) + |a0| (ρL)N(r) = 0.(14)

The maximum number of positive roots of (14) is two because maximum numberof changes of sign in g(t) = 0 is two and if it is less, less by two. Clearly, t = 1is a positive root of g(t) = 0. Therefore, g(t) = 0 must have exactly one positiveroot other than 1. Let the positive root of g(t) be t1. Let us take t0 = max 1, t1 .Clearly, for t > t0, g(t) > 0. If not for some t2 > t0, g(t2) < 0. Also g(∞) > 0.Therefore g(t) = 0 has another positive root in (t2,∞) which gives a contradiction.So, for t > t0, g(t) > 0. Also t0 ≥ 1. Therefore, |Q(z)| > 0 if |z| > t0. So, Q(z)does not vanish in |z| > t0. Hence, all the zeros of Q(z) lie in |z| ≤ t0.

Let z = z0 be a zero of P (z). So, P (z0) = 0. Clearly, z0 = 0 as a0 = 0. Puttingz = ρLz0 in Q(z), we get that

Q(ρLz0) =(ρL)N(r)

P (z0) =(ρL)N(r)

.0 = 0.

Therefore, z = ρLz0 is a zero of Q(z). So,∣∣ρLz0∣∣ ≤ t0 or |z0| ≤

1

ρLt0. As z0 is an

arbitrary zero of P (z),

all the zeros of P (z) lie in the region |z| ≤ 1

ρLt0.(15)

In order to prove the lower bound of Theorem 3, let us consider

R(z) = (ρL)N(r)zN(r)P

(1

ρLz

).

Then

R(z) = (ρL)N(r)zN(r)

(a0 +

a1ρLz

+ · · ·+ aN(r)1

(ρL)N(r)zN(r)

)= a0(ρ

L)N(r)zN(r) + a1(ρL)N(r)−1zN(r)−1 + · · ·+ aN(r).

Now

|R(z)| ≥ |a0| (ρL)N(r) |z|N(r) −∣∣a1(ρL)N(r)−1zN(r)−1 + · · ·+ aN(r)

∣∣ .


Also∣∣a1(ρL)N(r)−1zN(r)−1 + · · ·+ aN(r)

∣∣ ≤ |a1| (ρL)N(r)−1 |z|N(r)−1 + · · ·+∣∣aN(r)

∣∣ .So, applying the condition |a0| (ρL)N(r) ≥ |a1| (ρL)N(r)−1 ≥ · · · ≥

∣∣aN(r)

∣∣, we getfrom above that

−∣∣a1(ρL)N(r)−1zN(r)−1 + · · ·+ aN(r)

∣∣≥ − |a1| (ρL)N(r)−1 |z|N(r)−1 − · · · −

∣∣aN(r)

∣∣≥ − |a1| (ρL)N(r)−1

(|z|N(r)−1 + · · ·+ 1

)= − |a1| (ρL)N(r)−1 |z|

N(r)−1 − 1

|z| − 1for |z| = 1.

Using (16), we get for |z| = 1 that

|R(z)| ≥ (ρL)N(r)−1

(|a0| ρL |z|N(r) − |a1|

|z|N(r)−1 − 1

|z| − 1

).(16)

Now

|R(z)| > 0 if (ρL)N(r)−1

(|a0| ρL |z|N(r) − |a1|

|z|N(r)−1 − 1

|z| − 1

)> 0

i.e., if |a0| ρL |z|N(r) − |a1||z|N(r)−1 − 1

|z| − 1> 0

i.e., if |a0| ρL |z|N(r) > |a1||z|N(r)−1 − 1

|z| − 1

i.e., if |a0| ρL |z|N(r) (|z| − 1) > |a1|(|z|N(r)−1 − 1

)i.e., if |a0| ρL |z|N(r)+1 −

(|a0| ρL + |a1|

)|z|N(r) + |a1| > 0.

Let us consider

f(t) ≡ |a0| ρLtN(r)+1 −(|a0| ρL + |a1|

)tN(r) + |a1| = 0.

Clearly, f(t) = 0 has two positive roots, because the number of changes of signof f(t) is two. If it is less, less by two. Also, t = 1 is the one of the positiveroots of f(t) = 0. Let us suppose that t = t2 be the other positive root. Also, lett′0 = max 1, t2 and so t′0 ≥ 1. Now t > t′0 implies f(t) > 0. If not, then thereexists some t3 > t′0 such that f(t3) < 0. Also, f(∞) > 0. Therefore, there existsanother positive root in (t3,∞) which is a contradiction. So, |R(z)| > 0 if |z| > t′0.Thus R(z) does not vanish in |z| > t′0. In other words, all the zeros of R(z) lie in|z| ≤ t′0.


Let z = z0 be any zero of P (z). So, P (z0) = 0. Clearly, z0 = 0 as a0 = 0.

Putting z =1

ρLz0in R(z), we get that

R

(1

ρLz0

)= (ρL)N(r)

(1

ρLz0

)N(r)

P (z0) =

(1

z0

)N(r)

· 0 = 0

Therefore,1

ρLz0is a root of R(z). So,

∣∣∣∣ 1

ρLz0

∣∣∣∣ ≤ t′0 implies |z0| ≥1

ρLt′0. As z0 is

an arbitrary zero of P (z) = 0,

all the zeros of P (z) lie in |z| ≥ 1

ρLt′0.(17)

From (15) and (18) we have all the zeros of P (z) lie in the ring shaped regiongiven by

1

ρLt′0≤ |z| ≤ 1

ρLt0

where t0 and t′0 are the greatest positive roots of g(t) = 0 and f(t) = 0 respectively.


In the line of Theorem 3, we may state the following theorem in view ofLemma 2:

Theorem 4. Let P (z) be an entire function having L∗-order ρL∗. For sufficiently

large r in the disc |z| ≤ [reL(r)], the Taylor’s series expansion of P (z) be given by

P (z) = a0 + a1z + · · ·+ aN(r)zN(r), a0 = 0.

Further, for ρL∗> 0,

|a0| (ρL∗)N(r) ≥ |a1| (ρL

∗)N(r)−1 ≥ · · · ≥

∣∣aN(r)

∣∣ .Then all the zeros of P (z) lie in the ring shaped region.

1

ρL∗t′0< |z| < 1

ρL∗ t0

where t0 and t′0 are the greatest roots of

g(t) ≡∣∣aN(r)

∣∣ tN(r)+1 −(∣∣aN(r)

∣∣+ (ρL∗)N(r) |a0|

)tN(r) + (ρL

∗)N(r) |a0| = 0

and

f(t) ≡ |a0| ρL∗tN(r)+1 −

(|a0| ρL

∗+ |a1|

)tN(r) + |a1| = 0.

The proof is omitted.


Corollary 3. From Theorem 3, we can easily conclude that all the zeros of

P (z) = a0 + a1z + · · ·+ anzn

of degree n with property |a0| ≥ |a1| ≥ · · · ≥ |an| lie in the ring shaped region

1

t′0≤ |z| ≤ t0

where t0 and t′0 are the greatest positive roots of

g(t) ≡ |an| tn+1 − (|an|+ |a0|)tn + |a0| = 0

andf(t) ≡ |a0| tn+1 − (|a0|+ |a1|)tn + |a1| = 0

respectively by putting ρL = 1.

Corollary 4. From Theorem 4, we can easily conclude that all the zeros of

P (z) = a0 + a1z + · · ·+ anzn

of degree n with property |a0| ≥ |a1| ≥ · · · ≥ |an| lie in the ring shaped region

1

t′0≤ |z| ≤ t0


g(t) ≡ |an| tn+1 − (|an|+ |a0|)tn + |a0| = 0

andf(t) ≡ |a0| tn+1 − (|a0|+ |a1|)tn + |a1| = 0

respectively by putting ρL∗= 1.

Corollary 5. Under the conditions of Theorem 3 and

P (z) = a0 + ap1zp1 + · · ·+ apmz

pm + aN(r)zN(r)

with1 ≤ p1 ≤ p2 ≤ · · · ≤ pm ≤ N(r)− 1,

where pi’s are integers and a0, ap1 , ..., aN(r) are non vanishing coefficients with


∣∣then we can show that all the zeros of P (z) lie in

1

ρLt′0≤ |z| ≤ 1

ρLt0


g(t) ≡∣∣aN(r)

∣∣ tN(r)+1 −(∣∣aN(r)

∣∣+ |a0| (ρL)N(r))tN(r) + |a0| (ρL)N(r) = 0

and

f(t) ≡ |a0| (ρL)p1tN(r)+1 −(|a0| (ρL)p1 + |ap1 |

)tN(r) − |ap1 | = 0 respectively.


Corollary 6. Under the conditions of Theorem 4 and

P (z) = a0 + ap1zp1 + .....+ apmz

pm + aN(r)zN(r)

with1 ≤ p1 ≤ p2 ≤ · · · ≤ pm ≤ N(r)− 1,

where pi’s are integers and a0, ap1 , ..., aN(r) are non vanishing coefficients with

|a0| (ρL∗)N(r) ≥ |ap1 | (ρL

∗)N(r)−p1 ≥ · · · ≥ |apm| (ρL

∗)N(r)−pm ≥

∣∣aN(r)

∣∣then we can show that all the zeros of P (z) lie in

1

ρL∗t′0≤ |z| ≤ 1

ρL∗ t0


g(t) ≡∣∣aN(r)

∣∣ tN(r)+1 −(∣∣aN(r)

∣∣+ |a0| (ρL∗)N(r)

)tN(r) + |a0| (ρL

∗)N(r) = 0

and

f(t) ≡ |a0| (ρL∗)p1tN(r)+1 −

(|a0| (ρL

∗)p1 + |ap1 |

)tN(r) − |ap1 | = 0 respectively.

Corollary 7. If we put ρL = 1 in Corollary 5, then all the zeros of

P (z) = a0 + ap1zp1 + · · ·+ apmz

pm + anzn

lie in the ring shaped region1

t′0≤ |z| ≤ t0


g(t) ≡ |an| tn+1 − (|an|+ |a0|) tn + |a0| = 0

andf(t) ≡ |a0| tn+1 − (|a0|+ |ap1 |) tn − |ap1 | = 0 respectively

provided|a0| ≥ |ap1 | ≥ ..... ≥ |apm| ≥ |an| .

Corollary 8. If we put ρL∗= 1 in Corollary 6, then all the zeros of

P (z) = a0 + ap1zp1 + · · ·+ apmz

pm + anzn

lie in the ring shaped region1

t′0≤ |z| ≤ t0



g(t) ≡ |an| tn+1 − (|an|+ |a0|) tn + |a0| = 0

andf(t) ≡ |a0| tn+1 − (|a0|+ |ap1 |) tn − |ap1 | = 0 respectively

provided|a0| ≥ |ap1 | ≥ ..... ≥ |apm| ≥ |an| .

References

[1] Chattopadhyay, A., Das, S., Jain, V.K., Konwer, H., Certain gene-ralization of Enestrom-Kakeya theorem, J. Indian Math. Soc., vol. 72, no. 1-4(2005), 147-156.

[2] Datt, B., Govil, N.K., On the location of the zeros of polynomial,J. Approximation Theory, vol. 24 (1978), 78-82.

[3] Das, S., Datta, S.K., On Cauchy’s proper bound for zeros of a polynomial,International J. of Math. Sci. and Engg. Appls. (IJMSEA), vol. 2, no. IV(2008), 241-252.

[4] Govil, N.K., Rahaman, Q.I., On the Enestrom-Kakeya theorem, TohokuMath. J., vol. 20 (1968), 126-136.

[5] Joyal, A., Labelle, G., and Rahaman, Q.I., On the location of zerosof polynomials, Canad. Math.Bull., vol. 10 (1967), 53-63.

[6] Jain, V.K., On the location of zeros of polynomials, Ann. Univ. Marie Curie-Sklodowska, Lublin-Polonia Sect. A, vol. 30 (1976), 43-48.

[7] Jain, V.K., On Cauchy’s bound for zeros of a polynomial, Turk. J. Math.,vol. 30 (2006), 95-100.

[8] Marden, M., Geometry of polynomials, Amer. Math. Soc. Providence, R.I.,1966.

[9] Mohammad, Q.G., Location of zeros of polynomials, Amer. Math. Monthly,vol. 74 (1967), 290-292.

[10] Singh, S.K., Barker, G.P., Slowly changing functions and their applica-tions, Indian J. Math., vol. 19, no. 1 (1977), 1-6.

[11] Somasundaram, D., Thamizharasi, R., A note on the entire functionsof L-bounded index and L-type, Indian J. Pure Appl. Math., vol. 19, no. 3(1988), 284-293.


[12] Sun, Y.J., Hsieh, J.G., A note on circular bound of polynomial zeros, IEEETrans. Circuit Syst., vol. 143 (1996), 476-478.

[13] Valiron, G., Lectures on the general theory of integral functions, ChelseaPublishing Company (1949).

[14] Zilovic, M.S., Roytman, L.M., Combetts, P.L., Swami, M.N.S.,A bound for the zeros of polynomials, ibid vol. 39 (1992), 476-478.

Accepted: 9.12.2013


SOFT INTERSECTION h-IDEALS OF HEMIRINGSAND ITS APPLICATIONS

Xueling Ma

Jianming Zhan1

Department of MathematicsHubei University for NationalitiesEnshi, Hubei Province 445000China

Abstract. In this paper, we introduce a new kind of soft hemirings called soft inter-

section hemirings and obtain some related properties. Some basic operations are also

investigated. Finally, we describe some characterizations of h-hemiregular hemirings by

means of SI-h-ideals.

Keywords and phrases: Soft set; soft intersection hemiring; soft intersection h-ideal;

soft h-sum; soft h-product; h-hemiregular hemiring.

2000 Mathematics Subject Classification: 16Y60; 13E05; 16Y99.

1. Introduction

The complexities of modeling uncertain data in economics, engineering, environ-mental science, sociology and many other fields can not be successfully dealt withby classical methods. In order to overcome these difficulties, Molodtsov [17] in-troduced the concept of soft sets as a new mathematical tool for dealing withuncertainties. Maji [15] discussed further soft set theory. Ali et al. [3] proposedsome new operations on soft sets. In the same time, this theory has proven usefulin many different fields such as decision making [5], [6], [7], [9], [16], data analysis,forecasting and so on. Recently, the algebraic structures of soft sets have beenstudied increasingly, such as soft rings [1], soft-int groups [4], soft semirings [8],soft BCK/BCI-algebras [11], soft intersection near-rings [18] and so on.

On the other hand, semirings have been found useful for dealing with problemsin different areas of applied mathematics and information sciences, as the semiringstructure provides an algebraic framework for modeling and investigating the keyfactors in these problems. We know that ideals in the semiring S do not ingeneral coincide with the usual ring ideals if S is a ring, and so many results inring theory have no analogues in semirings using only ideals. Consequently, some

1Corresponding author. Tel/Fax: 0086-718-8437732. E-mails: [email protected] (X. Ma);[email protected] (J. Zhan)

302 x. ma, j. zhan

more restricted concepts of ideals such as k-ideals and h-ideals [2], [19], [10], [15],[16], [14], [20], [21] have been investigated. Nowadays, many researchers discussedthis theory including their application. In application, hemirings are useful inautomata and formal languages.

In this paper, we introduce a new kind of soft hemirings called soft inter-section hemirings and obtain some related properties. Some basic operations arealso investigated. Finally, we describe some characterizations of h-hemiregularhemirings by means of SI-h-ideals.

2. Preliminaries

A semiring (S,+, ·) with zero is called a hemiring if (S,+) is commutative. Asubhemiring of a hemiring S is a subset A of S closed under addition and multi-plication. A left (resp., right) ideal of a hemiring S is a subset A of S closed underaddition such that SA ⊆ A (resp., AS ⊆ A). A subset A is called an ideal if it isboth a left ideal and a right ideal. A subhemiring (left ideal, right ideal, ideal) A ofS is called an h-subhemiring (left h-ideal, right h-ideal, h-ideal) of S, respectively,if for any x, z ∈ S, and a, b ∈ A, x+a+z = b+z implies x ∈ A. The h-closure A ofa subset A of S is defined as A = x ∈ S|x+a+z = b+z for some a, b ∈ A, z ∈ S.

Throughout this section, S is a hemiring, U is an initial universe, E is a setof parameters, P (U) is the power set of U and A,B,C ⊆ E.

Definition 2.1 [17] A soft set fA of U is a set defined by fA : E → P (U) suchthat fA(x) = ∅ if x /∈ A. Here fA is also called an approximate function. A softset over U can be represented by the set of ordered pairs fA = (x, fA(x))|x ∈ E,fA(x) ∈ P (U). It is clear to see that a soft set is a parameterized family of subsetsof the set U . Note that the set of all soft sets over U will be denoted by S(U).

Definition 2.2 [6] Let fA, fB ∈ S(U), then

(i) The intersection of fA and fB, denoted by fA∩fB, is defined as fA∩fB =fA∩B, where fA∩B(x) = fA(x) ∩ fB(x), for all x ∈ E;

(ii) The union of fA and fB, denoted by fA∪fB, is defined as fA∪fB = fA∪B,where fA∪B(x) = fA(x) ∪ fB(x), for all x ∈ E.

Definition 2.3 [6] Let fA, fB ∈ S(U). Then ∧-product and ∨-product of fAand fB, denoted by fA ∧ fB and fA ∨ fB, are defined by fA∧B(x, y) = fA(x) ∩fB(y), fA∨B(x, y) = fA(x) ∪ fB(y) for all x, y ∈ E, respectively.

Definition 2.4 [4] Let fA be a soft set over U and α ⊆ U . Then, upper α-inclusion of fA, denoted by U(fA;α), is defined as U(fA;α) = x ∈ A|fA(x) ⊇ α.

3. SI-hemirings (SI-h-ideals)

In this paper, we introduce the concepts of soft intersection hemirings (soft inter-section h-ideals) and obtain some related properties.

soft intersection h-ideals of hemirings and its applications 303

Definition 3.1 A soft set fS over U is called a soft intersection hemiring (briefly,SI-hemiring) of S over U if it satisfies:

(SI1) fS(x+ y) ⊇ fS(x) ∩ fS(y),

(SI2) fS(xy) ⊇ fS(x) ∩ fS(y),

(SI3) fS(x) ⊇ fS(a) ∩ fS(b) with x+ a+ z = b+ z for all x, a, b, z ∈ S.

Example 3.2 Let U = S = Z6 = 0, 1, 2, 3, 4, 5 be the hemiring of non-negativeintegers module 6. Define a soft set fS over U by fS(0) = fS(2) = fS(4) =0, 1, 2, 3, 4, 5 and fS(1) = fS(3) = fS(5) = 0, 2, 4. Then one can easily checkthat fS is an SI-hemiring of S over U .

From the above definition, we can obtain the following:

Proposition 3.3 If fS is an SI-hemiring of S over U , then fS(0) ⊇ fS(x) forall x ∈ S.

Definition 3.4 A soft set fS over U is called a soft intersection left (right) h-ideal(briefly, SI-left(right) h-ideal) of S over U if it satisfies (SI1), (SI3) and:

(SI4) fS(xy) ⊇ fS(y) (fS(xy) ⊇ fS(x)), for all x, y ∈ S.

A soft set over U is called a soft intersection h-ideal (briefly, SI-h-ideal) of Sif it is both an SI-left h-ideal and an SI-right h-ideal of S over U .

Example 3.5 Assume that U = Z+ is the universal set and S = Z6 is the set ofparameters. Define a soft set fS as fS(0) = n|n ∈ Z+, fS(1) = fS(5) = 6n|n ∈ Z+, fS(2) = fS(4) = 2n|n ∈ Z+ and fS(3) = 3n|n ∈ Z+. Then, onecan easily check that fS is an SI-h-ideal of S over U .

Proposition 3.6 Let fS1 and fS2 be two SI-hemirings of S1 and S2 over U ,respectively. Then fS1 ∧ fS2 is an SI-hemiring of S1 × S2 over U .

Proof. Let fS1 and fS2 be two SI-hemirings of S1 and S2 over U , respectively.Then, for all (x1, y1), (x2, y2) ∈ S1 × S2, we have

(i) fS1∧S2((x1, y1) + (x2, y2)) = fS1∧S2(x1 + x2, y1 + y2)= fS1(x1 + x2) ∩ fS2(y1 + y2)⊇ (fS1(x1) ∩ fS1(x2)) ∩ (fS2(y1) ∩ fS2(y2))= (fS1(x1) ∩ fS2(y1)) ∩ (fS1(x2) ∩ fS2(y2))= fS1∧S2(x1, y1) ∩ fS1∧S2(x2, y2).

(ii) is similar to (i).

304 x. ma, j. zhan

(iii) Let (a1, a2), (b1, b2), (x1, x2), (z1, z2) ∈ S1 × S2 be such that (x1, x2) +(a1, a2)+(z1, z2) = (b1, b2)+(z1, z2), and so x1+a1+z1 = b1+z1 and x2+a2+z2 =b2 + z2. Then

fS1∧S2(x1, x2) = fS1(x1) ∩ fS2(x2)⊇ (fS1(a1) ∩ fS1(b1)) ∩ (fS2(a2) ∩ fS2(b2))= (fS1(a1) ∩ fS2(a2)) ∩ (fS1(b1) ∩ fS2(b2))= fS1∧S2(a1, a2) ∩ fS1∧S2(b1, b2).

Hence, fS1∧S2 is an SI-hemiring of S1 × S2 over U .

Similarly, we can obtain the following result:

Proposition 3.7 Let fS1 and fS2 be two SI-h-ideals of S1 and S2 over U , respec-tively. Then fS1 ∧ fS2 is an SI-h-ideal of S1 × S2 over U .

Remark 3.8 Note that fS1 ∨ fS2 is not an SI-hemiring or SI-h-ideal over U .

Example 3.9 Let U = S3, symmetric group, be the universal set, S1 = Z5 =0, 1, 2, 3, 4 and S2 = Z2 = 0, 1 be two hemirings of non-negative integersmodule 5 and module 2, respectively. Define two soft sets fS1 and fS2 overU by fS1(0) = S3, fS1(1) = fS1(4) = (1), (12), (132), fS1(2) = fS1(3) =(12), (123), (132), fS2(0) = S3, fS2(1) = (1), (12), (132). It is clear that fS1

and fS2 are two SI-hemirings over U . However, we have

fS1∨S2((3, 1) + (1, 0)) = fS1∨S2(4, 1)= fS1(4) ∪ fS2(1)= (1), (12), (132),

but

fS1∨S2(3, 1) ∩ fS1∨S2(1, 0) = (fS1(3) ∪ fS2(1)) ∩ (fS1(1) ∪ fS2(0))= (1), (12), (123), (132) ∩ S3

= (1), (12), (123), (132).

This implies that fS1∨S2((3, 1)+ (1, 0)) + fS1∨S2(3, 1)∩ fS1∨S2(1, 0). Hence, fS1∨S2

is not an SI-hemiring over U .

Theorem 3.10 Let fS and gS be two SI-hemirings of S over U . Then fS∩gS isalso an SI-hemiring of S over U .

Proof. Let fS and gS be two SI-hemirings of S over U . Then for all x, y ∈ S,we have

(i) (fS∩gS)(x+ y) = fS(x+ y) ∩ gS(x+ y)⊇ (fS(x) ∩ fS(y)) ∩ (gS(x) ∩ gS(y))= (fS(x) ∩ gS(x)) ∩ (fS(y) ∩ gS(y))= (fS∩gS)(x) ∩ (fS∩gS)(y).


(ii) is similar to (i).

(iii) Let a, b, x, z ∈ S be such that x+ a+ z = b+ z. Then

(fS∩gS)(x) = fS(x) ∩ gS(x)⊇ (fS(a) ∩ fS(b)) ∩ (gS(a) ∩ gS(b))= (fS(a) ∩ gS(a)) ∩ (fS(b) ∩ gS(b))= (fS∩gS)(a) ∩ (fS∩gS)(b).

Hence fS∩gS is an SI-hemiring of S over U .

Similarly, we can obtain the following theorem:

Theorem 3.11 Let fS and gS be two SI-h-ideals of S over U , then fS∩gS is alsoan SI-h-ideal of S over U .

4. h-hemiregular hemirings

In this section, we describe the characterizations of h-hemiregular hemirings bymeans of SI-h-ideals.

Definition 4.1 [22] A hemiring S is called h-hemiregular if for each x ∈ S, theseexist a1, a2, z ∈ S such that x+ xa1x+ z = xa2x+ z.

Lemma 4.2 [22] If A and B, are, respectively, a right and a left h-ideal of S,then AB ⊆ A ∩B.

Lemma 4.3 [22] A hemiring S is h-hemiregular if and only if for any right h-idealA and any left h-ideal B, we have AB = A ∩B.

Definition 4.4 Let fS, gS ∈ S(U). Define soft h-sum and soft h-product of fSand gS as follows:

(1) (fS +h gS)(x) =∪

x+a1+b1+z=a2+b2+z

(fS(a1) ∩ fS(a2) ∩ gS(b1) ∩ gS(b2)) and

(fS +h gS)(x) = ∅ if x cannot be expressed as x+ a1 + b1 + z = a2 + b2 + z.

(2) (fS h gS)(x) =∪

x+a1b1+z=a2b2+z

(fS(a1) ∩ fS(a2) ∩ gS(b1) ∩ gS(b2)) and (fS h

gS)(x) = ∅ if x cannot be expressed as x+ a1b1 + z = a2b2 + z.

Lemma 4.5 Let fS and gS be an SI-right h-ideal and an SI-left h-ideal of Sover U , respectively, then fS h gS⊆fS∩gS.

306 x. ma, j. zhan

Proof. If (fS h gS)(x) = ∅, then it is clear that fS h gS⊆fS∩gS. Otherwise, wehave

(fS h gS)(x) =∪

x+a1b1+z=a2b2+z

(fS(a1) ∩ fS(a2) ∩ gS(b1) ∩ gS(b2))

⊆∪

x+a1b1+z=a2b2+z

(fS(a1b1) ∩ fS(a2b2) ∩ gS(a1b1) ∩ gS(a2b2))

⊆∪

x+a1b1+z=a2b2+z

(fS(x) ∩ gS(x))

= fS(x) ∩ gS(x)= (fS ∩ gS)(x),

which implies, fS h gS⊆fS∩gS.

Definition 4.6 Let A ⊆ S. We denote SA the soft characteristic function of Aand define as

SA(x) =

U if x ∈ A,∅ if x /∈ A.

The following proposition is obvious and we omit the details.

Proposition 4.7 Let A,B ⊆ S. Then the following hold:

(1) A ⊆ B ⇒ SA⊆SB.

(2) SA∩SB = SA∩B.

(3) SA h SB = SAB.

Theorem 4.8 For any hemiring S, then the following are equivalent:

(1) S is h-hemiregular;

(2) fS h gS = fS∩gS for any SI-right h-ideal fS and SI-left h-ideal gS of Sover U .

Proof. (1) =⇒ (2): Let S be an h-hemiregular hemiring, fS and gS an SI-right h-ideal and an SI-left h-ideal of S over U , respectively. By Lemma 4.5,we have fS h gS⊆fS∩gS. Let x ∈ S, then there exist a, a′, z ∈ S such thatx+ xax+ z = xa′x+ z since S is h-hemiregular. Thus, we have

(fS h gS)(x) =∪

x+a1b1+z=a2b2+z

(fS(a1) ∩ fS(a2) ∩ gS(b1) ∩ gS(b2))

⊇ fS(xa) ∩ fS(xa′) ∩ gS(x)

⊇ fS(x) ∩ gS(x)

= (fS ∩ gS)(x),

which implies fS h gS⊇fS∩gS. Thus, fS h gS = fS∩gS.


(2) =⇒ (1): Let R and L be any right h-ideal and left h-ideal of S, respec-tively. Then, by Lemma 4.2, we have RL ⊆ R ∩ L. Moreover, it is easy tocheck that SR and SL are an SI-right h-ideal and an SI-left h-ideal of S over U ,respectively. Let x ∈ R ∩ L, then, by Proposition 4.8, we have

SRL(x) = (SR h SL)(x) = (SR∩SL)(x) = SR∩L(x) = U,

and so x ∈ RL. Then, R ∩ L ⊆ RL. Thus, R ∩ L = RL. It follows from Lemma4.3 that S is h-hemiregular.

Acknowledgements. This research is partially supported by a grant of NationalNatural Science Foundation of China (61175055), Natural Science Foundation ofHubei Province (2012FFB01101) and Natural Science Foundation of EducationCommittee of Hubei Province (D20131903).

References

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CHARACTERIZATIONS OF REGULAR ABEL-GRASSMANN’SGROUPOIDS BY THE PROPERTIES OF THEIR (∈,∈ ∨qk)-FUZZYIDEALS

Xueling Ma

Jianming Zhan1


Madad Khan

Tariq Aziz

Department of MathematicsCOMSATS Institute of Information TechnologyAbbottabadPakistan

Abstract. In this paper, we characterize some properties of regular Abel-Grassmann’s

groupoid in terms of its (∈,∈ ∨qk)-fuzzy ideals.

Keywords and phrases: AG-groupoid, left invertive law, medial law, paramedial law

and (∈,∈ ∨qk)-fuzzy ideals.

1. Introduction

Fuzzy set theory and its applications are growing day by day in various branchesof Science like mathematics, computer science, engineering, physics, managementsciences, medical science, operational research, artificial intelligence, robotics, ex-pert system and various other fields of sciences. Fuzzy mappings are used in fuzzyimage processing, fuzzy decision making, fuzzy linear programming and fuzzydata bases. It is used in mechanical engineering, industrial engineering, computerengineering and civil engineering. Also the uses of fuzzification can be found infuzzy systems, genetic algorithms mechanics and economics.

Mordeson et al. [20] has discovered the grand exploration of fuzzy semigroups,where theory of fuzzy semigroups is explored along with the applications of fuzzysemigroups in fuzzy coding, fuzzy finite state mechanics and fuzzy languages andthe use of fuzzification in automata and formal language has widely been explored.

1Corresponding author. Tel/Fax: 0086-718-8437732. E-mail: [email protected]

310 x. ma, j. zhan, m. khan, t. aziz

Moreover the complete l-semigroups have wide range of applications in the theoriesof automata, formal languages and programming. It is worth mentioning thatsome recent investigations of l-semigroups are closely connected with algebraiclogic and non-classical logic. Fuzzy sets are also closely related to other softcomputing models such as rough sets [23], random sets [7] and soft sets [5], [6], [19].Zadeh further discussed the relationships between fuzzy set theory and probabilitytheory [31].

An AG-groupoid is a mid structure between a groupoid and a commuta-tive semigroup. If an AG-groupoid contains left identity then this left identityis unique. However an AG-groupoid with right identity becomes a commutativesemigroup with identity. Moreover every commutative AG-groupoid becomes acommutative semigroup. Mostly an AG-groupoid works like a commutative semi-group. For instance a2b2 = b2a2, for all a, b holds in a commutative semigroup,while this equation also holds for an AG-groupoid with left identity e, moreoverab = (ba)e for any subset a, b of an AG-groupoid. Now our aim is to discoversome logical investigations for regular AG-groupoids using the new generalizedconcept of fuzzy sets. It is therefore concluded that this research work will givea new direction for applications of fuzzy set theory particularly in algebraic logic,non-classical logics, fuzzy coding, fuzzy finite state mechanics and fuzzy languages.

In [21], Murali gave the idea of belongingness of a fuzzy point to a fuzzy subsetunder a natural equivalence on a fuzzy subset. The idea of quasi-coincidence of afuzzy point with a fuzzy set is defined in [27]. Bhakat and Das [1], [2], gave theidea of (α, β)-fuzzy subgroups by using the “belongs to” relation ∈ and “quasi-coincident with” relation q between a fuzzy point and a fuzzy subgroup, andintroduced the concept of an (∈,∈ ∨q)-fuzzy subgroups, where α, β ∈ ∈, q,∈∨q,∈ ∧q and α =∈ ∧q. Davvaz defined (∈,∈ ∨q)-fuzzy subnearrings and idealsof a near ring in [4]. Jun and Song initiated the study of (α, β)-fuzzy interiorideals of a semigroup in [10]. In [29], regular semigroups are characterized by theproperties of their (∈,∈ ∨q)-fuzzy ideals. In [28], semigroups are characterized bythe properties of their (∈,∈ ∨qk)-fuzzy ideals.

In this paper we have introduced (∈,∈ ∨qk)-fuzzy ideals in a new non-associative algebraic structure, that is, in an AG-groupoid and developed somenew results. We have defined a regular AG-groupoid and characterized it by theproperties of its (∈,∈ ∨qk)-fuzzy ideals.

2. AG-groupoids

A groupoid (S, .) is called AG-groupoid, if its elements satisfy left invertive law:(ab)c = (cb)a. In an AG-groupoid medial law [12], (ab)(cd) = (ac)(bd), holds forall a, b, c, d ∈ S. It is also known that in an AG-groupoid with left identity, theparamedial law: (ab)(cd) = (db)(ca), holds for all a, b, c, d ∈ S. If an AG-groupoidcontains left identity, the following law holds,

(1) a(bc) = b(ac), for all a, b, c ∈ S.

characterizations of regular abel-grassmann’s groupoids ... 311

Let S be an AG-groupoid. By AG-subgroupoid of S we means a non-emptysubset A of S such that A2 ⊆ A, by a left (right) ideal of S we mean a non-emptysubset L of S such that SL ⊆ L (RS ⊆ R) and by a quasi-ideal of S we mean anon-empty subset Q of S such that QS ∩ SQ ⊆ Q. By two-sided ideal or simplyideal, we mean a non-empty subset of S which is both a left and a right ideal ofS. An AG-subgroupoid B of S is called bi-ideal of S if (BS)B ⊆ B. A subset Bof S is called generalized bi-ideal of S if (BS)B ⊆ B.

A fuzzy subset f of a given set S is described as an arbitrary functionf : G −→ [0, 1], where [0, 1] is the usual closed interval of real numbers. For anytwo fuzzy subsets f and g of S, f ≤ g means that, f(x) ≤ g(x) for all x in S. Thesymbols f ∩ g and f ∪ g will means that the following fuzzy subsets of S

(f ∩ g)(x) = minf(x), g(x) = f(x) ∧ g(x)

(f ∪ g)(x) = maxf(x), g(x) = f(x) ∨ g(x)

for all x in S.Let f and g be any fuzzy subsets of an AG-groupoid S, then the product f g

is defined by

(f g) (a) =

∨a=bc

f(b) ∧ g(c) , if there exist b, c ∈ S, such that a = bc

0, otherwise.

A fuzzy subset f of an AG-groupoid S is called a fuzzy AG-subgroupoid ofS if f(xy) ≥ f(x) ∧ f(y) for all x, y ∈ S.

A fuzzy subset f of an AG-groupoid S is called fuzzy left (right) ideal of S iff(xy) ≥ f(y) (f(xy) ≥ f(x)) for all x, y ∈ S.

A fuzzy subset f of an AG-groupoid S is called fuzzy two-sided ideal of S ifit is both fuzzy left and fuzzy right ideal of S.

A fuzzy AG-subgroupoid f of an AG-groupoid S is called fuzzy bi-ideal of Sif f((xy)z) ≥ f(x) ∧ f(z), for all x, y and z ∈ S.

A fuzzy subset f of an AG-groupoid S is called fuzzy generalized bi-ideal ofS if f((xy)z) ≥ f(x) ∧ f(z), for all x, y and z ∈ S.

A fuzzy subset f of an AG-groupoid S is called fuzzy quasi-ideal of S if(f S) (x) ∧ (S f) (x) ≤ f(x), for all x ∈ S.

Let F (S) denote the collection of all fuzzy subsets of an AG-groupoid S withleft identity, then (F (S), ) becomes an AG-groupoid with left identity S, that is(F (S), ) satisfies left invertive law, medial law, paramedial law and property (1).Note that S can be considered as a fuzzy subset of S itself and we write S = CS,that is, S(x) = 1 for all x ∈ S. Moreover S S = S.

The characteristic function CA for a subset A of an AG-groupoid S is de-fined by

CA(x) =

1, if x ∈ A,0, if x /∈ A.

The proof of the following three lemmas are the same as in [20].


Definition 1 A fuzzy subset f of an AG-groupoid S is called an (∈,∈ ∨q)-fuzzyAG-subgroupoid of S if

xt ∈ f, yr ∈ f ⇒ (xy)t∧r ∈ ∨qf

for all x, y ∈ S and t, r ∈ (0, 1].

Theorem 1 Let f be a fuzzy subset of S. Then f is an (∈,∈ ∨q)-fuzzy AG-subgroupoid of S if and only if f(xy) ≥ minf(x), f(y), 1−k

2 for all x, y ∈ S.

Proof. It is similar to the proof of Theorem 12 in [28].

Definition 2 A fuzzy subset f of an AG-groupoid S is called an (∈,∈ ∨q)-fuzzyleft (resp. right) ideal of S if it satisfies the following condition:

yt ∈ f ⇒ (xy)t ∈ ∨qf (resp. yt ∈ f ⇒ (yx)t ∈ ∨qf)

for all x, y ∈ S and t ∈ (0, 1].

A fuzzy subset f of S is called an (∈,∈ ∨q)-fuzzy ideal of S if it is both an(∈,∈ ∨q)-fuzzy left ideal and an (∈,∈ ∨q)-fuzzy right ideal of S.

Theorem 2 A fuzzy subset f of S is an (∈,∈ ∨q)-fuzzy left (resp. right) ideal ofS, if and only if

f(xy) ≥ min

f(y),

1− k

2

(resp. f(xy) ≥ min

f(x),

1− k

2

).

Proof. It is similar to the proof of Lemma 5 in [28].

Definition 3 Let S be an AG-groupoid, and f be a fuzzy subset of S. Then f isan (∈,∈ ∨q)-fuzzy generalized bi-ideal of S, if for all x, y,z ∈ S and t, r ∈ (0, 1],we have

xt ∈ f, zr ∈ f ⇒ ((xy)z)t∧r ∈ ∨qf.

An (∈,∈ ∨q)-fuzzy generalized bi-ideal of S is called an (∈,∈ ∨q)-fuzzy bi-idealof S if it is also an (∈,∈ ∨q)-fuzzy AG-subgroupoid of S.

Definition 4 A fuzzy subset f of an AG-groupoid S is called an (∈,∈ ∨q)-fuzzyquasi-ideal of S if it satisfies f(x) ≥ min(f CS(x), CS f(x), 1−k

2), where CS is

the fuzzy subset of S mapping every element of S on 1.

Theorem 3 Let f be a fuzzy subset of an AG-groupoid S. Then f is an (∈,∈ ∨q)-fuzzy bi-ideal of S if and only if it satisfies:

(i) f(xy) ≥ minf(x), f(y), 1−k2, for all x, y ∈ S.

(ii) f((xy)z) ≥ minf(x), f(z), 1−k2, for all x, y, z ∈ S.


Definition 5 An element a of an AG-groupoid S is called regular if there exist xin S such that a = (ax)a and S is called regular, if every element of S is regular.

Definition 6 An element a of an AG-groupoid S is called intra-regular if thereexist x, y ∈ S such that a = (xa2)y and S is called intra-regular, if every elementof S is intra-regular.

Lemma 1 For an AG-groupoid S, the following holds.

(i) A non empty subset I of AG-groupoid S is an ideal if and only if (CI)kis (∈,∈ ∨qk)-fuzzy ideal.

(ii) A non empty subset L of AG-groupoid S is left ideal if and only if (CL)kis (∈,∈ ∨qk)-fuzzy left ideal.

(iii) A non empty subset R of AG-groupoid S is right ideal if and only if (CR)kis (∈,∈ ∨qk)-fuzzy right ideal.

(iv) A non empty subset B of AG-groupoid S is bi-ideal if and only if (CB)kis (∈,∈ ∨qk)-fuzzy bi-ideal.

(v) A non empty subset Q of AG-groupoid S is quasi-ideal if and only if (CQ)kis (∈,∈ ∨qk)-fuzzy quasi-ideal.

Lemma 2 Let A,B be non empty subsets of an AG-groupoid S. Then the fol-lowing holds.

(i) (CA∩B)k = (CA ∧k CB) .

(ii) (CA∪B)k = (CA ∨k CB) .

(iii) (CAB)k = (CA k CB) .

Lemma 3 Let S be an AG-groupoid. If a = a(ax), for some x in S. Thena = a2y, for some y in S.

Proof. Using the medial law, we get

a = a(ax) = [a(ax)](ax) = (aa)((ax)x) = a2y, where y = (ax)x.

Lemma 4 Let S be an AG-groupoid with left identity. If a = a2x, for some xin S. Then a = (ay)a, for some y in S.

Proof. Using the medial law, the left invertive law, (1), the paramedial law andthe medial law, we get

a = a2x = (aa)x = ((a2x)(a2x))x = ((a2a2)(xx))x = (xx2)(a2a2)

= a2((xx2)a2)) = ((xx2)a2)a)a = ((aa2)(xx2))a = ((x2x)(a2a))a

= [a2(x2x)a]a = [a(x2x)(aa)]a = [a(a(x2x)a)]a= (ay)a, where y = a(x2x)a.


Lemma 5 In AG-groupoid S, with left identity, the following holds.

(i) (aS) a2 = (aS) a.

(ii) (aS) ((aS) a) = (aS) a.

(iii) S ((aS) a) = (aS) a.

(iv) (Sa) (aS) = a (aS) .

(v) (aS) (Sa) = (aS) a.

(vi) [a(aS)]S = (aS)a.

(vii) [(Sa)S](Sa) = (aS)(Sa).

(viii) (Sa)S = (aS).

(ix) S(Sa) = Sa.

(x) Sa2 = a2S.

Proof. It is easy.

Lemma 6 Every intra-regular AG-groupoid with left identity is regular but theconverse is not true.

Proof. It is easy.For the converse of Lemma 6, see the following example.

Example 1 Let us consider an AG-groupoid S = 1, 2, 3 in the following mul-tiplication table.

1 2 31 1 1 12 1 1 33 1 2 1

It is easy to check that 1, 2 is the quasi-ideal of S. Clearly S is regular because1 = 1 1, 2 = (2 3) 2 and 3 = (3 2) 3. But is not intra-regular AG-groupoid.

Let us define a fuzzy subset f on S as follows:

f(x) =

0.9 for x = 10.8 for x = 20.6 for x = 3

Then, clearly, f is an (∈,∈ ∨qk)-fuzzy ideal of S.

Theorem 4 For an AG-groupoid S, with left identity, the following are equiva-lent.

(i) S is regular.

(ii) For bi-ideal B, ideal I and left ideal L of S, B ∩ I ∩ L ⊆ (BI)L.

(iii) B [a] ∩ I [a] ∩ L [a] ⊆ (B [a] I [a])L [a] , for some a in S.


Proof. (i) ⇒ (ii) Assume that B, I and L are bi-ideal, ideal and left ideal of aregular AG-groupoid S respectively. Let a ∈ B ∩ I ∩ L. This implies that a ∈ I,a ∈ B and a ∈ L. Since S is regular so for a ∈ S there exist x ∈ S such that usingleft invertive law and (1) , we have, a = (ax) a = (((ax) a)x) a = ((xa) (ax)) a =(a ((xa)x)) a = (B ((SI)S))L = (BI)L. Thus, B ∩ I ∩ L ⊆ (BI)L.

(ii) ⇒ (iii) is obvious.

(iii) ⇒ (i) B [a] = a ∪ a2 ∪ (aS) a, I [a] = a ∪ Sa ∪ aS and L [a] = a ∪ Saare principle bi-ideal, principle ideal and principle left ideal of S generated by arespectively. Thus, by (iii), Lemma 5, (1), left invertive law and paramedial lawwe have,

(a ∪ a2 ∪ (aS) a) ∩ (a ∪ Sa ∪ aS) ∩ (a ∪ Sa)⊆ ((a ∪ a2 ∪ (aS) a) (a ∪ Sa ∪ aS)) (a ∪ Sa)⊆ S (a ∪ Sa ∪ aS) (a ∪ Sa)⊆ Sa ∪ S (Sa) ∪ S (aS) (a ∪ Sa)= (Sa ∪ aS) (a ∪ Sa)= (Sa) a ∪ (Sa) (Sa) ∪ (aS) a ∪ (aS) (Sa)= a2S ∪ a2S ∪ (aS) a ∪ (aS) a= a2S ∪ (aS) a.

Hence S is regular.

Theorem 5 For an AG-groupoid S, with left identity, the following are equivalent.

(i) S is regular.

(ii) For (∈,∈ ∨qk)-fuzzy bi-ideal f , (∈,∈ ∨qk)-fuzzy ideal g, and (∈,∈ ∨qk)-fuzzy left ideal h of S, (f ∧k g) ∧k h ≤ (f k g) k h.

(iii) For (∈,∈ ∨qk)-fuzzy generalized bi-ideal f , (∈,∈ ∨qk)-fuzzy ideal g, and(∈,∈ ∨qk)-fuzzy left ideal h of S, (f ∧k g) ∧k h ≤ (f k g) k h.

Proof. (i) ⇒ (iii) Assume that f , g and h are (∈,∈ ∨qk)-fuzzy generalizedbi-ideal, (∈,∈ ∨qk)-fuzzy ideal and (∈,∈ ∨qk)-fuzzy left ideal of a regular AG-groupoid S, respectively. Now since S is regular so for a ∈ S there exist x ∈ Ssuch that using left invertive law and (1) , we have, a = (ax) a = (((ax) a)x) a =((xa) (ax)) a = (a ((xa) x)) a. Thus,

((f k g) k h)(a) =∨a=pq

(f k g)(p) ∧ h(q) ∧ 1− k

2

=∨a=pq

( ∨p=uv

f(u) ∧ g(v) ∧ 1− k

2

∧ h(q) ∧ 1− k

2

)

=∨

a=(uv)q

(f(u) ∧ g(v) ∧ h(q) ∧ 1− k

2

)


=∨

a=(a((xa)x))a=(uv)q

(f(u) ∧ g(v) ∧ h(q) ∧ 1− k

2

)≥ f(a) ∧ g ((xa)x) ∧ h(a) ∧ 1− k

2

≥f(a) ∧

(g(a) ∧ 1− k

2

)∧ h(a) ∧ 1− k

2

= f (a) ∧ g (a) ∧ 1− k

2 ∧ h (a) ∧ 1− k

2= ((f ∧k g) ∧k h) (a) .

Therefore (f ∧k g) ∧k h ≤ (f k g) k h.(iii) ⇒ (ii) is obvious.(ii) =⇒ (i) Assume that B, I and L are bi-ideal, ideal and left ideal of S

respectively. Then, by Lemma 1, (CB)k, (CI)k and (CL)k are (∈,∈ ∨qk)-fuzzybi-ideal, (∈,∈ ∨qk)-fuzzy ideal and (∈,∈ ∨qk)-fuzzy left ideal of S respectively.Therefore, by Lemma 2, we have, (CB∩I∪L)k = (CB ∧k CI) ∧k CL ≤ (CB k CI) kCL = (C(BI)L)k = (C(BI)L)k. Therefore B ∩ I ∩ L ⊆ (BI)L. Hence, by Theorem4, S is regular.


(i) S is regular.

(ii) For left ideal L, ideal I and quasi-ideal Q of S, L ∩ I ∩Q ⊆ (LI)Q.

(iii) L [a] ∩ I [a] ∩Q [a] ⊆ (L [a] I [a])Q [a] , for some a in S.

Proof. (i) ⇒ (ii) Assume that L, I and Q are left ideal, ideal and quasi-idealof regular AG-groupoid S. Let a ∈ L ∩ I ∩ Q. This implies that a ∈ L, a ∈ Iand a ∈ Q. Now since S is regular so for a ∈ S there exist x ∈ S such that usingleft invertive law and (1) , we have, a = (ax) a = (((ax) a) x) a = ((xa) (ax)) a =(a ((xa)x)) a ∈ (L ((SI)S))Q ⊆ (LI)Q. Thus L ∩ I ∩Q ⊆ (LI)Q.

(ii) ⇒ (iii) is obvious.(iii) ⇒ (i) L [a] = a∪Sa, I [a] = a∪Sa∪aS and Q [a] = a∪(Sa ∩ aS) are left

ideal, ideal and quasi-ideal of S generated a respectively. Thus, by (iii), Lemma5 and medial law we have,

(a ∪ Sa) ∩ (a ∪ Sa ∪ aS) ∩ (a ∪ (Sa ∩ aS)) ⊆ ((a ∪ Sa) (a ∪ Sa ∪ aS))

(a ∪ (Sa ∩ aS))

⊆ (a ∪ Sa)S (a ∪ aS)

= aS ∪ (Sa)S (a ∪ aS)

= (aS) (a ∪ aS)

= (aS) a ∪ (aS) (aS)

= (aS) a ∪ a2S.

Hence S is regular.



(i) S is regular.

(ii) For (∈,∈ ∨qk)-fuzzy left ideal f , (∈,∈ ∨qk)-fuzzy ideal g, and (∈,∈ ∨qk)-fuzzy quasi-ideal h of S, (f ∧k g) ∧k h ≤ (f k g) k h.

Proof. (i) ⇒ (ii) Assume that f , g and h are (∈,∈ ∨qk)-fuzzy left ideal, (∈,∈ ∨qk)-fuzzy ideal and (∈,∈ ∨qk)-fuzzy quasi-ideal of a regular AG-groupoid S,respectively. Now, since S is regular so for a ∈ S there exist x ∈ S such that usingleft invertive law and (1) , we have, a = (ax) a = (((ax) a)x) a = ((xa) (ax)) a =(a ((xa)x)) a. Thus,

((f k g) k h)(a) =∨a=pq

(f k g)(p) ∧ h(q) ∧ 1− k

2

=∨a=pq

( ∨p=uv

f(u) ∧ g(v) ∧ 1− k

2

∧ h(q) ∧ 1− k

2

)

=∨

a=(uv)q

(f(u) ∧ g(v) ∧ h(q) ∧ 1− k

2

)

=∨

a=(a((xa)x))a=(uv)q

(f(u) ∧ g(v) ∧ h(q) ∧ 1− k

2

)≥ f(a) ∧ g ((xa) x) ∧ h(a) ∧ 1− k

2

≥f(a) ∧

(g(a) ∧ 1− k

2

)∧ h(a) ∧ 1− k

2

= f (a) ∧ g (a) ∧ 1− k

2 ∧ h (a) ∧ 1− k

2= ((f ∧k g) ∧k h) (a) .

Therefore, (f ∧k g) ∧k h ≤ (f k g) k h.(ii) =⇒ (i) Assume that L, I and Q are left ideal, ideal and quasi-ideal of S

respectively. Thus, by Lemma 1, (CL)k, (CI)k and (CQ)k are (∈,∈ ∨qk)-fuzzy leftideal, (∈,∈ ∨qk)-fuzzy ideal and (∈,∈ ∨qk)-fuzzy quasi-ideal of S respectively.Therefore, by Lemma 2, we have, (CL∩I∪Q)k = (CL ∧k CI) ∧k CQ ≤ (CL k CI) kCQ = (C(LI)Q)k = (C(LI)Q)k. Therefore L∩ I ∩Q ⊆ (LI)Q. Hence by Theorem 6,S is regular.


(i) S is regular.

(ii) For bi-ideal B, ideal I and quasi-ideal Q of S, B ∩ I ∩Q ⊆ (BI)Q.

(iii) B [a] ∩ I [a] ∩Q [a] ⊆ (B [a] I [a])Q [a] , for some a in S.


Proof. (i) ⇒ (ii) Assume that B, I and Q are bi-ideal, ideal and quasi-ideal ofregular AG-groupoid S. Let a ∈ B ∩ I ∩ Q. This implies that a ∈ B, a ∈ I anda ∈ Q. Now, since S is regular so for a ∈ S there exist x ∈ S such that usingleft invertive law and (1) , we have, a = (ax) a = (((ax) a)x) a = ((xa) (ax)) a =(a ((xa)x)) a ∈ (B ((SI)S))Q ⊆ (BI)Q. Thus, B ∩ I ∩Q ⊆ (BI)Q.

(ii) ⇒ (iii) is obvious.(iii) ⇒ (i) Since B [a] = a ∪ a2 ∪ (aS) a, I [a] = a ∪ Sa ∪ aS and Q [a] =

a ∪ (Sa ∩ aS) are principle bi-ideal, principle ideal and principle quasi-ideal of Sgenerated by a respectively. Thus, by (ii) and Lemma 5, (1), medial law and leftinvertive law, we have,

(a ∪ a2 ∪ (aS) a) ∩ (a ∪ Sa ∪ aS) ∩ (a ∪ (Sa ∩ aS))⊆ ((a ∪ a2 ∪ (aS) a) (a ∪ Sa∪aS)) (a ∪ (Sa ∩ aS))⊆ (S(a ∪ Sa ∪ aS)) (a ∪ aS)= (Sa ∪ S (Sa) ∪ S (aS)) (a ∪ aS)= (Sa ∪ S (Sa) ∪ S (aS)) (a ∪ aS)= (aS ∪ Sa) (a ∪ aS)= (aS) a ∪ (aS) (aS) ∪ (Sa) a ∪ (Sa) (aS)= (aS) a ∪ a2S ∪ a (aS) .

Hence S is regular.


(i) S is regular.

(ii) For (∈,∈ ∨qk)-fuzzy bi-ideal f , (∈,∈ ∨qk)-fuzzy ideal g, and (∈,∈ ∨qk)-fuzzyquasi ideal h of S, (f ∧k g) ∧k h ≤ (f k g) k h.

(iii) For (∈,∈ ∨qk)-fuzzy generalized bi-ideal f , (∈,∈ ∨qk)-fuzzy ideal g, and(∈,∈ ∨qk)-fuzzy quasi ideal h of S, (f ∧k g) ∧k h ≤ (f k g) k h.

Proof. (i) ⇒ (iii) Assume that f , g and h are (∈,∈ ∨qk)-fuzzy generalizedbi-ideal, (∈,∈ ∨qk)-fuzzy ideal and (∈,∈ ∨qk)-fuzzy quasi ideal of a regular AG-groupoid S, respectively. Now, since S is regular so for a ∈ S there exist x ∈ Ssuch that using left invertive law and (1) , we have, a = (ax) a = (((ax) a)x) a =((xa) (ax)) a = (a ((xa) x)) a. Thus,

((f k g) k h)(a) =∨a=pq

(f k g)(p) ∧ h(q) ∧ 1−k2

=∨a=pq

( ∨p=uv

f(u) ∧ g(v) ∧ 1−k2

∧ h(q) ∧ 1−k

2

)=

∨a=(uv)q

(f(u) ∧ g(v) ∧ h(q) ∧ 1−k

2

)=

∨a=(a((xa)x))a=(uv)q

(f(u) ∧ g(v) ∧ h(q) ∧ 1−k

2

)


≥ f(a) ∧ g ((xa)x) ∧ h(a) ∧ 1−k2

≥f(a) ∧

(g(a) ∧ 1−k

2

)∧ h(a) ∧ 1−k

2

= f (a) ∧ g (a) ∧ 1−k2 ∧ h (a) ∧ 1−k

2

= ((f ∧k g) ∧k h) (a) .

Therefore, (f ∧k g) ∧k h ≤ (f k g) k h.(iii) ⇒ (ii) is obvious.(ii) =⇒ (i) Assume that B, I and Q be bi-ideal, ideal and quasi-ideal of S

respectively. Then, by Lemma 1, (CB)k, (CI)k and (CQ)k are (∈,∈ ∨qk)-fuzzybi-ideal, (∈,∈ ∨qk)-fuzzy ideal and (∈,∈ ∨qk)-fuzzy quasi-ideal of S respectively.Therefore, by Lemma 2, we have, (CB∩I∪Q)k = (CB ∧k CI)∧k CQ ≤ (CB k CI) kCQ = (C(BI)Q)k = (C(BI)Q)k. Therefore B ∩ I ∩Q ⊆ (BI)Q. Hence, by Theorem8, S is regular.


(i) S is regular.

(ii) For an ideals I1, I2 and I3 of S, I1 ∩ I2 ∩ I3 ⊆ (I1I2) I3.

(iii) I [a] ∩ I [a] ∩ I [a] ⊆ (I [a] I [a]) I [a] , for some a in S.

Proof. (i) ⇒ (ii) Assume that I1, I2, and I2 are ideals of a regular AG-groupoidS. Let a ∈ I1 ∩ I2 ∩ I3. This implies that a ∈ I1, a ∈ I2 and a ∈ I3. Now, sinceS is regular so for a ∈ S there exist x ∈ S such that using left invertive lawand (1) , we have, a = (ax) a = (((ax) a) x) a = ((xa) (ax)) a = (a ((xa)x)) a ∈(I1 ((SI2)S)) I3 ⊆ (I1I2) I3. Thus, I1 ∩ I2 ∩ I3 ⊆ (I1I2) I3.

(ii) ⇒ (iii) is obvious.(iii) ⇒ (i)Since I [a] = a ∪ Sa ∪ aS is a principle ideal of S generated by a. Thus, by

(iii), Lemma 5, left invertive law, medial law and paramedial law we have,

(a ∪ Sa ∪ aS) ∩ (a ∪ Sa ∪ aS) ∩ (a ∪ Sa ∪ aS)

⊆ ((a ∪ Sa ∪ aS) (a ∪ Sa ∪ aS))

(a ∪ Sa ∪ aS)

⊆ (a ∪ Sa ∪ aS)S (a ∪ Sa ∪ aS)

= aS ∪ (Sa)S ∪ (aS)S (a ∪ Sa ∪ aS)

= aS ∪ Sa (a ∪ Sa ∪ aS)

= (aS) a ∪ (aS) (Sa) ∪ (aS) (aS) ∪ (Sa) a

∪ (Sa) (Sa) ∪ (Sa) (aS)

= (aS) a ∪ a2S.

Hence S is regular.



(i) S is regular.

(ii) For quasi-ideals Q1, Q2 and ideal I of S, Q1 ∩ I ∩Q2 ⊆ (Q1I)Q2.

(iii) Q [a] ∩ I [a] ∩Q [a] ⊆ (Q [a] I [a])Q [a] , for some a in S.

Proof. (i) ⇒ (ii) Assume that Q1 and Q are quasi-ideal and I is an ideal of aregular AG-groupoid S. Let a ∈ Q1 ∩ I ∩ Q2. This implies that a ∈ Q1, a ∈ Iand a ∈ Q2. Now, since S is regular so for a ∈ S there exist x ∈ S such that usingleft invertive law and (1) , we have, a = (ax) a = (((ax) a)x) a = ((xa) (ax)) a =(a ((xa)x)) a ∈ (Q1 ((SI)S))Q2 ⊆ (Q1I)Q2. Thus, Q1 ∩ I ∩Q2 ⊆ (Q1I)Q2.

(ii) ⇒ (iii) is obvious.

(iii) ⇒ (i) Q [a] = a ∪ (Sa ∩ aS) and I [a] = a ∪ Sa ∪ aS are principle quasi-ideal and principle ideal of S generated by a respectively. Thus by (iii), leftinvertive law, medial law and Lemma 5, we have,

(a ∪ (Sa ∩ aS)) ∩ (a ∪ Sa ∪ aS) ∩ (a ∪ (Sa ∩ aS))

⊆ ((a ∪ (Sa ∩ aS)) (a ∪ Sa ∪ aS))

(a ∪ (Sa ∩ aS))

⊆ (a ∪ aS)S (a ∩ aS)

= aS ∪ (aS)S (a ∩ aS)

= (aS ∪ Sa) (a ∩ aS)

= (aS) a ∪ (aS) (aS) ∪ (Sa) a ∪ (Sa) aSa

= (aS) a ∪ a2S ∪ a (aS) .

Hence, S is regular.


(i) S is regular.

(ii) For (∈,∈ ∨qk)-fuzzy quasi-ideals f, h, and (∈,∈ ∨qk)-fuzzy ideal g, of S,(f ∧k g) ∧k h ≤ (f k g) k h.

Proof. (i) ⇒ (ii) Assume that f , h are (∈,∈ ∨qk)-fuzzy quasi-ideal and g is(∈,∈ ∨qk)-fuzzy ideal of a regular AG-groupoid S, respectively. Now since S isregular so for a ∈ S there exist x ∈ S such that using left invertive law and (1) ,we have, a = (ax) a = (((ax) a) x) a = ((xa) (ax)) a = (a ((xa)x)) a. Thus,


((f k g) k h)(a) =∨a=pq

(f k g)(p) ∧ h(q) ∧ 1− k

2

=∨a=pq

( ∨p=uv

f(u) ∧ g(v) ∧ 1− k

2

∧ h(q) ∧ 1− k

2

)

=∨

a=(uv)q

(f(u) ∧ g(v) ∧ h(q) ∧ 1− k

2

)

=∨

a=(a((xa)x))a=(uv)q

(f(u) ∧ g(v) ∧ h(q) ∧ 1− k

2

)≥ f(a) ∧ g ((xa) x) ∧ h(a) ∧ 1− k

2

≥f(a) ∧

(g(a) ∧ 1− k

2

)∧ h(a) ∧ 1− k

2

= f (a) ∧ g (a) ∧ 1− k

2 ∧ h (a) ∧ 1− k

2= ((f ∧k g) ∧k h) (a) .

Therefore, (f ∧k g) ∧k h ≤ (f k g) k h.(ii) =⇒ (i) Assume that Q1 and Q2 are quasi-ideals and I is an ideal of

S respectively. Thus, by Lemma 1, (CQ1)k, (CI)k and (CQ2)k are (∈,∈ ∨qk)-fuzzy quasi-ideal, (∈,∈ ∨qk)-fuzzy ideal and (∈,∈ ∨qk)-fuzzy quasi-ideal of Srespectively. Therefore, by Lemma 2, we have,

(CQ1∩I∪Q2)k = (CQ1∧kCI)∧kCQ2 ≤ (CQ1kCL)kCQ2 = (C(Q1I)Q2)k = (C(Q1I)Q2)k.

Therefore Q1 ∩ I ∩Q2 ⊆ (Q1I)Q2. Hence, by Theorem 11, S is regular.

Theorem 13 For an AG-groupoid S with left identity, the following are equiva-lent.

(i) S is regular.

(ii) For bi-ideal B, B = (BS)B.

(iii) For generalized bi-ideal B, B = (BS)B.

Proof. The proof is straightforward.


(i) S is regular.

(ii) For (∈,∈ ∨qk)-fuzzy bi-ideal f , of S, fk = (f k S) k f .

(iii) For (∈,∈ ∨qk)-fuzzy generalized bi-ideal f , of S, fk = (f k S) k f .


Proof. (i) ⇒ (iii) Assume that f is (∈,∈ ∨qk)-fuzzy generalized bi-ideal of aregular AG-groupoid S. Since S is regular so for b ∈ S there exist x ∈ S suchthat b = (bx) b. Therefore, we have,

((f k S) k f)(b) =∨b=pq

(f k S)(p) ∧ f(q) ∧ 1− k

2

=∨b=pq

( ∨p=uv

f(u) ∧ S(v) ∧ 1− k

2

∧ f(q) ∧ 1− k

2

)

=∨

b=(uv)q

(f(u) ∧ S(v) ∧ f(q) ∧ 1− k

2

)

=∨

b=(bx)b=(uv)q

(f(u) ∧ S(v) ∧ f(q) ∧ 1− k

2

)≥ f(b) ∧ S (x) ∧ f(b) ∧ 1− k

2

≥ f(b) ∧ 1 ∧ f(b) ∧ 1− k

2

= f (b) ∧ 1− k

2= fk (b) .

Thus (f kS)k f ≥ fk. Since f is (∈,∈ ∨qk)-fuzzy generalized bi-ideal of a regularAG-groupoid S. So we have,

((f k S) k f)(b) =∨b=pq

(f k S)(p) ∧ f(q) ∧ 1− k

2

=∨b=pq

( ∨p=uv

f(u) ∧ S(v) ∧ 1− k

2

∧ f(q) ∧ 1− k

2

)

=∨b=pq

( ∨p=uv

f(u) ∧ 1

∧ f(q) ∧ 1− k

2

)

=∨b=pq

( ∨p=uv

f(u) ∧ f(q) ∧ 1− k

2

)

=∨

b==pq

∨p=uv

(f(u) ∧ f(q) ∧ 1− k

2

)

≤∨

b==(uv)q

(f ((uv)q) ∧ 1− k

2

)= f (b) ∧ 1− k

2= fk (b) .

This implies that (f k S) k f ≤ fk. Thus (f k S) k f = fk.(iii) ⇒ (ii) is obvious.


(ii) =⇒ (i) Assume that B is a bi-ideal of S. Then, by Lemma 1, (CB)k,is an (∈,∈ ∨qk)-fuzzy bi-ideal of S. Therefore, by (ii) and Lemma 2, we have,(CB)k = (CBkCS)kCB = (C(BS)B)k. Therefore B = (BS)B. Hence, by Theorem13, S is regular.

Acknowledgements. This research is partially supported by a grant of NationalNatural Science Foundation of China (61175055), Natural Science Foundation ofHubei Province (2012FFB01101) and Natural Science Foundation of EducationCommittee of Hubei Province (D20131903).

References

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[2] Bhakat, S.K., Das, P., (∈,∈ ∨q)-fuzzy subgroups, Fuzzy Sets and Systems,80 (1996), 359-368.

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[4] Davvaz, B., (∈,∈ ∨q)-fuzzy subnearrings and ideals, Soft Comput., 10(2006), 206-211.

[5] Feng, F., Li, C.X., Davvaz, B., Ali, M.I., Soft sets combined withfuzzy sets and rough sets: a tentative approach, Soft Computing, 14 (2010),899–911.

[6] Feng, F., Liu, X.Y., Leoreanu-Fotea, V., Jun, Y.B., Soft sets andsoft rough sets, Inform. Sci., 181 (2011), 1125–1137.

[7] Goodman, I.R., Fuzzy sets as equivalence classes of random sets.In: Recent Developments in Fuzzy Sets and Possibility Theory (R. Yager,Ed.), Pergamon, New York (1982).

[8] Iseki, K., A characterization of regular semigroups, Proc. Japan Acad., 32(1965), 676-677.

[9] Jun, Y.B., Generalizations of (∈,∈ ∨q)-fuzzy subalgebras in BCK/BCI-algebra, Comput. Math. Appl., 58 (2009), 1383-1390.

[10] Jun, Y.B., Song, S.Z., Generalized fuzzy interior ideals in semigroups,Inform. Sci., 176 (2006), 3079-3093.

[11] Kazancı, O., Yamak, S., Generalized fuzzy bi-ideals of semigroups, SoftComput., 12 (2008), 1119-1124.

[12] Kazim, M.A., Naseeruddin, M., On almost semigroups, The Alig. Bull.Math., 2 (1972), 1-7.

[13] Kehayopulu, N., Tsingelis, M., Regular ordered semigroups in terms offuzzy subsets, Inform. Sci., 176 (2006), 3675-3693.


[14] Kuroki, N., Fuzzy semiprime quasi ideals in semigroups, Inform. Sci., 75(3) (1993), 201-211.

[15] Lajos, S., A note on semilattice of groups, Acta Sci. Math. Szeged, 31(1982), 179-180.

[16] Khan, M., Ahmad, N., Characterizations of left almost semigroups by theirideals, Journal of Advanced Research in Pure Mathematics, 2 (2010), 61-73.

[17] Khan, M., Jun, Y.B., Ullah, K., Characterizations of right regular Abel-Grassmann’s groupoids by their (∈,∈ ∨qk)-fuzzy ideals, submitted.

[18] Khan, M., Faisal, Amjid, V., On some classes of Abel-Grassmann’sgroupoids, Journal of Advanced Research in Pure Mathematics, 3, 4 (2011),109-119.

[19] Molodtsov, D., Soft set theory–First results, Comput. Math. Appl., 37 (1999),19–31.

[20] Mordeson, J.N., Malik, D.S., Kuroki, N., Fuzzy semigroups, Springer-Verlag, Berlin, Germany, (2003).

[21] Murali, V., Fuzzy points of equivalent fuzzy subsets, Inform. Sci., 158(2004), 277-288.

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[26] Protic, P.V., Stevanovic, N., AG-test and some general properties ofAbel-Grassmann’s groupoids, PU.M.A., 4 (6) (1995), 371-383.

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[31] Zadeh, L.A., Fuzzy sets, Inform. Control, 8 (1965), 338-353.

[32] Zhan, J., Jun, Y.B., Generalized fuzzy interior ideals of semigroups, NeuralComput. Applic., 19 (2010), 515-519.



SECRET KEY DISTRIBUTION TECHNIQUE USING THEORYOF NUMBERS

S. Srinivasan

P. Muralikrishna

School of Advanced SciencesVIT UniversityVellore – 632014Indiae-mails: [email protected]

[email protected]

N. Chandramowliswaran

Visiting FacultyIndian Institute of Management IndoreIndore – 453 331Indiae-mail: [email protected]

Abstract. A group key distribution protocol can enable members of a group to share

a secret group key and use it for secret communications. In this article we review the

number of share holders require to reconstruct the secret grouped among them. The

confidentiality of our proposed protocol is unconditionally secure.

Key words: Key distribution, Secret Sharing.

AMS Classification: 94A60, 94A62.

1. Introduction

Secret sharing is a technique for protecting sensitive data, such as cryptographickeys. It is used to distribute a secret value to a number of parts or shares that haveto be combined together to access the original value. These shares can then begiven to separate parties that protect them using standard means, e.g., memorize,store in a computer or in a safe. Secret sharing is used in modern cryptographyto lower the risks associated with compromised data.

A threshold scheme enables a secret to be shared among a group of ℓ membersproviding each member with a share. The scheme has a threshold t + 1 if any

326 s. srinivasan, p. muralikrishna, n. chandramowliswaran

subset with cardinality t+1 out of the ℓ shares enables the secret to be recovered.We will use the notation (t + 1, ℓ) to refer to such a scheme. Ideally, in a (t + 1)threshold scheme, t shares should not give any information on the secret. We willdiscuss later how to express this information. Sharing a secret spreads the riskof compromising the value across several parties. Standard security assumptionsof secret sharing schemes state that if an adversary gains access to any numberof shares lower than some defined threshold, it gains no information of the se-cret value. The first secret sharing schemes were proposed by Shamir [1] andBlakley [5].

Cryptography is the collection of methods and approaches for concealing in-formation in communications from the access by uninvited or unauthorized par-ties. A logical art for dealing with this problem is known from early antiquity andit developed along the centuries, mostly in the frame in which two parties, say no-bleman and general, or concealed lovers, communicated in written by sending eachother messages which could only be understood when knowing some additionaldata, secret keys and the details for the procedure of encrypting and decryptingthe messages algorithm. Algorithms were often assembled from a collection ofuseful basic ideas, known by tradition.

The group communication has been developed extensively in many applica-tions currently. Ensuring the security of a group communication has become oneof the most important issues in the development. Generally speaking, the secu-rity properties of a group communication include two basic aspects, that is, (i) themessages transmitted within the group can only be shared by authorized groupmembers, but not by any unauthorized users and (ii) the transmitted messagesmust be able to be authenticated by members. The security properties mentionedpreviously imply that a group session key must be used by authorized groupmembers to encrypt and authenticate the messages.

Suppose we have a large number of people, processes, or systems that want tocommunicate with one another in a secure fashion. Let’s further add this group ofpeople or processes or systems is not static, meaning that the individual entitiesmay join or leave the group at any time. In such case, if the owner of the secretcan fix the number of share holders, then it will be useful to distribute the shares.

This paper is organized as follows. In Section 2, we provide a technique tofix the number of share holders to reconstruct the distributed secret among theshare holders and Conclusion is given in Section 3.

2. Main result

In this section, we consider a secret S and attain a protocol to identify the numberof share holders, those who are having the distributed shares among them.

Theorem 2.1. For any positive integer k ≥ 2, there exists k +k(k − 1)

2share

holders, sharing the common secret S.

secret key distribution technique using theory of numbers 327

Proof. Select the secret positive integers α, β, αi and βi with 1 ≤ i ≤ k. Define

M = β

(k∑

j=1

β2j +

∑i<j

βiβj

)− 1,

where i, j ∈ 1, 2, ..., k, a = αM + β, aj = αjM + βj for j ∈ 1, 2, ..., k and

N =

a

(k∑

j=1

a2j +∑i<j

aiaj

)− 1

M.

Select a secret S such that (S,N) = 1.Define Yi ≡ Saa2i (mod N) for i ∈ 1, 2, ..., k.

Xi,j ≡ Saaiaj (mod N) for i < j and i, j ∈ 1, 2, ..., k.

Now, it is easy to observe

k∑i=1

Yi +∑i<j

Xi,j ≡ S (mod N).

This completes the proof.

3. Conclusion

Cryptography was born in early ages as a skill of mental combinations put atthe service of privacy and military protection. It developed along time into ahighly mathematized discipline, which unites the science of concealing with theanalysis of attacks into one single unit, cryptorology. There have been numerousinteresting attempts to use the large list of NP complete problems. This paperhardly deals the expected number of share holder in order to share a secret withpositive integer k ≥ 2.

References

[1] Adi Shamir, How to share a secret, Communications of the ACM, 22 (11)(1979), 612-613.

[2] Barnard, S., Child, J.M., Higher Algebra, Macmillan and Co., 1952.

[3] Beimel, A., Secret-sharing schemes: a survey, Proceedings of the Thirdinternational conference on Coding and cryptology, Berlin, Heidelberg, 2011,Springer-Verlag, IWCC’11, 11-46.

[4] Berlekamp, E.R., Algebraic Coding Theory, NY, McGraw-Hill, 1968.

328 s. srinivasan, p. muralikrishna, n. chandramowliswaran

[5] Blakley, G.R., Safeguarding cryptographic keys, Proceedings of the Na-tional Computer Conference 48 (1979), 313-317.

[6] Herstein, I.N., Topics in Algebra, 2nd Edition, John Wiley, 1975.

[7] Mignotte, M., How to share a secret, Advances in Cryptology - Euro-crypt82, LNCS, Springer-Verlag, 149 (1983), 371-375.

[8] Muralikrishna, P., Srinivasan, S., Chandramowliswaran, N.,Secure Schemes for Secret Sharing and Key Distribution using Pell’s equa-tion, International Journal of Pure and Applied Mathematics, 85 (5) (2013),933-937.

[9] Srinivasan, S., Muralikrishna, P., Chandramowliswaran, N.,Authenticated Multiple Key Distribution using Simple Continued Fraction,International Journal of Pure and Applied Mathematics, 87 (2) (2013),349-354.

[10] Srinivasan, S., Muralikrishna, P., Chandramowliswaran, N.,Authenticated Key Distribution using given set of Primes for SecretSharing, Submitted.

[11] Niven, I., Zuckerman, H.S., Montgomery, H.L., An Introduction tothe Theory of Numbers, John Wiley.

[12] Apostol, T.M., Introduction to Analytic Number Theory, Springer.



SOME CHARACTERIZATIONS OF INTRA-REGULARABEL GRASSMANN’S GROUPOIDS

Xueling Ma

Jianming Zhan1


Madad Khan

Tariq Aziz

Department of MathematicsCOMSATS Institute of Information TechnologyAbbottabadPakistan

Abstract. In this paper, we give some characterizations of a new non-associative

structure, namely intra-regular AG-groupoids by the properties of its (∈γ ,∈γ ∨qδ)-fuzzysubset, (∈γ ,∈γ ∨qδ)-fuzzy left (right) ideals and (∈γ ,∈γ ∨qδ)-fuzzy bi-ideals.

Keywords and phrases: AG-groupoid, left invertive law, medial law, paramedial law

and (∈γ ,∈γ ∨qδ)-fuzzy ideal.

1. Introduction

The fuzzy set theory was developed by Zadeh in 1965 [15]. This theory plays animportant role in the real life problems involving uncertainties. Fuzzy set theorycan be applied to several basic notations of algebras. Rosenfeld in 1971, introducedthe concept of fuzzy set theory in groups [13]. Mordeson et, al. [7] have discussedthe applications of fuzzy set theory in fuzzy coding, fuzzy automata and finitestate machines. In today’s world, many theories have been developed to deal withsuch uncertainties like fuzzy set theory, theory of vague sets, theory of soft ideals,theory of intuitionistic fuzzy sets and theory of rough sets. The theory of softsets (see [4, 5]) has many applications in different fields such as the smoothnessof functions, game theory, operations research, Riemann integration etc.

1Corresponding author. Tel/Fax: 0086-718-8437732. E-mail: [email protected]


Fuzzy set theory on semigroups has already been developed. In [8] Murali definedthe concept of belongingness of a fuzzy point to a fuzzy subset under a naturalequivalence on a fuzzy subset. The idea of quasi-coincidence of a fuzzy pointwith a fuzzy set was defined in [10]. Bhakat and Das [1], [2] gave the concept of(α, β)-fuzzy subgroups by using the “belongs to” relation ∈ and “quasi-coincidentwith” relation q between a fuzzy point and a fuzzy subgroup, and introduced theconcept of an (∈,∈ ∨q)-fuzzy subgroups, where α, β ∈ ∈, q,∈ ∨q,∈ ∧q andα =∈ ∧q. In [12] regular semigroups were characterized by the properties of their(∈,∈ ∨q)-fuzzy ideals. In [11] semigroups were characterized by the properties oftheir (∈,∈ ∨q)-fuzzy ideals.

An AG-groupoid is a mid structure between a groupoid and a commuta-tive semigroup. Mostly it works like a commutative semigroup. For instance,a2b2 = b2a2, for all a, b holds in a commutative semigroup, while this equationalso holds for an AG-groupoid with left identity e, moreover ab = (ba)e for anysubset a, b of an AG-groupoid. Now our aim is to discover some logical inves-tigations for regular and intra-regular AG-groupoids using the new generalizedconcept of fuzzy sets.

In this paper, we discuss the (∈γ,∈γ ∨qδ)-fuzzy ideals and (∈γ,∈γ ∨qδ)-fuzzybi-ideals in a new non-associative algebraic structure, that is, in AG-groupoidsand develop some new results. We characterize intra-regular AG-groupoids bythe properties of their (∈γ,∈γ ∨qδ)-fuzzy ideals.

2. Preliminaries

A groupoid (S, .) is called an AG-groupoid (LA-semigroup in some articles [9]), ifits elements satisfy left invertive law: (ab)c = (cb)a. In an AG-groupoid mediallaw [3], (ab)(cd) = (ac)(bd), holds for all a, b, c, d ∈ S. It is also known that inan AG-groupoid with left identity, the paramedial law: (ab)(cd) = (db)(ca), holdsfor all a, b, c, d ∈ S. If an AG-groupoid contains left identity, the following lawholds,

(1) a(bc) = b(ac), for all a, b, c ∈ S.

Let S be an AG-groupoid. By AG-subgroupoid of S we means a non-empty subsetA of S such that A2 ⊆ A, and by a left (right) ideal of S we mean a non-emptysubset L of S such that SL ⊆ L (LS ⊆ L). By two-sided ideals or simply ideal,we mean a non-empty subset of S which is both a left and a right ideal of S. AnAG-subgroupoid B of S is called a bi-ideal of S if (BS)B ⊆ B. A non-emptysubset B of S is called a generalized bi-ideal of S if (BS)B ⊆ B.

A fuzzy subset f of a given set S is described as an arbitrary functionf : S → [0, 1], where [0, 1] is the usual closed interval of real numbers. Forany two fuzzy subsets f and g of S, f ≤ g means that, f(x) ≤ g(x) for all x in S.Let f and g be any fuzzy subsets of an AG-groupoid S, then the product f g isdefined by

some characterizations of intra-regular ag groupoids 331

(f g) (a) =

∨a=bc

f(b) ∧ g(c) if there exist b, c ∈ S, such that a = bc,

0 otherwise.

The following definitions are available in [7].A fuzzy subset f of an AG-groupoid S is called a fuzzy AG-subgroupoid of

S if f(xy) ≥ f(x) ∧ f(y) for all x, y ∈ S.A fuzzy subset f of an AG-groupoid S is called a fuzzy left (right) ideal of S

if f(xy) ≥ f(y) (f(xy) ≥ f(x)) for all x, y ∈ S.A fuzzy subset f of an AG-groupoid S is called a fuzzy ideal of S if it is both

fuzzy left and fuzzy right ideal of S.A fuzzy subset f of an AG-groupoid S is called a fuzzy generalized bi-ideal

of S if f((xy)z) ≥ f(x) ∧ f(z), for all x, y and z ∈ S.A fuzzy generalized bi-ideal f of S is called a fuzzy bi-ideal of S if it is fuzzy

AG-subgroupoid of S. Let F (S) denote the collection of all fuzzy subsets of anAG-groupoid S with left identity, then (F (S), ) becomes an AG-groupoid withleft identity S, that is (F (S), ) satisfies left invertive law, medial law, paramediallaw and property (1). Note that S can be considered as a fuzzy subset of S itselfand we write S = CS, that is, S(x) = 1 for all x ∈ S. Moreover, S S = S.

Definition 1 [15] Let X be a non-empty set. A fuzzy subset f of X is define asa mapping from X into [0, 1], where [0, 1] is the usual interval of real numbers.We denote by F(X) the set of all fuzzy subsets of X.

A fuzzy subset f of X of the form

f(y) =

r( = 0) if y = x,0 otherwise,

is said to be a fuzzy point with support x and value r and is denoted by xr, wherer ∈ (0, 1].

3. (∈γ,∈γ ∨qδ)-fuzzy ideals in AG-groupoids

The following definitions are available in [14].In what follows, let γ, δ ∈ [0, 1] be such that γ < δ. For any B ⊆ A, we

define XδγB be the fuzzy subset of X by Xδ

γB(x) ≥ δ for all x ∈ B and XδγB(x) ≤ γ

otherwise. Clearly, XδγB is the characteristic function of B if γ = 0 and δ = 1.

For a fuzzy point xr and a fuzzy subset f of X, we say that

(1) xr ∈γ f if f(x) ≥ r > γ.(2) xrqδf if f(x) + r > 2δ.(3) xr ∈γ ∨qδf if xr ∈γ f or xrqδf.

Now, we introduce a new relation on F(X), denoted as “⊆ ∨q(γ,δ)”, as follows.


For any f, g ∈ F(X), by f ⊆ ∨q(γ,δ)g we mean that xr ∈γ f implies xr ∈γ ∨qδgfor all x ∈ X and r ∈ (γ, 1]. Moreover, f and g are said to be (γ, δ)-equal, denotedby f =(γ,δ) g, if f ⊆ ∨q(γ,δ)g and g ⊆ ∨q(γ,δ)f .

The proofs of the following lemmas are the same as in [14].

Lemma 1 Let f and g are fuzzy subsets of F(X). Then f ⊆ ∨q(γ,δ)g if and onlyif maxf(x), γ ≥ ming(x), δ for all x ∈ X.

Lemma 2 Let f, g, h ∈ F(X). If f ⊆ ∨q(γ,δ)g and g ⊆ ∨q(γ,δ)h, then f ⊆ ∨q(γ,δ)h.

It is shown in [14] that “=(γ,δ)” is an equivalence relation on F(X). It is alsonotified that f =(γ,δ) g if and only if maxminf(x), δ, γ = maxming(x), δ, γfor all x ∈ X.

Lemma 3 Let A, B be any non empty subsets of an AG-groupoid S with leftidentity. Then we have

(1) A ⊆ B if and only if XδγA ⊆ ∨q(γ,δ)Xδ

γB, where r ∈ (γ, 1] and γ, δ ∈ [0, 1].(2) Xδ

γA ∩XδγB =(γ,δ) X

δγ(A∩B).

(3) XδγA Xδ

γB =(γ,δ) Xδγ(AB).

Proof. It is the same in [14].

Definition 2 A fuzzy subset f of an AG-groupoid S is called an (∈γ,∈γ ∨qδ)-fuzzy AG-subgroupoid of S if for all x, y ∈ S and t, s ∈ (γ, 1], it satisfies xt ∈γ f,ys ∈γ f implies that (xy)mint,s ∈γ ∨qδf.

Theorem 1 Let f be a fuzzy subset of an AG groupoid S. Then f is an (∈γ,∈γ

∨qδ)-fuzzy AG subgroupoid of S if and only if maxf(xy), γ ≥ minf(x), f(y), δ,where γ, δ ∈ [0, 1].

Proof. Let f be a fuzzy subset of an AG-groupoid S which is (∈γ,∈γ ∨qδ)-fuzzysubgroupoid of S. Assume that there exist x, y ∈ S and t ∈ (γ, 1], such that

maxf(xy), γ < t ≤ minf(x), f(y), δ.

Then maxf(xy), γ < t. This implies that f(xy) < t, which further impliesthat (xy)min t∈γ ∨qδf and minf(x), f(y), δ ≥ t. Therefore minf(x), f(y) ≥ t⇒ f(x) ≥ t > γ, f(y) ≥ t > γ, implies that xt ∈γ f , ys ∈γ f. But (xy)mint,s∈γ ∨qδfa contradiction to the definition. Hence

maxf(xy), γ ≥ minf(x), f(y), δ for all x, y ∈ S.

Conversely, assume that there exist x, y ∈ S and t, s ∈ (γ, 1] such that xt ∈γ f ,ys ∈γ f but (xy)mint,s∈γ ∨qδf, then f(x) ≥ t > γ, f(y) ≥ s > γ, f(xy) <minf(x), f(y), δ and f(xy) + mint, s ≤ 2δ. It follows that f(xy) < δ andso maxf(xy), γ < minf(x), f(y), δ, this is a contradiction. Hence xt ∈γ f ,ys ∈γ f implies that (xy)mint,s ∈γ ∨qδf for all x, y in S.


Definition 3 A fuzzy subset f of an AG-groupoid S is called an (∈γ,∈γ ∨qδ)-fuzzy left (respt-right) ideal of S if for all x, y ∈ S and t, s (γ, 1] it satisfies yt ∈γ fimplies that (xy)t ∈γ ∨qδf (respt xt ∈γ f implies ((xy)t ∈γ ∨qδf).

Example 1 Consider an AG-groupoid S = 1, 2, 3 in the following multiplica-tion table.

1 2 31 1 1 12 1 1 33 1 2 1

Define a fuzzy subset f on S as follows:

f(x) =

0.41 if x = 1,0.44 if x = 2,0.42 if x = 3.

Then, we have

• f is an (∈0.1,∈0.1 ∨q0.11)-fuzzy right ideal,

• f is not an (∈,∈ ∨q0.11)-fuzzy right ideal.

Example 2 Let S = 1, 2, 3 and the binary operation be defined on S asfollows:

1 2 31 2 2 22 2 2 23 1 2 2

Then, clearly, (S, ) is an AG-groupoid. Define a fuzzy subset f on S as follows:

f(x) =

0.3 if x = 1,0.7 if x = 2,0.8 if x = 3.

Then, we have

• f is an (∈0.4,∈0.4 ∨q0.45)-fuzzy AG-subgroupoid of S,

• f is not an (∈0.4,∈0.4 ∨q0.45)-fuzzy right ideal of S.

Example 3 Let S = 1, 2, 3 and the binary operation ” · ” be defined on S asfollows:

· 1 2 31 1 1 12 1 1 13 1 2 1


Then clearly (S, ·) is an AG-groupoid. Define a fuzzy subset f on S as follows:

f(x) =

0.8 if x = 1,0.6 if x = 2,0.5 if x = 3.

Then it is easy to see that f is an (∈0.3,∈0.3 ∨q0.4)-fuzzy ideal of S.

Theorem 2 A fuzzy subset f of an AG-groupoid S is an (∈γ,∈γ ∨qδ)-fuzzy left(respt right) ideal of S if and only if

maxf(xy), γ ≥ minf(y), δ (respt maxf(xy), γ ≥ minf(x), δ).

Proof. Let f be an (∈γ,∈γ ∨qδ)-fuzzy left ideal of S. If exist x, y ∈ S andt ∈ (γ, 1] such that

maxf(xy), γ < t ≤ minf(y), δ.

Then maxf(xy), γ < t ≤ γ this implies that (xy)t∈γf which further impliesthat (xy)t∈γ ∨qδf . As minf(y), δ ≥ t > γ which implies that f(y) ≥ t > γ,this implies that yt ∈γ f . But (xy)t∈γ ∨qδf a contradiction to the definition.Thus

maxf(xy), γ ≥ minf(y), δ.Conversely, assume that there exist x, y ∈ S and t, s ∈ (γ, 1] such that ys ∈γ fbut (xy)t∈γ ∨qδf, then f(y) ≥ t > γ, f(xy) < minf(y), δ and f(xy) + t ≤ 2δ.It follows that f(xy) < δ and so maxf(xy), γ < minf(y), δ which is a con-tradiction. Hence yt ∈γ f this implies that (xy)mint,s ∈γ ∨qδf (respt xt ∈γ fimplies (xy)mint,s ∈γ ∨qδf) for all x, y in S.

Definition 4 A fuzzy subset f of an AG-groupoid S is called an (∈γ,∈γ ∨qδ)-fuzzy bi-ideal of S if for all x, y and z ∈ S and t, s ∈ (γ, 1], the following conditionshold.

(1) if xt ∈γ f and ys ∈γ f implies that (xy)mint,s ∈γ ∨qδf.(2) if xt ∈γ f and zs ∈γ f implies that ((xy)z)mint,s ∈γ ∨qδf.

Example 4 Define a fuzzy subset f on S in example 2 as follows:

f(x) =

0.44 if x = 1,0.6 if x = 2,0.7 if x = 3.

Then, we have

• f is an (∈0.4,∈0.4 ∨q0.45)-fuzzy bi-ideal of S,

• f is not an (∈0.4,∈0.4 ∨q0.45)-fuzzy right ideal of S.

Theorem 3 A fuzzy subset f of an AG-groupoid S is (∈γ,∈γ ∨qδ)-fuzzy bi-idealof S if and only if


(I) maxf(xy), γ ≥ minf(x), f(y), δ.

(II) maxf((xy)z), γ ≥ minf(x), f(z), δ.

Proof. (1) ⇔ (I) is the same as Theorem 1.(2) ⇒ (II) Assume that x, y ∈ S and t, s ∈ (γ, 1] such that

maxf((xy)z), γ < t ≤ minf(x), f(z), δ.

Then maxf((xy)z), γ < t which implies that f((xy)z) < t this implies that((xy)z)t∈γf which further implies that ((xy)z)t∈γ ∨qδf . Also minf(x), f(z), δ ≥t > γ, this implies that f(x) ≥ t > γ, f(z) ≥ t > γ implies that xt ∈γ f , zt ∈γ f .But ((xy)z)t∈γ ∨qδf, a contradiction. Hence

maxf((xy)z), γ ≥ minf(x), f(z), δ.

(II) ⇒ (2) Assume that x, y in S and t, s ∈ (γ, 1], such that xt ∈γ f, zs ∈γ fbut ((xy)z)mint,s∈γ ∨qδf , then f(x) ≥ t > γ, f(z) ≥ s > γ, f((xy)z) <minf(x), f(y), δ and f((xy)z) + mint, s ≤ 2δ. It follows that f((xy)z) < δand so maxf((xy)z), γ < minf(x), f(y), δ, a contradiction. Hence xt ∈γ f ,zs ∈γ f implies that ((xy)z)mint,s ∈γ ∨qδf for all x, y in S.

Lemma 4 Let f be any (∈γ,∈γ ∨qδ)-fuzzy AG-subgroupoid and g be (∈γ,∈γ ∨qδ)-fuzzy left ideal of an AG-groupoid S. Then (f g) is an (∈γ,∈γ ∨qδ)-fuzzy leftideal of S.

Proof. Let f and g be an (∈γ,∈γ ∨qδ)-fuzzy AG-subgroupoid an (∈γ,∈γ ∨qδ)-fuzzy left ideal of an AG-groupoid S with left identity,respectively. So for any yin S, there exist a and b in S such that y = ab. Therefore, xy = x(ab) = a(xb).

Then

minf g(y), δ = min

∨y=ab

f(a) ∧ g(b), δ

=

∨y=ab

minminf(a), δ,ming(b), δ

≤∨

xy=a(xb)

minmaxf(a), γ,maxg(xb), γ

=∨

xy=a(xb)

maxminf(a), g(xb), γ

≤∨

xy=ac

maxminf(a), g(c), γ

= max(f g)(xy), γ.


Corollary 1 Let f and g be (∈γ,∈γ ∨qδ)-fuzzy left ideals of an AG-groupoid Swith left identity. Then (f g) is an (∈γ,∈γ ∨qδ)-fuzzy left ideal of S.

Theorem 4 Let f and g be (∈γ,∈γ ∨qδ)-fuzzy right ideals of an AG-groupoid Swith left identity. Then (f g) is an (∈γ,∈γ ∨qδ)-fuzzy right ideal of S.

Proof. Let f and g be (∈γ,∈γ ∨qδ)-fuzzy right ideals of an AG-groupoid S. SinceS is an AG-groupoid, there exist x, y in S such that x = ab and y = y1y2, forsome a, b, y1, y2 in S. Then by medial law we get

xy = (ab)y = (ab)(y1y2) = (ay1)(by2).

Then we have

minf g(x), δ = min

∨x=ab

f(a) ∧ g(b), δ

=

∨x=ab

minminf(a), δ,ming(b), δ

≤∨

xy=(ay1)(by2)

minmaxf(ay1), γ,maxg(by2), γ

=∨

xy=(ay1)(by2)

maxminf(ay1), g(by2), γ

≤∨

xy=cd

maxminf(c), g(d), γ

= max(f g)(xy), γ.

Theorem 5 Let f and g be (∈γ,∈γ ∨qδ)-fuzzy bi-ideals of an AG-groupoid S.Then (f g) is an (∈γ,∈γ ∨qδ)-fuzzy bi-ideal of S.

Proof. Assume that f and g are (∈γ,∈γ ∨qδ)-fuzzy bi-ideals of an AG-groupoid S.Consider on the contrary that there exist a, b ∈ S and t, s ∈ (γ, 1] such that

max(f g)(ab), γ < t ≤ min(f g)(a), (f g)(b), δ.

Then min(f g)(a), (f g)(b), δ ≥ t implies that (f g)(a) ≥ t > γ, (f g)(b) ≥t > γ, this implies that at ∈γ (f g), bt ∈γ (f g) and max(f g)(ab), γ < timplies that (f g)(ab) < t implies that (f g)(ab) + s < 2δ further implies that(ab)t∈γ ∨qδ(f g). But (ab)t∈γ ∨qδ(f g), a contradiction. Hence

max(f g)(ab), γ ≥ min(f g)(a), (f g)(b), δ.

Assume that max(f g)((ax)b), γ < t ≤ min(f g)(a), (f g)(b), δ, thenmin(f g)(a), (f g)(b), δ ≥ t implies that (f g)(a) ≥ t, (f g)(b) ≥ t,this implies that at ∈γ (f g), bt ∈γ (f g) and max(f g)((ax)b), γ < t


implies that (f g)((ax)b) < t further implies that ((ax)b)t∈γ(f g) implies that((ax)b)t∈γ ∨qδ(f g). But ((ax)b)t∈γ ∨qδ(f g), a contradiction. Hence

max(f g)((ax)b), γ ≥ min(f g)(a), (f g)(b), δ.

Hence (f g) is an (∈γ,∈γ ∨qδ)-fuzzy bi-ideal of S.

Lemma 5 A subset B of an AG-groupoid S is a bi-ideal if and only if XδγB is an

(∈γ,∈γ ∨qδ)-fuzzy bi-ideal of S.

Proof. (i) Let B be a bi-ideal and assume that x, y ∈ B then for any a in Swe have (xa)y ∈ B, thus Xδ

γB((xa)y) ≥ δ. Now since x, y ∈ B so XδγB(x) ≥ δ,

XδγB(y) ≥ δ which clearly implies that minXδ

γB(x), XδγB(y) ≥ δ. Thus

maxXδγB((xa)y), γ = Xδ

γB((xa)y).

minXδγB(x), X

δγB, δ = δ.

Hence maxXδγB((xa)y), γ ≥ minXδ

γB(x), XδγB(y), δ.

(ii) Let x ∈ B, y /∈ B, then (xa)y /∈ B, for all a in S. This implies thatXδ

γB((xa)y) ≤ γ, XδγB(x) ≥ δ and Xδ

γB(y) < γ. Therefore

maxXδγB((xa)y), γ = γ and minXδ

γB(x), XδγB(y), δ = Xδ

γB(y).


γB(x), Xδ≥γB(y), δ.

(iii) Let x /∈ B, y ∈ B implies that (xa)y /∈ B, for all a in S. This implies thatmaxXδ

γB((xa)y), γ ≥ δ, XδγB(x) < γ,Xδ

γB(y) ≥ δ and minXδγB(x), X

δγB(y), δ =

XδγB(x). Therefore

maxXδγB((xa)y), γ ≥ minXδ


(iv) Let x, y /∈ B which implies that (xa)y /∈ B, for all a in S. This impliesthat minXδ

γB(x), XδγB(y) ≤ γ, Xδ

γB((xa)y) ≤ γ. Thus

maxXδγB((xa)y), γ = γ and

minXδγB(x), X

δγB(y), δ ≤ minXδ

γB(x), XδγB(y) ≤ γ.



If (xa)y ∈ B, then minXδγB(x), X

δγB(y) ≤ γ, Xδ

γB((xa)y) ≥ δ. Thus

maxXδγB((xa)y), γ = Xδ

γB((xa)y) and

minXδγB(x), X

δγB(y), δ ≤ minXδ

γB(x), XδγB(y) ≤ γ.



Conversely, let (b1s)b2 ∈ (BS)B, where b1, b2 ∈ B and s ∈ S.Now, by hypothesis maxXδ

γB((b1s)b2), γ ≥ minXδγB(b1), X

δγB(b2), δ.


Since b1, b2 ∈ B, therefore XδγB(b1) ≥ δ and Xδ

γB(b2) ≥ δ which implies thatminXδ

γB(b1), XδγB(b2), δ = δ. Thus

maxXδγB((b1s)b2), γ ≥ δ.

This clearly implies that maxXδγB((b1s)b2) ≥ δ. Therefore (b1s)b2 ∈ B. Hence B

is a bi-ideal of S.Similarly, we can prove the following lemmas.

Lemma 6 A subset A of an AG-groupoid S is closed if and only if XδγA is an

(∈γ,∈γ ∨qδ)-fuzzy AG-subgroupoid of S.

Lemma 7 For any fuzzy subset f of an AG-groupoid S with left identity,f ⊆ ∨q(γ,δ)S f .

Proof. Let f be a fuzzy subset of an AG-groupoid S with left identity. Therefore,for any a in S there exist p and q in S such that a = pq. Thus

max(S f)(a), γ = max

∨a=pq

S(p) ∧ f(q), γ

≥ maxmin S(e) ∧ f(a), γ= maxmin 1 ∧ f(a), γ= maxf(a), γ≥ min f(a), δ .

Hence by Lemma 1, f ⊆ ∨q(γ,δ)S f .

4. Intra-regular AG-groupoids

An element a of an AG-groupoid S is called intra-regular if there exist x, y ∈ Ssuch that a = (xa2)y and S is called intra-regular, if every element of S isintra-regular. In this section, we discuss the characterizations of intra-regularAG-groupoids.

Example 5 Let S = 1, 2, 3, 4, 5, 6, the following table shows that S is an intra-regular AG-groupoid.

1 2 3 4 5 61 2 1 1 1 1 12 1 2 2 2 2 23 1 2 6 3 4 54 1 2 5 6 3 45 1 2 4 5 6 36 1 2 3 4 5 6

It is easy to see that (S, ) is an AG-groupoid and is non-commutative and non-associative structure because (3 4) = (4 3) and (3 6) 4 = 3 (6 4). Also1 = (3 12) 1, 2 = (2 22) 2, 3 = (4 32) 5, 4 = (4 42) 6, 5 = (6 52) 5,6 = (6 62) 6. Therefore, (S, ) is an intra-regular AG-groupoid.


Theorem 1 Let S be an AG-groupoid with left identity, then the following con-ditions are equivalent.

(i) S is intra-regular.

(ii) For every subset L and bi-ideal B of S, L ∩B ⊆ LB.

(iii) For every (∈γ,∈γ ∨qδ) fuzzy subset f and (∈γ,∈γ ∨qδ)-fuzzy bi-ideal g,f ∩ g ⊆ ∨q(γ,δ)f g.

Proof. (i) ⇒ (iii) Let f and g be an (∈γ,∈γ ∨qδ)-fuzzy subset and an (∈γ,∈γ ∨qδ)-fuzzy bi-ideal of an intra-regular AG-groupoid S with left identity, respectively.Now since S is intra-regular, therefore, for any a in S there exist x, y in S suchthat a = (xa2)y. Since S = S2, so for any y there exists y1, y2 such that y = y1y2.Now, using (1) and the left invertive law, the medial law and the paramedial law,we get

a = (xa2)y = (x(aa))y = (a(xa))y = (y(xa))a

= (y (xa))((xa2

)y)=

(y(xa2

))((xa) y) =

((y1y2)

(xa2

))((xa) (y1y2))

=((a2x

)(y2y1)

)((y2y1) (ax)) =

((a2x

)(y2y1)

)(a ((y2y1)x))

= a[(a2x

)(y2y1)(y2y1)x] = a[((y2y1)x) a2 ((y2y1) x)]

= a[a2 (x (y2y1))(y2y1) x] = a[(y2y1) x (x (y2y1))a2]= a[ua2], where u = (y2y1)x (x (y2y1)) .

Now

ua2 = u[(xa2)y

]2= u

[(xa2)2y2

]= (xa2)2

[uy2

]= [x2(a2)2]

[uy2

]= [(a2)2x2]

[uy2

]= [(uy2)x2](a2a2) = (a2a2)[x2(uy2)]

= ([x2(uy2)]a2)a2 = (a2[x2(uy2)])a2

= (a2q)a2, where q = x2(uy2).

Thus a = a((a2q)a2), where q = x2(uy2) and u = (y2y1)x (x (y2y1)).For any a in S, there exist p and q in S such that a = pq, then

max(f g)(a), γ = max

∨a=pq

f(p) ∧ g(q), γ

≥ maxminf(a)), g((a2q)a2), γ= minmaxf(a)), g((a2q)a2), γ= minmaxf(a)), γ,maxg((a2q)a2), γ≥ min minf(a), δ,ming(a), δ= min(f ∩ g)(a), δ.

Thus, by Lemma 1, f ∩ g ⊆ ∨q(γ,δ)f g.


(iii) ⇒ (ii) Let L and B be any subset and bi-ideal of S, respectively. Then,by Lemma 3 and (iii), we get

Xδγ(L∩B) =(γ,δ) X

δγL ∩Xδ

γB ⊆ ∨q(γ,δ)XδγL Xδ

γB =(γ,δ) XδγLB.

Hence, by Lemma 3, we get L ∩B ⊆ LB.(ii) ⇒ (i) Since Sa is an bi-ideal and any subset of S containing a. Therefore,

by (ii) medial law, paramedial law and (1), we get

a ∈ Sa ∩ Sa ⊆ (Sa)(Sa) = (SS)(aa) = (a2S)S

= ((aa)(SS))S = ((SS)(aa))S = (Sa2)S.

Hence S is intra-regular.

Remark 1 If S is an intra-regular AG-groupoid with left identity, then for every(∈γ,∈γ ∨qδ) fuzzy subset f of S, f ⊆ ∨q(γ,δ)f S.

Theorem 2 Let S be an AG-groupoid with left identity, then the following con-ditions are equivalent.


(ii) For every bi-ideal B and any subset A of S, B ∩ A ⊆ BA.

(iii) For every (∈γ,∈γ ∨qδ)-fuzzy bi-ideal f and (∈γ,∈γ ∨qδ)-fuzzy subset g of S,f ∩ g ⊆ ∨q(γ,δ)f g.

(iv) For every (∈γ,∈γ ∨qδ) generalized fuzzy bi-ideal f and (∈γ,∈γ ∨qδ)-fuzzysubset g of S, f ∩ g ⊆ ∨q(γ,δ)f g.

Proof. (i) ⇒ (iv) Let f and g be an (∈γ,∈γ ∨qδ)-fuzzy generalized bi-ideal and(∈γ,∈γ ∨qδ)-fuzzy subset of an intra-regular AG-groupoid S with left identity,respectively. Now since S is intra-regular, therefore for any a in S, there exist x, yin S such that a = (xa2)y. Now using (1), the left invertive law, the medial lawand the paramedial law, we get

(xa2)y = (a(xa))y = (y(xa))a = [(y1y2)(xa)]a = [(ax)(y2y1)]a

= [((y2y1)x)a]a = [((y2y1)x)((xa2)y)]a

= [(tx)((xa2)y)]a, where t = y2y1

(tx)((xa2)y) = (xa2)((tx)y) = (a(xa))((tx)y)

= [((tx)y)(xa))]a = [(u(xa)]a, where u = (tx)y

u(xa) = u[x(xa2)y)] = u[(xa2)(xy)] = (xa2)(u(xy))

= ((xy)u)(a2x) = a2[(xy)ux] = [x(xy)u]a2 = a[x((xy)u)a]= a(pa), where p = (x((xy)u))a.

Thus a = [(a(pa))a]a = [(aq)a]a, where q = pa.


For any a in S, there exist p and q in S such that a = pq. Then

maxf g(a), γ = max

∨a=pq

f(p) ∧ g(q), γ

= max

∨a=pq

minf(p), g(q), γ

≥ min maxf((aq)a), g(a), γ= minmaxf((aq)a), γ,maxg(a), γ≥ minminf(a), δ,ming(a), δ= min(f ∩ g)(a), δ.

Thus, by Lemma 1, f ∩ g ⊆ ∨q(γ,δ)f g.(iv) ⇒ (iii) It is obvious.(iii) ⇒ (ii) Let A be any subset and B be an bi- ideal of S. Then, by Lemma

3 and (iii), we get

Xδγ(B∩A) =(γ,δ) X

δγB ∩Xδ

γA ⊆ ∨q(γ,δ)XδγB Xδ

γA =(γ,δ) XδγBA

Hence, by Lemma 3, we get B ∩ A ⊆ BA.(ii) ⇒ (i) Since Sa is a bi-ideal containing a, so using (ii), we get

a ∈ Sa ∩ Sa = Sa2 = (Sa2)S.

Hence S is intra-regular.

Corollary 2 For an AG-groupoid S with left identity, the following conditionsare equivalent.


(ii) For all bi-ideal B1 and B2, B1 ∩B2 ⊆ B1B2.

(iii) f ∩ g ⊆ ∨q(γ,δ)f g, for all (∈γ,∈γ ∨qδ) fuzzy bi-ideals f and g.

(iv) f ∩ g ⊆ ∨q(γ,δ)f g, for all (∈γ,∈γ ∨qδ)-fuzzy generalized bi-ideals f and g.

Definition 5 A fuzzy subset f of an AG-groupoid S is called an (∈γ,∈γ ∨qδ)-fuzzy semiprime ideal if x2

t ∈γ f implies that xt ∈γ ∨qδ for all x ∈ S and t ∈ (γ, 1].

Example 6 Define a fuzzy subset f on S in Example 2 as given:

f(x) =

0.3 if x = 1,0.2 if x = 2,0.4 if x = 3.

Then, it is easy to see that f is an (∈0.2,∈0.2 ∨q0.3)-fuzzy semiprime ideal of S.


Example 7 Define a fuzzy subset f on S in Example 5 as given:

f(x) =

0.7 if x = 1,0.6 if x = 2,0.68 if x = 3,0.63 if x = 4,0.52 if x = 5,0.5 if x = 6.

Then, it is easy to see that f is an (∈0.4,∈0.4 ∨q0.5)-fuzzy semiprime ideal of S.

Theorem 3 A fuzzy subset f of an AG-groupoid S is an (∈γ,∈γ ∨qδ)-fuzzysemiprime ideal if and only if maxf(x), γ ≥ minf(x2), δ.

Proof. Let f be an (∈γ,∈γ ∨qδ)-fuzzy semiprime ideal of S. If exist x, y ∈ S andt ∈ (γ, 1] such that maxf(x), γ < t ≤ minf(x2), δ. Then maxf(x), γ < timplies that xt∈γf implies that xt∈γ ∨qδf . As minf(x2), δ ≥ t > γ this impliesthat f(x2) ≥ t > γ implies that x2

t ∈γ f . But xt∈γ ∨qδf, a contradiction to thedefinition of semiprime ideals. Thus, we have maxf(x), γ ≥ minf(x2), δ.

Conversely, assume that there exist x, y in S and t ∈ (γ, 1] such that x2t ∈γ f

but xt∈γ ∨qδf , then f(x2) ≥ t > γ, f(x) < minf(x2), δ and f(x)+t ≤ 2δ. It fol-lows that f(x) < δ and so maxf(x), γ < minf(x2), δ which is a contradictionto the definition of semiprime ideals. Hence, x2

t ∈γ f implies that (x2)t ∈γ ∨qδf ,for all x, y in S.

Theorem 4 For a non empty subset I of an AG-groupoid S with left identity,the following conditions are equivalent.

(i) I is semiprime.

(ii) XδγI is an (∈γ,∈γ ∨qδ)-fuzzy semiprime.

Proof. (i) ⇒ (ii) Let I be a semiprime ideal of an AG-groupoid S. Let abe any element of S such that a ∈ I, then I is an ideal, so a2 ∈ I. HenceXδ

γI(a), XδγI(a

2) ≥ δ which implies that maxXδγI(a), γ ≥ minXδ

γI(a2), δ.

Now, let a /∈ I, since I is semiprime, thus a2 /∈ I. This implies thatXδ

γI(a) ≤ γ and XδγI(a

2) ≤ γ. Therefore, maxXδγI(a), γ ≥ minXδ

γI(a2), δ.

Hence, we have maxXδγI(a), γ ≥ minXδ

γI(a2), δ for all a ∈ S.

(ii) ⇒ (i) Let XδγI is fuzzy semiprime. If a2 ∈ I, for some a in S, this implies

that XδγI(a

2) ≥ δ. Now, since XδγI is an (∈γ,∈γ ∨qδ)-fuzzy semiprime. Thus,

maxXδγI(a), γ ≥ minXδ

γI(a2), δ. Therefore maxXδ

γI(a), γ ≥ δ. But δ > γ,so Xδ

γI(a) ≥ δ. Thus, a ∈ I. Hence, I is semiprime.

Lemma 8 Let f be a fuzzy subset of an AG-groupoid S. Then f is an (∈γ,∈γ

∨qδ)-fuzzy bi-ideal of S if and only if maxf(a), γ ≥ min((f S) f)(a), δ.


Proof. Assume that f is an (∈γ,∈γ ∨qδ)-fuzzy bi-ideal of an AG-groupoid S. Ifa ∈ S, then there exist c, d, p and q in S such that a = pq and p = cd. Since f is an(∈γ,∈γ ∨qδ)-fuzzy bi-ideal of S, we have maxf((cd)q), γ ≥ minf(c), f(q), δ.Therefore,

min((f S) f)(a), δ = min

∨a=pq

(f S)(p) ∧ f(q), δ

= min

∨a=pq

∨p=cd

f(c) ∧ S(d) ∧ f(q), δ

= min

∨a=(cd)q

f(c) ∧ 1 ∧ f(q), δ

= min

∨a=(cd)q

f(c) ∧ f(q), δ

=

∨a=(cd)q

minf(p), f(q), δ

≤∨

a=(cd)q

maxf((cd)q), γ

= maxf(a), γ.

Hence, maxf(a), γ ≥ min((f S) f)(a), δ.Conversely, assume that maxf(a), γ ≥ min((f S) f)(a), δ. Let a in S,

there exist c, d and q in S such that a = (cd)q. Then we have

maxf((cd)q), γ = maxf(a), γ≥ min((f S) f)(a), δ

= min

∨a=bc

(f S)(b) ∧ f(c), δ

≥ min(f S)(cd) ∧ f(q), δ

= min

∨cd=st

f(s) ∧ S(t) ∧ f(q), δ

≥ minmin(f(c), f(q), δ= min(f(c), f(q), δ.

Hence maxf((cd)q), γ ≥ minf(c), f(q), δ.

Lemma 9 Every (∈γ∈γ∨qδ)-fuzzy right ideal of an AG-groupoid S is an (∈γ∈γ∨qδ)-fuzzy bi-ideal of S.


Proof. Let f be an (∈γ∈γ ∨qδ)-fuzzy right ideal of an AG-groupoid S. For anya in S, there exist p, q, s and t in S, such that

min((f S) f)(a), δ = min∨a=pq

(f S)(p) ∧ f(q), δ

= min∨a=pq

∨p=st

f(s) ∧ S(t) ∧ f(q), δ

= min∨

a=(st)q

f(s) ∧ f(q), δ

= min∨

a=(st)q

[minf(s), f(q)], δ

=∨

a=(st)q

min[minf(s), δ,minf(q), δ]

≤∨

a=(st)q=(qt)s

minmaxf(st)q, γ,maxf(qt)s, γ

= minmaxf(a), γ,maxf(a), γ= maxf(a), γ.

Hence maxf(a), γ ≥ min((f S) f)(a), δ. Hence, by Lemma 8, f is an(∈γ∈γ ∨qδ)-fuzzy bi-ideal of S.

Theorem 5 Let S be an intra-regular AG-groupoid with left identity, then thefollowing conditions are equivalent.


(ii) For every ideal of S is semiprime.

(iii) For every (∈γ,∈γ ∨qδ)-fuzzy ideal of S is fuzzy semiprime.

(iv) For every (∈γ,∈γ ∨qδ)-fuzzy right ideal of S is fuzzy semiprime.

(v) For every (∈γ,∈γ ∨qδ)-fuzzy bi-ideal of S is fuzzy semiprime.

Proof. (i) ⇒ (v) Let f be an (∈γ,∈γ ∨qδ)-fuzzy right ideal of an intra-regularAG-groupoid S with left identity. Now, since S is intra-regular so for each a ∈ Sthere exist x, y in S such that a = (xa2)y. Now, using (1) and the left invertivelaw, we get

a = (xa2)y = [x(aa)]y = [a(xa)]y = [y(xa)]a

= [yx((xa2)y)]a = [y(xa2)(xy)]a= [(xa2)y(xy)]a = [(xy)y(a2x)]a = [a2((y2x)x)]a

= [a((y2x)x)]a2 = [(xa2)y((y2x)x)]a2

= [((y2x)x)y(xa2)]a2 = [(a2x)y((y2x)x)]a2

= [y((y2x)x)xa2]a2 = [a2xy((y2x)x)]a2

= (a2q)a2, where q = ((y2x)x)xy.


Thus

maxf(a), γ = maxf((a2q)a2), γ≥ minf(a2), f(a2), δ= minf(a2), δ.

(v) ⇒ (iv) and (iv) ⇒ (iii) are obvious.(iii) ⇒ (ii) It follows from Lemma 3.(ii) ⇒ (i) Assume that every ideal is semiprime and since Sa2 is an ideal

containing a2. Thus, we have

a ∈ Sa2 = (SS)a2 = (a2S)S = (Sa2)S.

Hence S is an intra-regular AG-groupoid.

5. Conclusions

In this paper, we characterize intra-regular AG-groupoids with left identity usingthe properties of their (∈γ,∈γ ∨qδ)-fuzzy bi-deals. In our future research workwe will focus on considering other types of (∈γ,∈γ ∨qδ)-fuzzy ideals intra-regularAG-groupoids. We remark that this research work will give a new direction forapplications of fuzzy set theory particularly in algebraic logic, non-classical logic,fuzzy coding, fuzzy finite state mechanics and fuzzy languages.

Acknowledgements. This research is partially supported by a grant of NaturalScience Foundation of Education Committee of Hubei Province(D20131903).

References

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[7] Mordeson, J.N., Malik, D.S., Kuroki, N., Fuzzy semigroups, Springer-Verlag, Berlin, Germany, 2003.

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SOME DOUBLE LACUNARY SEQUENCE SPACES

Kuldip Raj

Sunil K. Sharma

Seema Jamwal

School of MathematicsShri Mata Vaishno Devi UniversityKatra-182320, J & KIndiae-mail: [email protected]

[email protected]

Abstract. The purpose of this paper is to introduce some double lacunary sequence

spaces defined by a sequence of Orlicz functions. We also make an effort to study some

topological properties and inclusion relations between these sequence spaces.

Keywords: paranorm space, Orlicz function, solid, analytic sequences, double se-

quences.

2000 Mathematics Subject Classification: 40A05, 40C05, 40D05.


The initial work on double sequences is found in Bromwich [4]. Later on, itwas studied by Hardy [7], Moricz [16], Moricz and Rhoades [17], Tripathy ([30],[31]), Basarir and Sonalcan [2] and many others. Hardy [7] introduced the notionof regular convergence for double sequences. Quite recently, Zeltser [33] in herPh.D thesis has essentially studied both the theory of topological double sequencespaces and the theory of summability of double sequences. Mursaleen and Edely[21] have recently introduced the statistical convergence and Cauchy convergencefor double sequences and given the relation between statistical convergent andstrongly Cesaro summable double sequences. Nextly, Mursaleen [18] and Mur-saleen and Edely [22] have defined the almost strong regularity of matrices fordouble sequences and applied these matrices to establish a core theorem and in-troduced the M -core for double sequences and determined those four dimensionalmatrices transforming every bounded double sequences x = (xk,l) into one whosecore is a subset of the M -core of x. More recently, Altay and Basar [1] have de-fined the spaces BS, BS(t), CSp, CSbp, CSr and BV of double sequences consistingof all double series whose sequence of partial sums are in the spaces Mu, Mu(t),

348 k. raj, s.k. sharma, s. jamwal

Cp, Cbp, Cr and Lu, respectively and also examined some properties of these se-quence spaces and determined the α-duals of the spaces BS, BV , CSbp and theβ(v)-duals of the spaces CSbp and CSr of double series. Recently, Basar and Sever[3] have introduced the Banach space Lq of double sequences corresponding tothe well known space ℓq of single sequences and examined some properties of thespace Lq. By the convergence of a double sequence we mean the convergence inthe Pringsheim sense, i.e., a double sequence x = (xk,l) has Pringsheim limit L(denoted by P − limx = L) provided that given ϵ > 0 there exists n ∈ N suchthat |xk,l − L| < ϵ whenever k, l > n see [25]. We shall write more briefly asP -convergent. The double sequence x = (xk,l) is bounded if there exists a positivenumber M such that |xk,l| < M for all k and l.

An Orlicz function M is a function, which is continuous, non-decreasing andconvex with M(0) = 0, M(x) > 0 for x > 0 and M(x) −→ ∞ as x −→ ∞.

Lindenstrauss and Tzafriri [10] used the idea of Orlicz function to define thefollowing sequence space. Let w be the space of all real or complex sequencesx = (xk), then

ℓM =x ∈ w :

∞∑k=1

M( |xk|

ρ

)< ∞

which is called as an Orlicz sequence space. The space ℓM is a Banach space withthe norm

||x|| = infρ > 0 :

∞∑k=1

M( |xk|

ρ

)≤ 1

.

It is shown in [10] that every Orlicz sequence space ℓM contains a subspace iso-morphic to ℓp(p ≥ 1). The ∆2-condition is equivalent to M(Lx) ≤ kLM(x) forall values of x ≥ 0, and for L > 1. The notion of difference sequence spaces wasintroduced by Kızmaz [8], who studied the difference sequence spaces l∞(∆), c(∆)and c0(∆). The notion was further generalized by Et and Colak [5] by introducingthe spaces l∞(∆m), c(∆m) and c0(∆

m).Letm,n be non-negative integers, then for Z = c, c0 and l∞, we have sequence

spacesZ(∆m

n ) = x = (xk) ∈ w : (∆mn xk) ∈ Z,

where ∆mn x = (∆m

n xk) = (∆m−1n xk − ∆m−1

n xk+1) and ∆0xk = xk for all k ∈ N,which is equivalent to the following binomial representation

∆mn xk =

m∑v=0

(−1)v(

mv

)xk+nv.

Taking m = n = 1, we get the spaces l∞(∆), c(∆) and c0(∆) studied by Kızmaz[8]. Taking n = 1, we get the spaces l∞(∆m), c(∆m) and c0(∆

m) studied by Etand Colak [5]. Similarly, we can define difference operators on double sequencespaces as:

∆xk,l = (xk,l − xk,l+1)− (xk+1,l − xk+1,l+1)

= xk,l − xk,l+1 − xk+1,l + xk+1,l+1,

some double lacunary sequence spaces 349

∆mxk,l = ∆m−1xk,l −∆m−1xk,l+1 −∆m−1xk+1,l +∆m−1xk+1,l+1

and

∆mn xk,l = ∆m−1

n xk,l −∆m−1n xk,l+1 −∆m−1

n xk+1,l +∆m−1n xk+1,l+1.

A double sequence x = (xk,l) of real numbers is called almost P -convergent to alimit L if

P − limp,q→∞

supm,n≥0

1

pq

m+p−1∑k=m

n+q−1∑l=n

|xk,l − L| = 0

i.e., the average value of (xk,l) taken over any rectangle

(k, l) : m ≤ k ≤ m+ p− 1, n ≤ l ≤ n+ q − 1

tends to L as both p and q tends to ∞, and this P -convergence is uniform in mand n.

By a lacunary sequence θ = (ir), r = 0, 1, 2, ..., where i0 = 0, we shall meanan increasing sequence of non-negative integers hr = (ir − ir−1) → ∞ (r → ∞).The intervals determined by θ are denoted by Ir = (ir−1, ir] and the ratio ir/ir−1

will be denoted by qr. The space of lacunary strongly convergent sequences Nθ

was defined by Freedman et al. [6] as follows:

Nθ =

x = (xk) : lim

r→∞

1

gr

∑k∈Ir

|xk − L| = 0 for some L

.

The double sequence θr,s = (kr, ls) is called double lacunary if there exist twoincreasing sequences of integers such that

k0 = 0, hr = kr − kr−1 → ∞ as r → ∞

andl0 = 0, hs = ls − ls−1 → ∞ as s → ∞.

Let kr,s = krls, hr,s = hrhs and θr,s is determined by Ir,s = (k, l) : kr−1 < k ≤ kr

and ls−1 < l ≤ ls, qr =krkr−1

, qs =lsls−1

and qr,s = qrqs.

A sequence space E is said to be solid if (αk,lxk,l) ∈ E, whenever (xk,l) ∈ Eand for all sequence (αk,l) of scalars with |αk,l| ≤ 1, for all k, l ∈ N.

A sequence space E is said to be symmetric if (xk,l) ∈ E implies (xπ(k,l)) ∈ E,where π is a permutation of N.

A sequence space E is said to be convergence free if (yk,l) ∈ E whenever(xk,l) ∈ E and xk,l = 0 implies yk,l = 0.

Let X be a linear metric space. A function p : X → R is called paranorm, if

1. p(x) ≥ 0, for all x ∈ X,

2. p(−x) = p(x), for all x ∈ X,


3. p(x+ y) ≤ p(x) + p(y), for all x, y ∈ X,

4. if (λn) is a sequence of scalars with λn → λ as n → ∞ and (xn) is a sequenceof vectors with p(xn−x) → 0 as n → ∞, then p(λnxn−λx) → 0 as n → ∞.

A paranorm p for which p(x) = 0 implies x = 0 is called total paranorm and thepair (X, p) is called a total paranormed space. It is well known that the metric ofany linear metric space is given by some total paranorm (see [32], Theorem 10.4.2,P-183). For more details about sequence spaces see ([9], [11], [12], [13], [14], [19],[23], [24], [26], [27], [28], [29]).

Let M = (Mk,l) be a sequence of Orlicz functions, p = (pk,l) be a boundedsequence of positive real numbers and u = (uk,l) be any sequence of positive realnumbers. Let X be a seminormed space over the field of complex numbers withthe seminorm q and w(X) denotes the space of all sequences x = (xk,l), wherexk,l ∈ X. Now, we define the following sequence spaces in this paper:

[Nθ,M,∆mn , p, q, u]1

=x = (xk,l) ∈ w(X) : lim

r,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆mn xk,l − L)

ρ

)]pk,l= 0,

for some L and ρ > 0,

[Nθ,M,∆mn , p, q, u]0

=x = (xk,l) ∈ w(X) : lim

r,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆mn xk,l)

ρ

)]pk,l= 0,

for some ρ > 0

[Nθ,M,∆mn , p, q, u]∞

=x = (xk,l) ∈ w(X) : sup

r,s

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆mn xk,l)

ρ

)]pk,l< ∞,

for some ρ > 0.

If we take M(x) = x, we get the following spaces:

[Nθ,∆mn , p, q, u]1

=x = (xk,l) ∈ w(X) : lim

r,s→∞

1

hr,s

∑k,l∈Ir,s


ρ

)pk,l= 0,


[Nθ,∆mn , p, q, u]0

=x = (xk,l) ∈ w(X) : lim

r,s→∞

1

hr,s

∑k,l∈Ir,s

(q(uk,l∆mn xk,l)

ρ

)pk,l= 0,

for some ρ > 0


[Nθ,∆mn , p, q, u]∞

=x = (xk,l) ∈ w(X) : sup

r,s

1

hr,s

∑k,l∈Ir,s

(q(uk,l∆mn xk,l)

ρ

)pk,l< ∞,

for some ρ > 0.

If we take p = (pk,l) = 1, we get the spaces like:

[Nθ,M,∆mn , q, u]1

=x = (xk,l) ∈ w(X) : lim

r,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l


ρ

)]= 0,


[Nθ,M,∆mn , q, u]0

=x = (xk,l) ∈ w(X) : lim

r,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆mn xk,l)

ρ

)]= 0,

for some ρ > 0

[Nθ,M,∆mn , q, u]∞

=x = (xk,l) ∈ w(X) : sup

r,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆mn xk,l)

ρ

)]< ∞,

for some ρ > 0.

The following inequality will be used throughout the paper.

If 0 ≤ pk,l ≤ sup pk,l = H, K = max(1, 2H−1), then

(1.1) |ak,l + bk,l|pk,l ≤ K|ak,l|pk,l + |bk,l|pk,l

for all k, l and ak,l, bk,l ∈ C. Also |a|pk,l ≤ max(1, |a|H) for all a ∈ C.The main aim of the present paper is to study some topological properties

and inclusion relations between the above defined sequence spaces.

2. Main results

Theorem 2.1. Let M = (Mk,l) be a sequence of Orlicz functions, p = (pk,l)be a bounded sequence of positive real numbers and u = (uk,l) be a sequence ofstrictly positive real numbers. Then, [Nθ,M,∆m

n , p, q, u]1, [Nθ,M,∆mn , p, q, u]0

and [Nθ,M,∆mn , p, q, u]∞ are linear spaces over the set of complex numbers C.

Proof. Suppose x = (xk,l) and y = (yk,l) ∈ [Nθ,M,∆mn , p, q, u]0. Then, there

exist positive numbers ρ1, ρ2 such that

limr,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆mn (xk,l))

ρ1

)]pk,l= 0


and

limr,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆mn (yk,l))

ρ2

)]pk,l= 0.

Let ρ3 = max(2|α|ρ1, 2|β|ρ2). Since M = (Mk,l) is a non-decreasing and convexso by using inequality (1.1), we have

limr,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆mn (αxk,l + βyk,l))

ρ3

]pk,l≤ K lim

r,s→∞

1

hr,s

∑k,l∈Ir,s

1

2pk,l

[Mk,l


ρ1

)]pk,l+K lim

r,s→∞

1

hr,s

∑k,l∈Ir,s

1

2pk,l

[Mk,l


ρ2

)]pk,l≤ K lim

r,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l


ρ1

)]pk,l+K lim

r,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l


ρ2

)]pk,l= 0.

Thus, αx + βy ∈ [Nθ,M,∆mn , p, q, u]0. This prove that [Nθ,M,∆m

n , p, q, u]0 is alinear space. Similarly, we can prove other cases.

Theorem 2.2. Let M = (Mk,l) be a sequence of Orlicz functions, p = (pk,l)be a bounded sequence of positive real numbers and u = (uk,l) be a sequenceof strictly positive real numbers and θ = (kr,s) be a lacunary sequence. Then[Nθ,M,∆m

n , p, q, u]0 is a paranormed spaces with the paranorm

g(x) =

m,n∑i,j=1

|xi,j|+ infρpk,l/H : sup

r,s

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆mn xk,l

ρ

)]≤ 1,

for some ρ > 0 and r = 1, 2, 3, ...,

where H = max(1, supk,l

pk,l).

Proof. Clearly g(x) = g(−x). Since Mk,l(0) = 0, for all k, l ∈ N, we getg(0) = 0. Let x = (xk,l) and y = (yk,l) ∈ [Nθ,M,∆m

n , p, q, u]0 and let us chooseρ1 > 0 and ρ2 > 0 be such that

supr,s

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆mn xk,l)

ρ1

)]≤ 1, r = 1, 2, 3, ...,

and

supr,s

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆mn yk,l

ρ2

)]≤ 1, r = 1, 2, 3, ....


Let ρ = ρ1 + ρ2. We have

supr,s

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(ukl∆mn (xk,l + yk,l)

ρ

)]≤

( ρ1ρ1 + ρ2

)supr,s

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆mn xk,l)

ρ1

)]+( ρ2ρ1 + ρ2

)supr,s

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆mn yk,l)

ρ2

)]≤ 1.

Since ρ > 1, we have

g(x+ y)

=

m,n∑i,j=1

|xi,j + yi,j|+ infρ

pk,lH : sup

r,s

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆mn (xk,l + yk,l))

ρ

)]≤

m,n∑i,j=1

|xi,j|+ infρ

pk,lH : sup

r,s

1

hr,s

∑k,l∈Ir,s

[Mk,l


ρ

)]≤ 1, r=1, 2, 3, ...

+

m,n∑i,j=1

|yi,j|+ infρ

pk,lH : sup

r,s

1

hr,s

∑k,l∈Ir,s

[Mk,l


ρ

)]≤ 1, r=1, 2, 3, ...

= g(x) + g(y).

Finally, let λ be a given non-zero scalar in C. Then the continuity of the productfollows from the following expression

g(λx) =

m,n∑i,j=1

|λxi,j| + infρ

pk,lH : sup

r,s

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆mn (λxk,l))

ρ

)]≤ 1,

for some ρ > 0 and r = 1, 2, 3, ...

= λ

m,n∑i,j=1

|xi,j| + inf(|λ|η)

pk,lH : sup

r,s

1

hr,s

∑k,l∈Ir,s

[Mk,l


η

)]≤ 1,

for some ρ > 0 and r = 1, 2, 3, ...,

where η =ρ

|λ|> 0. This completes the proof of the theorem.

Theorem 2.3. Let M = (Mk,l) and M′ = (M ′k,l) be two sequences of Orlicz

functions and p = (pk,l) be a bounded sequence of positive real numbers. Then

(i) [Nθ,M,∆mn , p, q, u]Z ⊆ [Nθ,MM′,∆m

n , p, q, u]Z

(ii) [Nθ,M,∆mn , p, q, u]Z ∩ [Nθ,M′,∆m

n , p, q, u]Z ⊆ [Nθ,M + M′,∆mn , p, q, u]Z ,

where Z = 0, 1,∞.


Proof. The proof of the theorem is easy, so we omit it.

Theorem 2.4. The inclusion [Nθ,M,∆m−1n , p, q, u]Z ⊆ [Nθ,M,∆m

n , p, q, u]Zhold, for m ≥ 1. In general, [Nθ,M,∆i

n, p, q, u]Z ⊆ [Nθ,M,∆mn , p, q, u]Z , for

i = 1, 2, 3, ...,m− 1 and the inclusions are strict, where Z = 0, 1,∞.

Proof. Let x = (xk,l) ∈ [Nθ,M,∆m−1n , p, q, u]0. Then, there exists ρ > 0 such

that

limr,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆m−1n (xk,l))

ρ

)]pk,l= 0.

Since M is non-decreasing and convex, we have

limr,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l


2ρ

)]pk,l

= limr,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆m−1n (xk,l)− uk,l∆

m−1n (xk+1,l+1))

2ρ

)]pk,l≤ lim

r,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆m−1n (xk,l)

2ρ

)]pk,l+ lim

r,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆m−1n (xk+1,l+1)

2ρ

)]pk,l≤ lim

r,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆m−1n (xk,l)

ρ

)]pk,l+ lim

r,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆m−1n (xk+1,l+1)

ρ

)]pk,l−→ 0 as r, s → ∞,

i.e., x = (xk,l) ∈ [Nθ,M,∆mn , p, q, u]0. The other cases can be proved in the similar

way.

Theorem 2.5. Let θ = (kr, ls) be a lacunary sequence and let M = (Mk,l) be asequence of Orlicz functions. Then

(i) [Nθ,M,∆mn , p, q, u]0 ⊆ [Nθ,M,∆m

n , p, q, u]1 ⊆ [Nθ,M,∆mn , p, q, u]∞,

and the inclusion is strict.

(ii) If |uk,l| ≤ 1, then [Nθ,M,∆mn , p, q, u]Z ⊆ [Nθ,M,∆m

n , p, q, ]Zfor Z = 0, 1,∞.

Proof. (i) The inclusion [Nθ,M,∆mn , p, q, u]0⊆ [Nθ,M,∆m

n , p, q, u]1 is obvious.Let (xk,l) be an element of [Nθ,M,∆m

n , p, q, u]1. Then, there exists ρ > 0 suchthat

limr,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆mn (xk,l − L))

ρ

)]pk,l= 0.


Since (Mk,l) is non-decreasing and convex, we have

1

hr,s

∑k,l∈Ir,s

[Mk,l


ρ

)]pk,l≤ K

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,l∆mn (xk,l − L))

ρ

)]pk,l+K max

[1,Mk,l

(q(Lρ

))]H,

where H = supk,l

(pk,l), K = max(1, 2H−1). This completes the proof of the (i).

(ii) The proof of the (ii) is easy, so we omit it.

Example. Let

(pk,l) =

4, if k, l are even;

5, otherwise.

Letm,n ≥ 0 be given. Let uk,l = (kl),Mk,l(x) = x2, for all k, l ∈ N and q(x) = |x|.Let θ = (2rs) be a lacunary sequence. Consider a sequence (xk,l) defined by

(xk,l) = (kl)(mn), (kl)(mn), (kl)(mn), ....

Thus, the sequence (xk,l) belongs to [Nθ,M,∆mn , p, q, u]1, but (xk,l) does not be-

longs to [Nθ,M,∆mn , p, q, u]0.

Theorem 2.6. Let M = (Mk,l) and M′ = (M ′k,l) be two sequences of Orlicz

functions. If M and M′ are equivalent for each k, l ∈ N and θ = (kr, ls) be alacunary sequence. Then

[Nθ,M,∆mn , p, q, u]Z = [Nθ, ϕ,∆

mn , p, q, u]Z ,

where Z = 0, 1,∞.

Proof. The proof of the theorem is easy, so omitted.

Theorem 2.7. Let M = (Mk,l) be a sequence of Orlicz functions and let q1 andq2 be two seminorms. Then

(i) [Nθ,M,∆mn , p, q1, u]Z ∩ [Nθ,M,∆m

n , p, q2, u]Z ⊆ [Nθ,M,∆mn , p, q1 + q2, u]Z;

(ii) [Nθ,M,∆mn , p, q1, u]Z ⊆ [Nθ,M,∆m

n , p, q2, u]Z, where Z = 0, 1,∞.

Proof. The proof is easy, so omitted.

Proposition 2.8. The spaces [Nθ,M, p, q, u]0 and [Nθ,M, p, q, u]∞ are solid aswell as monotone. The spaces [Nθ,M,∆m

n , p, q, u]Z are not solid in general, forZ = 0, 1,∞.

Proof. Let (xk,l) ∈ [Nθ,M, p, q, u]0. Then, there exists ρ > 0 such that

limr,s→∞

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,lxk,l)

ρ

)]pk,l= 0.


Let (αk,l) be a sequence of scalers such that |αk,l| ≤ 1, for all k, l ∈ N. Since|αk,l| ≤ max(1, |αk,l|H) ≤ 1, for all k, l ∈ N, where H = sup

k,lpk,l < ∞, then, for

each r, s, we have

1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(αk,l(uk,lxk,l))

ρ

)]pk,l≤ 1

hr,s

∑k,l∈Ir,s

[Mk,l

(q(uk,lxk,l)

ρ

)]pk,l.

Therefore, (αk,lxk,l) ∈ [Nθ,M, p, q, u]0. Hence [Nθ,M, p, q, u]0 is solid. Therefore,the space [Nθ,M, p, q, u]0 is monotone.

Hence the space [Nθ,M, p, q, u]∞ is solid as well as monotone.In order to prove that the spaces [Nθ,M,∆m

n , p, q, u]1 and [Nθ,M,∆mn , p, q, u]∞

are not solid in general, we consider the following example.

Example. Let Mk,l(x) = xst, for all k, l ∈ N and s, t ≥ 1. Let (pk,l) = ( 1kl),

(uk,l) = (kl), for all k, l ∈ N and q(x) = |x|. Let θ = (2rs) be a lacunary sequence,for all k, l ∈ N. Consider a sequence (xk,l) defined by

(xk,l) = (kl)2, for all k, l ∈ N.

Then, (xk,l) belongs to [Nθ,M,∆mn , p, q, u]1 and [Nθ,M,∆m

n , p, q, u]∞ form,n = 1.Let (αk,l) = (−1)kl, for all k, l ∈ N. Then, (αk,lxk,l) does not belong to the

spaces [Nθ,M,∆mn , p, q, u]1 and [Nθ,M,∆m

n , p, q, u]∞.Hence the spaces [Nθ,M,∆m

n , p, q, u]1 and [Nθ,M,∆mn , p, q, u]∞ are not solid.

Therefore, the spaces [Nθ,M,∆mn , p, q, u]1 and [Nθ,M,∆m

n , p, q, u]∞ are not mono-tone.

Proposition 2.9. The spaces [Nθ,M,∆mn , p, q, u]Z are not symmetric in general,

for Z = 0, 1,∞.

Proof. The proof of the result follows from the following example.

Example. Let Mk,l(x) = x2, (pk,l) = (kl) and (uk,l) = (kl)2, for all k, l ∈ Nand q(x) = |x|. Let θ = (2rs) be a lacunary sequence for all k, l ∈ N. Considera sequence (xk,l) defined by (xk,l) = (kl)3, for all k, l ∈ N. Then (xk,l) belongto [Nθ,M,∆m

n , p, q, u]0, for m,n = 1. Consider the sequence (yk,l) which is therearrangement of the sequence (xk,l) defined by

(yk,l) = (x1,1, x2,2, x4,4, x3,3, x9,9, x5,5, x16,16, x6,6, x25,25, x7,7, ...).

Then, (yk,l) does not belongs to [Nθ,M,∆mn , p, q, u]Z .

Hence the spaces [Nθ,M,∆mn , p, q, u]Z are not symmetric in general.

Proposition 2.10. The space [Nθ,M,∆mn , p, q, u]0 is not convergence free.

Proof. The proof of the result follows from the following example.

Example. Let Mk,l(x) = x, (pk,l) = (kl), (uk,l) = (kl), for all k, l ∈ N andq(x) = |x|. Let θ = (2rs) be a lacunary sequence for all k, l ∈ N. Consider asequence (xk,l) defined by

(xk,l) =

2, if k, l are even;

0, otherwise.


Then (xk,l) belongs to [Nθ,M,∆mn , p, q, u]0, for m,n = 2.

Consider the sequence (yk,l) defined by

(yk,l) =

(kl)2, if k, l are even;

0, otherwise.

Then, (yk,l) does not belong to [Nθ,M,∆mn , p, q, u]0.

Hence the space [Nθ,M,∆mn , p, q, u]0 is not convergence free.

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A NEW APPROACH TO A CERTAIN GENERALIZED INTEGRALTRANSFORM FOR CERTAIN SPACE OF BOEHMIANS

S.K.Q. Al-Omari

Department of Applied SciencesFaculty of Engineering TechnologyAl-Balqa’ Applied UniversityAmman 11134Jordane-mail: [email protected]

Abstract. In this article, we introduce a generalization of Fourier and Hartely trans-

forms. The transform we have obtained has been investigated on certain space of distri-

butions. Two spaces of Boehmians are also established. The extended transform is then

obtained and is well-defined, linear, one-to-one and onto mapping. More properties are

also illustrated.

Keywords: ιβ,γα,R Transform; Fourier Transform Pair; Distribution Space; Generalized

Function; Boehmian; Hartley Transform Pair.

1. Introduction

Let R be the set of real numbers and g be an integrable function defined on R.The transform we consider in this article is given by the integral equation

(1) ιβ,γα,R (g) (ξ) =1√2π

∫Rg (τ) (α cos (γξτ) + β sin (γξτ)) dτ.

The inversion formula is recovered from our transform (1) as

(2) g (τ) =1√2π

∫Rιβ,γα,R (g) (ξ)

(γ

αcos (γξτ) +

γ

βsin (γξτ)

)dξ.

Let γ = α = β = 1, then the shortness of (1) and (2) reduces to the Hartleytransform pair [2], [8], [12]. On the other hand, a substitution of α = 1, β = i,γ = 1, in (1) and (2) , describes a Fourier transform pair [9].

360 s.k.q. al-omari

Denote by E′(R) the space of distributions of compact supports. Then, the

extention ιβ,γα,R of ιβ,γα,R to a distribution g ∈ E′(R) can be given as

(3) ιβ,γα,R (g) (ξ) =1√2π

⟨g (τ) , α cos (γξτ) + β sin (γξτ)⟩ .

This definition is indeed well-defined by the smoothness of α cos (γξτ)+β sin (γξτ).

Therefore, ιβ,γα,R justifies its following properties:

(i) ιβ,γα,R is linear;

(ii) ιβ,γα,R is continuous;

(iii) ιβ,γα,R is analytic.

The justification of those properties follows in fact from basic properties of distri-butions.

In what follows, we spread the discussion to further space of Boehmians overtwo sections.

In Section 2, we construct the image and preimage spaces of Boehmians. InSection 3, we discuss the transform in the context of Boehmian spaces and obtainsome properties.

2. The constructed spaces of Boehmian

We assume the reader is acquainted with the concept of Boehmian spaces. Forfurther constructions, reader can check the citations [1], [3], [4], [6], [7], [10], [11],[13] of this article.

Let us first define some auxiliary mappings that are useful to our next inves-tigation of Boehmian spaces.

Denote by D the Schwartz space of test functions of bounded support. Then,for every g ∈ E

′and v ∈ D we introduce the operation • defined by

(4) (g • v) ℓ (ς) = g (v ∗ ℓ) (ς) ,

where ℓ ∈ E, and ∗ is the usual convolution product of two functions of first kind[9], [5].

To establish the first space of Boehmians we are requested to establish thefollowing theorems.

Theorem 1. Let g ∈ E′and v ∈ D; then we have g • v ∈ E

′.

Proof. Let ℓ ∈ E and K be a compact subset of R; then, by (4) , we have

(5) (g • v) ℓ (ς) = g (v ∗ ℓ) (ς) .

a new approach to a certain generalized ... 361

So, to prove the theorem, it is sufficient to show that v∗ℓ ∈ E. By the property [9],

Dkf ∗ g = Dkf ∗ g

we derive thatdk

dςk(v ∗ ℓ) (ς) = v ∗ dk

dςkℓ (ς) .

Hence, allowing K traverse the compact subsets of R implies

supς∈K

∣∣∣∣ dkdςk(v ∗ ℓ) (ς)

∣∣∣∣ = supς∈K

∣∣∣v ∗ dk

dςkℓ (ς)

∣∣∣≤ η∗

∫K

∣∣∣∣ dkdςkℓ (ς − x)

∣∣∣∣ dx,η∗ is certain positive constant.

This gives

supς∈K

∣∣∣∣ dkdςk(v ∗ ℓ) (ς)

∣∣∣∣ < ∞.

Hence, v ∗ ℓ ∈ E. Thus, g • ℓ ∈ E′.


Theorem 2. Let g1, g2 ∈ E′and ℓ ∈ D, η ∈ C; then we have

η (g1 + g2) • ℓ = (ηg1 + ηg2) • ℓ.

The proof of this theorem is straightforward. Hence, we prefer to omit the details.

Theorem 3. Let gn → g in E′and v ∈ D; then gn • v → g • v as n → ∞.

Proof. For ℓ ∈ E, we can write

(6) (gn • v − g • v) ℓ = ((gn − g) • v) ℓ = (gn − g) (v ∗ ℓ) .

Right hand side of equation (6) is well defined by Theorem 1.Considering the limit as n → ∞, the right hand side of (6) tends to 0 as

n → ∞.Therefore,

(gn • v → g • v) ℓ → 0

as n → ∞.Hence gn • v → g • v as n → ∞.The proof of the theorem is completed.

As final in this construction, we merely need to establish the followingtheorem.

362 s.k.q. al-omari

Theorem 4. Let g ∈ E′and (δn) ∈ ∆; then we have g • δn → g as n → ∞.

Proof. Let ℓ ∈ E and (δn) ∈ ∆; then, since (δn) is delta sequence, we have δn ∗ ℓas n → ∞. Hence,

(7) (g • δn) ℓ = g (δn ∗ ℓ) → g as n → ∞.

Thus our theorem is completely proved.

The space B(E

′, D,∆, •

)is therefore considered as a Boehmian space.

The sum and multiplication by a scalar of two Boehmians can be defined ina natural way [

fnϵn

]+

[gnτn

]=

[fn • τn + gn • ϵn

ϵn • τn

]and

η

[fnϵn

]=

[ηfnϵn

], η ∈ C.

The operation • and the differentiation are defined by[fnϵn

]•[gnτn

]=

[fn • gnϵn • τn

]and Dα

[fnϵn

]=

[Dαfnϵn

].

If

[fnϵn

]∈ B

(E

′, D,∆, •

)and ϕ ∈ D, then we have

[fnϵn

]• ϕ =

[fn • ϕϵn

].

A sequence of Boehmians (βn) in B(E

′, D,∆, •

)is said to be δ-convergent

to a Boehmian β in B(E

′, D,∆, •

), denoted by βn

δ→ β, if there exists a delta

sequence (ϵn) such that (βn • ϵn) , (β • ϵn) ∈ E′, ∀k, n = 1, 2, 3, ...,and

(βn • ϵk) → (β • ϵk) as n → ∞, in E′, k = 1, 2, 3, ....

The following lemma is equivalent for the statement of δ-convergence:

βnδ→ β ( as n → ∞) in B

(E

′, D,∆, •

)if and only if there is fn,k, fk ∈ E

′

and (ϵk) ∈ ∆ such that βn =

[fn,kϵk

], β =

[fkϵk

]and for each k = 1, 2, 3, ...,

fn,k → fk as n → ∞ in E′.

A sequence (βn) of Boehmians in B(E

′, D,∆, •

)is said to be a ∆-convergent to

a Boehmian β in B(E

′, D,∆, •

), denoted by βn

∆→ β, if there exists a (ϵn) ∈ ∆such that

(βn − β) • ϵn ∈ E′,

∀n = 1, 2, 3, ... and (βn − β) • ϵn → 0 as n → ∞ in E′.


Let us now consider another space of Boehmians.Let W be the space of ιβ,γα,R transforms of distributions in E

′; then we define

a product ~ as

(8) (w ~ v) (ξ) =

∫Rw (ξ + x) v (x) dx,

ξ ∈ R. Then, (8) can be simply written as

(w ~ v) (ξ) = v ∗ w (ξ) ,

where w (ξ) = w (−ξ) .

Theorem 5. Let g ∈ E′, v ∈ D; then

ιβ,γα,R (g • v) = ιβ,γα,R (g)~ v.

Proof. Over compact subsets K of R we by (3) have that

ιβ,γα,R (g • v) (ξ) = ⟨(g • v) (τ) , α cos γξτ + β sin γξτ⟩

i.e. = ⟨g (τ) , ⟨v (x) , α cos γξ (τ + x) + β sin γξ (τ + x)⟩⟩

i.e. =

∫R⟨g (τ) , α cos γξ (τ + x) + β sin γξ (τ + x)⟩ v (x) dx.

Therefore, we have obtained

ιβ,γα,R (g • v) (ξ) =(ιβ,γα,Rg ~ v

)(ξ) .


Theorem 6. Let w ∈ W, v ∈ D; then w ~ v ∈ W .

Proof. The assumption that w ∈ W implies w = ιβ,γα,Rg for some g ∈ E′.

Therefore, we get

(9) w ~ v = ιβ,γα,Rg ~ v.

By Theorem 5, equation (9) gives

(10) w ~ v = ιβ,γα,R (g • v) .

Theorem 1, therefore, implies w ~ v ∈ E′.


The proofs of the following two theorems have similar techniques.

Theorem 7. wn, w ∈ W, (δn) ∈ ∆ and v ∈ D. Then we get

364 s.k.q. al-omari

(1) Let wn → w in W and v ∈ D; then wn ~ v → w ~ v.

(2) Let w ∈ W and (δn) ∈ ∆; then w ~ δn → w as n → ∞.

Theorem 8. Let w1, w2 ∈ W, v ∈ D; then for η1, η2 ∈ C, we get that

(η∗1w1 + η∗2w2)~ v = η∗1 (w1 ~ v)× η∗2 (w2 ~ v)

where v ∈ D .

The Boehmian space B (W,D,∆,~) is therefore constructed.Addition, multiplication by scalars and convergence on B (W,D,∆,~) are

similar to that of B(E

′, D,∆, •

). Hence details are avoided.

3. The generalized ιβ,γα,R transform of Boehmians

Let

[gnδn

]∈ B

(E

′, D,∆, •

); then, we define the generalized transform ιβ,γα,R

of

[gnδn

]as

(11) ιβ,γα,R

[gnδn

]=

[ιβ,γα,R (gn)

δn

]

in the space B (W,D,∆,~) .

Theorem 9. The mapping ιβ,γα,R is well-defined.

Proof. Let

[fnϵn

]=

[gnτn

]∈ B

(E

′, D,∆, •

); then we have

fnϵn

is equivalent tognτn

in B(E

′, D,∆, •

).

Therefore, fn • τm = gm • ϵn, ∀m,n ∈ N. The action of ιβ,γα,R jointly with Theorem 5imply

ιβ,γα,R (fn)~ τm = ιβ,γα,R (gm)~ ϵn,

∀m,n ∈ N. Hence

ιβ,γα,R (fn)

ϵnis equivalent to

ιβ,γα,R (gn)

τnin B (W,D,∆,~) .

Hence, we have obtained

ιβ,γα,R

[fnϵn

]= ιβ,γα,R

[gnτn

].



Theorem 10. The mapping ιβ,γα,R is linear.

Proof. Let

[fnϵn

],

[gnτn

]∈ B

(E

′, D,∆, •

), κ, η ∈ R; then

ιβ,γα,R

(κ

[fnϵn

]+ η

[gnτn

])= ιβ,γα,R

([κfnϵn

]+

[ηgnτn

])i.e. = ιβ,γα,R

([(κfn) • τn + (ηgn) • ϵn

ϵn • τn

])

i.e. =

[ιβ,γα,R ((κfn) • τn + (ηgn) • ϵn)

ϵn • τn

].

Linearity of ιβ,γα,R and (11) imply

ιβ,γα,R

(κ

[fnϵn

]+ η

[gnτn

])=

[κιβ,γα,R (fn • τn) + ηιβ,γα,R (gn • ϵn)

ϵn ~ τn

].

Theorem 5 gives

ιβ,γα,R

(κ

[fnϵn

]+ η

[gnτn

])=

[κιβ,γα,R (fn)~ τn + ηιβ,γα,R (gn)~ ϵn

ϵn ~ τn

].

Thus, we have

ιβ,γα,R

(κ

[fnϵn

]+ η

[gnτn

])=

[κηβ,γα,R (fn)

ϵn

]+

[ηηβ,γα,R (gn)

τn

]Hence, we got that

ιβ,γα,R

(κ

[fnϵn

]+ η

[gnτn

])= κ

[ιβ,γα,R (fn)

ϵn

]+ η

[ιβ,γα,R (gn)

τn

]The theorem is completely proved.

Theorem 11. The mapping ιβ,γα,R is an isomorphism from B(E

′, D,∆, •

)into

B (W,D,∆,~).

Proof. We first prove that ιβ,γα,R is injective mapping from B(E

′, D,∆, •

)into

B (W,D,∆,~) .

Assume that

ιβ,γα,R

[fnϵn

]= ιβ,γα,R

[gnτn

]

366 s.k.q. al-omari

in B (W,D,∆,~) .

Then, by (11) , we have[ιβ,γα,R (fn)

ϵn

]=

[ιβ,γα,R (gn)

τn

].

Hence, we derive that

ιβ,γα,R (fn)

ϵnand

ιβ,γα,R (gn)

τn

are equivalent quotients in B (W,D,∆,~) .

Thus, the concept of equivalent classes of B (W,D,∆,~) implies

ιβ,γα,R (fn)~ τm = ιβ,γα,R (gm)~ ϵn.

Theorem 5 then gives

ιβ,γα,R (fn ~ τn) = ιβ,γα,R (gn ~ ϵn) .

Injectivity of ιβ,γα,R and the concept of equivalent classes of B(E

′, D,∆, •

)implies

fnϵn

are equivalentgnτn

in B(E

′, D,∆, •

).

Therefore

[fnϵn

]and

[gnτn

]are equivalent.

This proves the first part.

Surjectivity of ιβ,γα,R is obvious.

The theorem is completely proved.

The proofs of the following theorems are obvious.

Theorem 12. If ιβ,γα,R

[gnδn

]= 0, then

[gnδn

]= 0

in the sense of B(E

′, D,∆, •

).

Theorem 13. If (βn) is sequence in B(E

′, D,∆, •

)such that βn

∆→ β as n → ∞,then


ιβ,γα,R (βn)∆→ ιβ,γα,R (β)

as n → ∞ in B (W,D,∆,~) on compact subsets.

Theorem 14. Let

[gnδn

]∈ B

(E

′, D,∆, •

); then

ιβ,γα,R

([gnδn

]~ ϵn

)= ιβ,γα,R

(ϵn ~

[gnδn

]),

(ϵn) ∈ ∆.

Readers can check the proofs from the citations given by the same author.

References

[1] Al-Omari, S.K.Q., Kilicman, A., Some Remarks on the ExtendedHartley-Hilbert and Fourier-Hilbert Transforms of Boehmians, Abstract andApplied Analysis, vol. 2013 (2013), 1-6.

[2] Al-Omari, S.K.Q., Notes for Hartley Transforms of Generalized Functions,Italian J. Pure Appl. Math., 28 (2011), 21-30.

[3] Al-Omari, S.K.Q., Kilicman, A., On Diffraction Fresnel Transforms forBoehmians, Abstract and Applied Analysis, Volume 2011, 1-15.

[4] Boehme, T.K., The Support of Mikusinski Operators, Tran. Amer. Math.Soc.176 (1973), 319-334.

[5] Banerji, P.K., Al-Omari, S.K.Q., Debnath, L., Tempered Distribu-tional Fourier Sine (Cosine) Transform, Integ. Trans. Spl. Funct., 17, 11(2006), 759-768.

[6] Al-Omari, S.K.Q., Kilicman, A., Note on Boehmians for Class of OpticalFresnel Wavelet Transforms, Journal of Function Spaces and Applications,vol. 2012 (2012), 1-12.

[7] Mikusinski, P., Convergence of Boehmians, Japanese Journal of Mathe-matics, 9 (1) (1983), 159–179.

[8] Millane, R.P., Analytic Properties of the Hartley Transform and their Ap-plications, Proc. IEEE, 82 (3) (1994), 413-428.

[9] Pathak, R.S., Integral Transforms of Generalized Functions and their Ap-plications, Gordon and Breach Science Publishers, Australia, Canada, India,Japan, 1997.

368 s.k.q. al-omari

[10] Al-Omari, S.K.Q., Loonker D., Banerji P.K., Kalla, S.L., FourierSine (Cosine) Transform for Ultradistributions and their Extensions to Tem-pered and UltraBoehmian Spaces, Integ. Trans. Spl. Funct., 19 (6) (2008),453–462.

[11] Al-Omari, S.K.Q., Kilicman, A., On Generalized Hartley-Hilbert andFourier-Hilbert Transforms, Advances in Difference Equations, vol. 2012(2012), 232-244.

[12] Bracewell, R.N., The Hartley Transform, New York, Oxford Univ., 1986.

[13] Al-Omari, S.K.Q., Kilicman, A., An Estimate of Sumudu Transform forBoehmians, Advances in Difference Equations, vol. 2013, (2013), 1-11.



ON SOLITARY WARE SOLUTIONS OF NONLINEARTIME-FRACTIONAL FORNBERG-WHITHAM EQUATION

Qazi Mahmood Ul Hassan

Syed Tauseef Mohyud-Din1

Department of MathematicsFaculty of SciencesHITEC University Taxila CanttPakistan

Abstract. This paper witnesses the combination of an efficient transformation and

Exp-function method to construct generalized solitary wave solutions of the nonlinear

time-fractional Fornberg-Whitham equation. Computational work and subsequent nu-

merical results re-confirm the efficiency of proposed algorithm. It is observed that

suggested scheme is highly reliable and may be extended to other nonlinear differential

equations of fractional order.

Keywords: Fornberg-Whitham equation, fractional calculus, exp-function method,

modified Riemann-Liouville derivative.

1. Introduction

The subject of factional calculus [1], [2] is a rapidly growing field of research, atthe interface between chaos, probability, differential equations, and mathematicalphysics. In recent years, nonlinear fractional differential equations (NFDEs) havegained much interest due to exact description of nonlinear phenomena of manyreal-time problems. The fractional calculus is also considered as a novel topic [3],[4]; has gained considerable popularity and importance during the recent past.It has been the subject of specialized conferences, workshops and treatises orso, mainly due to its demonstrated applications in numerous seemingly diverseand widespread fields of science and engineering. Some of the areas of present-day applications of fractional models [5]–[8] include fluid flow, solute transportor dynamical processes in self-similar and porous structures, diffusive transportakin to diffusion, material viscoelastic theory, electromagnetic theory, dynamicsof earthquakes, control theory of dynamical systems, optics and signal processing,bio-sciences, economics, geology, astrophysics, probability and statistics, chemicalphysics, and so on. As a consequence, there has been an intensive development ofthe theory of fractional differential equations, see [1–8] and the references therein.

1Corresponding author. E-mail: [email protected]


Recently, He and Wu [9] developed a very efficient technique which is called exp-function method for solving various nonlinear physical problems. The throughstudy of literature reveals that Exp-function method has been applied on a widerange of differential equations and is highly reliable. The exp-function method hasbeen extremely useful for diversified nonlinear problems of physical nature and hasthe potential to cope with the versatility of the complex nonlinearities of the pro-blems. The subsequent works have shown the complete reliability and efficiencyof this algorithm. He et.al. [10]–[11] used this scheme to find periodic solutions ofevolution equations; Mohyud-Din [12-15] extended the same for nonlinear phys-ical problems including higher-order BVPs; Oziz [16] tried this novel approachfor Fisher’s equation; Wu et.al. [17], [18] for the extension of solitary, periodicand compacton-like solutions; Yusufoglu [19] for MBBN equations, Zhang [20] forhigh-dimensional nonlinear evolutions; Zhu [21], [22] for the Hybrid-Lattice sys-tem and discrete m KdV lattice; Kudryashov [23] for exact soliton solutions of thegeneralized evolution equation of wave dynamics; Momani [24] for an explicit andnumerical solutions of the fractional KdV equation; Ebaid [25] for the improve-ment on the Exp-function method when balancing the highest order linear andnonlinear terms. The basic motivation of this paper is the development of an ef-ficient combination comprising an efficient transformation, exp-function methodusing Jumarie’s derivative approach [27]–[30] and its subsequent application toconstruct generalized solitary wave solutions of the nonlinear Fornberg-WhithamEquation [26] of fractional-order. It is to be highlighted that Ebaid [25] provedthat c = d and p = q are the only relations that can be obtained by applyingexp-function method to any nonlinear ordinary differential equation.

Theorem 1 [25] Suppose that u(r) and(u(γ)

)λare, respectively, the highest order

linear term and the highest order nonlinear term of a nonlinear ODE, where r andγ are both positive integers. Then the balancing procedure using the Exp-function

ansatz; U (η) =

∑dn=−c an exp (nη)∑qm=−p bm exp (mη)

, leads to c = d and p = q, ∀r, s, λ ≥ 1.

Theorem 2 [25] Suppose that u(r) and u(s)uk are, respectively, the highest orderlinear term and the highest order nonlinear term of a nonlinear ODE, where r, sand Ω are all positive integers. Then the balancing procedure using the Exp-function ansatz leads to c = d and p = q, ∀r, s, k ≥ 1.

Theorem 3 [25] Suppose that u(r) and(u(s))Ω are, respectively, the highest or-der linear term and the highest order nonlinear term of a nonlinear ODE, wherer, sand Ω are all positive integers. Then the balancing procedure using the Exp-function ansatz leads to c = d and p = q, ∀r, s ≥ 1,∀Ω ≥ 2.

Theorem 4 [25] Suppose that u(r) and (u(s))Ωuλ are, respectively, the highestorder linear term and the highest order nonlinear term of a nonlinear ODE, wherer, s,Ω and λ are all positive integers. Then the balancing procedure using the Exp-function ansatz leads to c = d and p = q, ∀r, s,Ω, λ ≥ 1.

on solitary ware solutions of nonlinear time-fractional... 371

2. Jumarie’s fractional derivative

Jumarie’s fractional derivative is a modified Riemann-Liouville derivative definedas [27]–[30]

(1) Dαt f (x) =

1

Γ (−α)

∫ x

0

(x− t)−α−1 (f (t)− f (0)) dt, α ≤ 0,

1

Γ (−α)

d

dx

∫ x

0

(x− t)−α (f (t)− f (0)) dt, 0 ≤ α ≤ 1

[fα−n (x)n]n, n ≤ α ≤ n+ 1, n ≥ 1

where f : R → R, x → f(x) denotes a continuous (but not necessarily differen-tiable) function. Some useful formulas and results of Jumarie’s modified Riemann–Liouville derivative were summarized in [27]–[30].

Dαxc = 0, α ≥ 0, c = constant(2)

Dαx [cf (x)] = cDα

xf (x) , α ≥ 0, c = constant(3)

Dαxx

β =Γ (1 + β)

Γ (1 + β − α)xβ−α, β ≥ α ≥ 0.(4)

Dαx [f (x) g (x)] = [Dα

xf (x) g (x) + f (x) [Dαx g (x)] .(5)

Dαxf (x (t)) = f

′x (x) .x

α (t) .(6)

3. Exp-function method [31]–[34]

We consider the general nonlinear FPDE of the type

(7) P (u, ut, ux, uxx uxxx, ..., Dαt u,D

αxu,D

αxxu, ...) = 0, 0 < α ≤ 1,

where Dαt u,D

αxu,D

αxxu are the modified Riemann-Liouville derivative of u with

respect to t, x, xx, respectively.Using a transformation [35]

(8) η = kx+ωtα

Γ (1 + α)+ η0, k, ω, η0 are all constants with k, ω = 0

we can rewrite equation (7) in the following nonlinear ODE;

(9) Q(u, u′, u′′, u′′′, uiv) = 0,

where the prime denotes derivative with respect to η.According to the Exp-function method, we assume that the wave solution can

be expressed in the following form

(10) u (η) =

∑dn−c an exp [nη]∑qm−p bm exp [mη]


where p, q, c and d are positive integers which are known to be further determined,an and bm are unknown constants. We can rewrite equation (4) in the followingequivalent form

(11) u (η) =ac exp (cη) + · · ·+ a−d exp (−dη)

bp exp (pη) + · · ·+ b−q exp (−qη).

This equivalent formulation plays an important and fundamental part for findingthe analytic solution of problems. To determine the value of c andp by using [25],

(12) p = c, q = d

4. Solution procedure

In this section, we apply the exp-function method for nonlinear time-fractionalFornberg-Whitham equation.

Example 4.1 Consider the nonlinear time-fractional Fornberg-Whitham equa-tion.

Dαt u+ ux − uxxt − uuxxx + uux − 3uxuxx = 0, t > 0, 0 < α < 1(13)

subject to initial condition u (x, t) = ex2

Using (8), equation (13) can be converted to an ordinary differential equation

(14) ωu′ + ku′2ω u

′′′3uu′′′ + kuu′3u′u′′ = 0,

where the prime denotes the derivative with respect to η. The solution of theequation (13) can be expressed in the form, equation (11). To determine thevalue of c andp, by using [25],

(15) p = c, q = d

Case 4.1.1. we can freely choose the values of c and d, but we will illustrate thatthe final solution does not strongly depend upon the choice of values of candd.For simplicity, we set p = c = 1 and q = d = 1 equation (11) reduces to

(16) u (η) =a1 exp [η] + a0 + a−1 exp [−η]

b1 exp [η] + a0 + b−1 exp [−η].

Substituting equation (16) into equation (14), we have

(17)1

A

c5 exp (5η) + c4 exp (4η) + c3 exp (3η) + c2 exp (2η)+c1 exp (η) + c0 + c−1 exp (−η) + c−2 exp (−2η)+c−3 exp (−3η) + c−4 exp (−4η) + c−5 exp (−5η)

= 0

A = 5 (b1 exp (η) + b0 + b−1 exp (−η))6 where ci (i = −5,−4, ..., 4, 5)


are constants obtained by Maple 16.

Equating the coefficients of exp (nη) to be zero, we obtain

(18)

(c−5 = 0, c−4 = 0, c−3 = 0, c−2 = 0, c−1 = 0, c0 = 0,

c1 = 0, c2 = 0, c3 = 0, c4 = 0, c5 = 0

)By solving (18), we get different solutions of (13).

1st Solution set:

(19)

ω = ω, a−1 =a1b−1

b1,

a1 = a1, b−1 = b−1, b0 = 0, b1 = b1, a0 = 0


(20) u (x, t) =

a1b−1

b1e−

xk+ωtα

Γ(1+α) + a1exk+ωtα

Γ(1+α)

b−1e−xk+ωtα

Γ(1+α) + b1exk+ωtα

Γ(1+α)

Figure 4.1(a) α = 0.25 Figure 4.1(b) α = 1

2nd Solution set:

(21)

ω = ω, a−1 =a1b−1

b1,

a1 = 0, b−1 = b−1, b0 = 0, b1 = 0, a0 = a0

We, therefore, obtained the following generalized solitary solution u (x, t)of equa-tion (13)

(22) u (x, t) =

a1b−1

b1e−

xk+ωtα

Γ(1+α) + a0

b−1e−xk+ωtα

Γ(1+α) + b0


Figure 4.1(c) α = 0.25 Figure 4.1(d) α = 1

3rd Solution set:

(23)

ω = ω, a−1 = a−1, a1 =

a−1b1b−1

, b−1 = b−1,

b0 = b0, b1 = b1, a0 =a−1b0b−1


(24) u (x, t) =a−1e

−xk+ωtα

Γ(1+α) + a−1b0b−1

+ a−1b1b−1

exk+ωtα

Γ(1+α)

b−1e−xk+ωtα

Γ(1+α) + b0 + b1e−xk+ωtα

Γ(1+α)

Figure 4.1(e) α = 0.25 Figure 4.1(f) α = 1

4th Solution set:

(25)

ω = ω, a−1 = 0, a0 =a1b0b1

,

a1 = a1, b−1 = 0, b0 = b0, b1 = b1

,

Therefore, we obtained the following generalized solitary solution u (x, t)of equa-tion (13)

(26) u (x, t) =a1e

xk+ωtα

Γ(1+α) + a1b0b1

b1exk+ωtα

Γ(1+α) + b0


Figure 4.1(g) α = 0.25 Figure 4.1(h) α = 1

Case 4.1.2 If p = c = 2, and q = d = 1 then equation (11) reduces to (27)

(27) u (η) =a2 exp [2η] + a1 exp [η] + a0 + a−1 exp [−η]

b2 exp [2η] + b1 exp [η] + a0 + b−1 exp [−η].

Proceeding as before, we obtain

1st Solution set:

(28)

b−1 = 0, a2 =

a0b2b0

, b2 = b2, b0 = b0, b1 = b1,

a1 =a0b1b0

, a−1 = 0, ω = ω, a0 = a0

,

Hence we get the generalized solitary wave solution of equation (13) as follows

(29) u (x, t) =a0b2b0

e2(xk+ωtα

Γ(1+α) ) + a0b1b0

exk+ωtα

Γ(1+α) + a0

b2e2(xk+ωtα

Γ(1+α) ) + b1exk+ωtα

Γ(1+α) + b0

Figure 4.1(i) α = 0.25 Figure 4.1(j) α = 1


2nd Solution set:Therefore, we obtained the Proceeding as before, we obtain

(30)

b−1 = b−1, a2 =a−1b2b−1

, b2 = b2, b0 = b0, b1 = b1,

a1 =a−1b1b−1

, a−1 = a−1, ω = ω, a0 =a−1b0b−1

,

Hence, we get the generalized solitary wave solution of equation (13) as follows

(31) u (x, t) =

a−1b2b−1

e2(xk+ωtα

Γ(1+α) ) + a−1b1b−1

exk+ωtα

Γ(1+α) + a−1b0b−1

+ a−1e−xk+ωtα

Γ(1+α)

b2e2(xk+ωtα

Γ(1+α) ) + b1exk+ωtα

Γ(1+α) + b0 ++b−1e−xk+ωtα

Γ(1+α)

Figure 4.1(k) α = 0.25 Figure 4.1(l) α = 1

We get the same soliton solutions which clearly illustrate that final solution doesnot strongly depends upon these parameters.

5. Conclusion

In this paper, we applied exp-function method to construct generalized solitarysolutions of the nonlinear fractional order Fornberg-Whitham equation. It is ob-served that the Exp-function method is very convenient to apply and is very usefulfor finding solutions of a wide class of nonlinear problems.

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GLOBAL EXPONENTIAL STABILITY OF IMPULSIVE HYBRIDDYNAMICAL SYSTEMS WITH ANY TIME DELAY1

Xingjie Wu2

Yang Liu

Zongmin Qiao

School of Mathematics and StatisticsHefei Normal UniversityHefei Anhui 230601P.R. China

Abstract. This present paper addresses global exponential stability for a class of more

general linear impulsive hybrid dynamical systems with any time delay. Combined

Lyapunov function methods with the Razumikhin technique, several criteria on global

exponential stability are derived, which are substantially extension and generalization

of the corresponding results in recent literature. Subsequently, two application exam-

ples and its numerical simulations demonstrate that the obtained stability criteria are

practical and effective.

Keywords: Impulsive dynamics, Any time delay, Hybrid systems, Global exponential

stability, Razumikhin technique.

1. Introduction and notation

In past few decades, the stability problems of dynamical systems with impulsesand delays have attracted a great deal of attention from a variety of applicationsin science and engineering, see [1]-[7] and references therein. In particular, specialattention has been focused on the exponential stability of such dynamical systemsbecause it has played an important role in practical applications of dynamicalsystems [8]-[10].

1This research is supported by the Natural Science Program of Anhui Higher EducationInstitution under Grant, (KJ2013B216, KJ2013B217); The 2014 scientific Research Project ofHefei Normal University, (“136 Talents Project”).

2Corresponding author. E-mail: [email protected].

380 x. wu, y. liu, z. qiao

At the same time, the impulsive hybrid dynamical system is regarded as aspecial but important of impulsive differential systems with variable structure, andit has recently emerged as a new challenging area of research due to its potentialapplications in various fields of engineering areas, including control technology,industrial robotics, and communication engineering, etc [11]-[17]. There are seve-ral research works appeared in the literature on exponential stability of impulsivehybrid dynamical systems with time delay [18]-[21]. For example, Liu and Shen[19] presented some criteria on uniform stability and uniform asymptotic stabilityof invariant sets for impulsive hybrid dynamical systems with time delay. In [20],Li, Ma, and Feng proved some general criteria on global exponential stability fornonlinear impulsive and switching delayed dynamical systems. However, there isa restrictive condition that the time delay is less than the length of all the impul-sive intervals, so they are generally inapplicable in some practical applications. Inaddition, Zhang and Sun [21] only given some results related to local uniformlystability for linear impulsive hybrid dynamical systems with time delay.

The main objective of this paper is further to investigate global exponentialstability for a class of more general linear impulsive hybrid dynamical systemswith any time delays. Based on the Lyapunov function methods combined withthe Razumikhin techniques, several criteria on global exponential stability arederived analytically, which are nature extension and generalization of the corres-ponding results existing in the literature. Compared with some existing works, adistinctive feature of this work is to address global exponential stability for linearor nonlinear impulsive hybrid dynamical systems with any time delays. Further-more, our results show that impulses do contribute to global exponential stabilityof dynamical systems with any time delays even if it may be chaotic or unstableitself, which can be usually used as an effective control strategy to stabilize theunderlying delayed dynamical systems in some practical applications.

Let R = (−∞,+∞) be the set of real numbers, R+ = [0,+∞) be the set ofnonnegative real numbers, and N = 0, 1, 2, . . . be the set of nonnegative integernumbers. For the vector u ∈ Rn, u⊤ denotes its transpose. The norm of the vectoru is defined as ∥u∥ =

√u⊤u. Rn×n stands for n× n the set of real matrices.

Consider the following impulsive delayed linear dynamical hybrid systems:x(t) = Ax(t) +Bx(t− τ(t)) +Mkxk, t ∈ [tk, tk+1),

x(t) = Ckx(t−), t = tk, k ∈ N,

(1.1)

where t ≥ t0, φ ∈ PC([−τ, 0], Rn), x(t) ∈ Rn, A, B, Ck, Mk ∈ Rn×n, C0 = I,Mkxk is hybrid term.

The time sequence tk+∞k=1 satisfy 0 = t0 < t1 < t2 < · · · < tk < · · · ,

limk→∞

tk = +∞, the time delay 0 ≤ τ(t) ≤ τ < +∞, x(t+) = lims→0+

x(t + s) and

x(t−) = lims→0−

x(t+ s).

Let PC([−τ, 0], Rn) = ϕ : [−τ, 0] → Rn, ϕ(t) is continuous everywhereexcept a finite number of points t at which ϕ(t+) and ϕ(t−) exist and ϕ(t+) = ϕ(t).

global exponential stability of impulsive hybrid... 381

For ψ ∈ PC([−τ, 0], Rn), the norm of ψ is defined by

∥ψ∥τ = sup−τ≤s≤0

∥ψ(s)∥.

xt, xt− ∈ PC([−τ, 0], Rn) are defined by xt(s) = x(t + s) and xt−(s) = x(t− + s)for s ∈ [−τ, 0], respectively.

We always assume that system (1.1) has a unique solution with respect toinitial conditions. Denote by x(t) = x(t, t0, φ) the solution of system (1.1) suchthat xt0 = φ. We further assume that all the solutions x(t) of system (1.1) arecontinuous except at tk, k ∈ N , at which x(t) is right continuous. Let xk = x(tk),∆tk = tk+1 − tk, k ∈ N . Obviously, x(t) = 0 is a solution of system (1.1), whichwe call the zero solution.

Definition 1.1. The zero solution of system (1.1) is said to be globally exponen-tially stable if there exist two constants λ > 0 and K ≥ 1 such that for any initialvalue xt0 = φ,

∥x(t, t0, φ)∥ ≤ K∥φ∥τe−λ(t−t0), t ≥ t0,

where (t0, φ) ∈ R+ × PC([−τ, 0], Rn).

Definition 1.2. Function V : R+ ×Rn → R+ is said to belong to the class ν0 if

(a) V is continuous in each of the sets [tk, tk+1) × Rn, and for each x ∈ Rn,t ∈ [tk, tk+1), k ∈ N , lim

(t,y)→(t−k ,x)V (t, y) = V (t−k , x) exists, and

(b) V (t, x) is locally Lipschitzian in all x ∈ Rn, and for all t ≥ t0, V (t, 0) ≡ 0.

Definition 1.3. Given a function V : R+ × Rn → R+, the upper right-handderivative of V with respect to system (1.1) is defined by

D+V (t, x(t)) = limδ→0+

sup1

δ

(V (t+ δ, x(t+ δ))− V (t, x(t))

).

2. Main results

In this section, by combining the Lyapunov function and Razumikhin technique,we shall present several criteria on global exponential stability for the delayedlinear hybrid dynamical system (1.1).

Theorem 2.1. Let the n × n matrix P be symmetric and positive definite, λ3be the largest eigenvalue of P−1(ATP + PA + 2PP ), λ4 be the largest eigenvalueof P−1BTB, λMk

be the largest eigenvalue of P−1CTk M

Tk MkCk, λCk

be the largesteigenvalue of P−1CT

k PCk, λ5 = supk∈N

λMk, λ6 = sup

k∈NλCk

and 0 < λ6 < 1. Assume

that there exist two constants λ > 0 and σ > 0, such that for all k ∈ N , thefollowing conditions are satisfied:


(i) F (λ) = σ − λ−(λ3 +

λ4λ6eλτ +

λ5λ6

)≥ 0;

(ii) lnλ6 < −(σ + λ)(tk+1 − tk).

Then, for any delay 0 ≤ τ(t) ≤ τ < +∞, the zero solution of the linear im-pulsive delayed hybrid dynamical system (1.1) is globally exponentially stable with

convergence rateλ

2.

Proof. Let x(t) = x(t, t0, φ) be any solution of the linear impulsive delayed hybriddynamical system (1.1) with xt0 = φ.

We construct a Lyapunov function as follow:

V (t, x(t)) = xT (t)Px(t).(2.1)

Let λ1 > 0 and λ2 > 0 are the smallest and the largest eigenvalues of Prespectively, so we have

(2.2) λ1∥x(t)∥2 ≤ V (t, x(t)) ≤ λ2∥x(t)∥2.

We shall prove that

V (t, x(t)) ≤ λ2K∥φ∥2τe−λ(t−t0), t ∈ [tk, tk+1), k ∈ N.(2.3)

For t ∈ [tk, tk+1), k ∈ N , we calculate the upper right derivative of V (t, x(t))along the solution of system (1.1) and have

D+V (t, x(t)) = xT (t)(ATP + PA)x(t)

+ 2xT (t− τ(t))BTPx(t) + 2xkMTk Px(t)

≤ xT (t)(ATP + PA+ 2PP )x(t)

+xT (t− τ(t))BTBx(t− τ(t)) + xTkMTk Mkxk

≤ λ3V (t, x(t)) + λ4V (t− τ(t), x(t− τ(t)))

+xTkMTk Mkxk.

(2.4)

From condition (ii), we can get

− lnλ6 + λτ − (σ + λ)(tk+1 − tk) > 0.(2.5)

From (2.5), we can choose K ≥ 1, such that

1 < e(σ+λ)(t1−t0) ≤ K ≤ − lnλ6eλτ−(σ+λ)(t1−t0)e(σ+λ)(t1−t0).(2.6)

Then

∥φ∥2τ < ∥φ∥2τeσ(t1−t0) ≤ K∥φ∥2τe−λ(t1−t0).(2.7)

We first prove

V (t, x(t)) ≤ λ2K∥φ∥2τe−λ(t−t0), t ∈ [t0, t1),(2.8)


To do this, we only need to prove that

V (t, x(t)) ≤ λ2K∥φ∥2τe−λ(t1−t0), t ∈ [t0, t1).(2.9)

If (2.9) is not true, from (2.2) and (2.8), then there exists some t ∈ (t0, t1) suchthat

V (t, x(t)) > λ2K∥φ∥2τe−λ(t1−t0) ≥ λ2∥φ∥2τeσ(t1−t0)

> λ2∥φ∥2τ ≥ V (t0 + s, x(t0 + s)), s ∈ [−τ, 0],

which implies that there exists some t∗ ∈ (t0, t) such that

V (t∗, x(t∗)) = λ2K∥φ∥2τe−λ(t1−t0),(2.10)

and

V (t, x(t)) ≤ V (t∗, x(t∗)), t ∈ [t0 − τ, t∗],(2.11)

then there exists t∗∗ ∈ [t0, t∗) such that

V (t∗∗, x(t∗∗)) = λ2∥φ∥2τ ,(2.12)

and

V (t∗∗, x(t∗∗)) ≤ V (t, x(t)), t ∈ [t∗∗, t∗].(2.13)

Hence, for any s ∈ [−τ, 0], we have

V (t+ s, x(t+ s)) ≤ λ2K∥φ∥2τe−λ(t1−t0)

≤ λ2λ6eλτ−(σ+λ)(t1−t0)e(σ+λ)(t1−t0)∥φ∥2τ

≤ eλτ

λ6V (t∗∗, x(t∗∗))

≤ eλτ

λ6V (t, x(t)), t ∈ [t∗∗, t∗],

(2.14)

From (2.3), (2.12) and (2.13), we can get

xT0MT0 M0x0 = xT0M

T0 M0P

−1Px0 ≤ λ5V (t0, x(t0))

≤ λ5λ2∥φ∥2τ ≤ λ5V (t, x(t)), t ∈ [t∗∗, t∗],(2.15)

Thus, from condition (i), (2.4), (2.14) and (2.15), we get

D+V (t, x(t)) ≤(λ3 +

λ4λ6eλτ +

λ5λ6

)V (t, x(t))

≤ (σ − λ)V (t, x(t)), t ∈ [t∗∗, t∗].(2.16)


From (2.7), (2.10), (2.11), (2.12) and (2.13) that

V (t∗, x(t∗)) ≤ V (t∗∗, x(t∗∗))e(σ−λ)(t∗−t∗∗)

< λ2∥φ∥2τeσ(t1−t0) ≤ V (t∗, x(t∗)),(2.17)

which is a contradiction. Hence (2.8) holds and then (2.3) is true for k = 0.

Suppose (2.3) holds for k = 0, 1, 2, . . . ,m (m ∈ N,m ≥ 0), i.e.,

V (t, x(t)) ≤ λ2K∥φ∥2τe−λ(t−t0), t ∈ [tk, tk+1), k = 0, . . . ,m.(2.18)

Next, we shall prove that (2.3) holds for k = m+ 1, i.e.,

V (t, x(t)) ≤ λ2K∥φ∥2τe−λ(t−t0), t ∈ [tm+1, tm+2).(2.19)

If (2.19) is not true, suppose

t = inft ∈ [tm+1, tm+2)|V (t, x(t)) > λ2K∥φ∥2τe−λ(t−t0).

From condition (ii) and (2.18), we have

V (tm+1, x(tm+1)) = xT (t−m+1)CTm+1PCm+1x(t

−m+1)

≤ λ6V (t−m+1, x(t−m+1))

≤ λ6λ2K∥φ∥2τe−λ(tm+1−t0)

< λ2K∥φ∥2τe−λ(t−t0),

(2.20)

and so t = tm+1. From the continuity of V (t, x(t)) in [tm+1, tm+2), we have

V (t, x(t)) = λ2K∥φ∥2τe−λ(t−t0), t ∈ [tm+1, t](2.21)

By (2.20), we know that there exists some t∗ ∈ (tm+1, t) such that

V (t∗, x(t∗)) = λ6λ2eλ(tm+2−tm+1)K∥φ∥2τe−λ(t−t0),(2.22)

and

V (t∗, x(t∗)) ≤ V (t, x(t)) ≤ V (t, x(t)), t ∈ [t∗, t].(2.23)

From (2.19), (2.20), (2.22) and (2.23), we can obtain

xTm+1MTm+1Mm+1xm+1 = x−

Tm+1C

Tm+1M

Tm+1Mm+1Cm+1x

−m+1

≤ λ5V (t−m+1, x(t−m+1))

= λ2eλ(tm+2−tm+1)K∥φ∥2τe−λ(t−t0)

=λ5λ6V (t∗, x(t∗))

≤ λ5λ6V (t, x(t)), t ∈ [t∗, t]

(2.24)


On the other hand, for any t ∈ [t∗, t], s ∈ [−τ, 0], then either t+ s ∈ [t0 − τ, tm+1)or t+ s ∈ [tm+1, t]. Two cases will be discussed as follows:

If t+ s ∈ [t0 − τ, tm+1), from (2.18), we have

V (t+ s, x(t+ s)) ≤ λ2K∥φ∥2τe−λ(t−t0)e−λs

≤ λ2eλτeλ(tm+2−tm+1)K∥φ∥2τe−λ(t−t0).(2.25)

While, if t+ s ∈ [tm+1, t], from (2.21), then

V (t+ s, x(t+ s)) ≤ λ2K∥φ∥2τe−λ(t−t0)

≤ λ2eλτeλ(tm+2−tm+1)K∥φ∥2τe−λ(t−t0).(2.26)

Form (2.25) and (2.26), in any case, we have for any s ∈ [−τ, 0],

V (t+ s, x(t+ s)) ≤ eλτ

λ6V (t∗, x(t∗))

≤ eλτ

λ6V (t, x(t)), t ∈ [t∗, t].(2.27)

Finally, from (i), (2.4), (2.24) and (2.27), we have

D+V (t, x(t)) ≤(λ3 +

λ4λ6eλτ +

λ5λ6

)V (t, x(t)) ≤ (σ − λ)V (t, x(t)).

Thus, in view of condition (ii), we have

V (t, x(t)) ≤ V (t∗, x(t∗))e(σ−λ)(t−t∗)

< λ2e−(σ+λ)(tm+2−tm+1)eλ(tm+2−tm+1)K∥φ∥2τe−λ(t−t0)e(σ−λ)(t−t∗)

< V (t, x(t)),

(2.28)

which is a contradiction. This implies the assumption is not true, and hence (2.3)holds for k = m+ 1. Therefore, we can obtain that (2.3) holds for any k ∈ N .

By (2.2) and (2.3), we have

∥x(t)∥ ≤√λ2λ1K∥φ∥τe−

λ2(t−t0), t ≥ t0,

which implies that the zero solution of the linear impulsive delayed hybrid dy-

namical system (1.1) is globally exponentially stable with convergence rateλ

2for

any time delay 0 ≤ τ(t) ≤ τ < +∞.Then, we complete the proof of Theorem 2.1.

Corollary 2.1. Let λ3, λ4, λ5, λ6 and condition (iii) be is precisely the same asthat of Theorem 2.1, and assume that for all k ∈ N ,


(iii)(λ3 +

λ4 + λ5λ6

)(tk+1 − tk) < − lnλ6.

Then, for any delay 0 ≤ τ(t) ≤ τ < +∞, the zero solution of the linear impulsivedelayed hybrid dynamical system (1.1) is globally exponentially stable.

Consider the following impulsive delayed nonlinear dynamical hybrid systems:x(t) = Ax(t) + f(t, x(t), x(t− τ(t))) +Mkxk, t ∈ [tk, tk+1),

x(t) = Ckx(t−), t = tk, k ∈ N,

xt0 = φ,

(2.29)

where f : R+ × Rn × Rn → Rn is a continuously vector-valued function,f(t, 0, 0) = 0, which satisfies the following condition

(H): ∥f(t, x(t), x(t − τ(t)))∥2 ≤ l1∥x(t)∥2 + l2∥x(t − τ(t))∥2, where l1 and l2 arepositive numbers.

In general, the condition (H) is very mild, since some general dissipative dy-namical systems, such as neural networks, coupled chaos oscillators, and networksof multi-agent, etc., can be included in condition (H).

Theorem 2.2. Let the n× n matrix P be symmetric and positive definite, λ3 bethe largest eigenvalue of P−1(ATP +PA+2PP + l1), λ4 be the largest eigenvalueof l2P

−1, λMkbe the largest eigenvalue of P−1CT

k MTk MkCk, λCk

be the largesteigenvalue of P−1CT

k PCk, λ5 = supk∈N

λMk, λ6 = sup

k∈NλCk

and 0 < λ6 < 1. Assume

both conditions (i) and (ii) are satisfied in Theorem 2.1.Then, for any time delay 0 ≤ τ(t) ≤ τ < +∞, the zero solution of the non-

linear impulsive delayed dynamical hybrid system (2.29) is globally exponentially

stable with convergence rateλ

2.

Proof. Similarly to the proof of Theorem 2.1, by the Lyapunov function (2.1),for t ∈ [tk, tk+1), we calculate the upper right derivative of V (t, x(t)) along thesolution of system (2.29) and have

D+V (t, x(t)) = xT (t)(ATP + PA)x(t)

+ 2xT (t)Pf(t, x(t), x(t− τ(t)) + 2Mkxkx(t)

≤ xT (t)(ATP + PA+ PP + l1P−1P )x(t)

+ l2xTx(t− τ(t))x(t− τ(t)) + xTkM

Tk Mkxk

≤ λ3V (t, x(t)) + λ4V (t− τ(t), x(t− τ(t))) + xTkMTk Mkxk.

(2.30)

The rest of the proof is precisely the same as Theorem 2.1, so we omit it. Theorem2.2 is proven.


Corollary 2.2. Let λ3, λ4, λ5 and λ6 be is precisely the same as that of Theorem2.2, and assume that for all k ∈ N ,

(iv)(λ3 +

λ4 + λ5λ6

)(tk+1 − tk) < − lnλ6.

Then, for any delay 0 ≤ τ(t) ≤ τ < +∞, the zero solution of the nonlinearimpulsive delayed hybrid dynamical system (2.29) is globally exponentially stable.

Remark 2.1. Obviously, Corollary 2.1 has extended the Theorem 1 of [21] con-cerning the local uniform stability to the global exponential stability for the li-near impulsive delayed hybrid dynamical system (1.1) under the same conditions.Therefore, Theorem 2.1 is an important improvement and generalization of themain results in [21]. In addition, it is interested that if the linear impulsive de-layed hybrid dynamical systems are local uniform stable, then it must be globallyexponentially stable.

Remark 2.2. Theorem 2.1, 2.2 and Corollary 2.1, 2.2 are valid for any timedelay 0 ≤ τ(t) ≤ τ < +∞. Therefore, our results are more practically applicablethan those in the literature since the restrictive condition that the time delays areless than the length of all the impulsive intervals is actually removed here (seefor example: Theorem 3.1 of [20]). Furthermore, our results show that impulseand hybrid term do contribute to the global exponential stability of linear ornonlinear impulsive delayed hybrid dynamical systems even if the correspondingsystems without impulses are chaotic or unstable.

Remark 2.3. Let h(λ) = (− lnλ6)/

(2λ+ λ3 +

λ4λ6eλτ +

λ5λ6

), in Theorem 2.1,

the satisfaction of both (i) and (ii) is equivalent to ∆tk < h(λ), if we require thatthe exponential convergence rate of system (1.1) or (2.29) is larger than or equal

to any givenλ∗

2> 0, we can choose any suitable ∆tk < t∗k, such that system

(1.1) or (2.29) is globally exponentially stable with exponential convergence rateλ

2≥ λ∗

2> 0, where ∆t∗k = h(λ∗).

3. Application examples

In this section, we shall discuss the applications of the above theoretic criteria.Two examples and their simulations are given to show that our main results arepractical.

Example 3.1. Consider the following linear impulsive delayed hybrid dynamicalsystems:


x(t) =

(1 1

214

5

)x(t) +

(2 213

6

)x(t− τ(t)) +

(1

k+10

0 1k+1

)xk,

t ∈ [tk, tk+1),

x(t) =

(3k+1k2+1

0

0 4k+1k2+1

)x(t−), t = tk, k ∈ N,

(3.1)

where x(t) = (x1(t), x2(t))T ∈ R2.

Let P =

(9 00 12

), then λ3 = 34.0371, λ4 = 3.4449, λ5 = 0.1302,

λ6 = 0.5208, which implies Theorem 2.1 hold. By condition (iii) of Corollary2.1, if ∆tk < 0.0159 the zero solution of system (3.1) is globally exponentiallystable for any time delay. The numerical simulation with ∆tk = 0.013, τ = 0.3and initial functions (φ1(t), φ2(t)), where

φ1(t) =

0, t ∈ [−0.3, 0),

−1, t = 0,

φ2(t) =

0, t ∈ [−0.3, 0),

1, t = 0,

(3.2)

is given in Fig. 1, the global exponential convergence rate is 0.9861. The system(3.1) without impulses and hybrid term is unstable with the initial functions (3.2)(see Fig. 2).

It should be noted that the global exponential stability cannot be derivedby applying existing criterion in [21], where only local uniform stability is provedunder the same conditions. Therefore, Theorem 2.1 or Corollary 2.1 substantiallyextends and improves the main results of [21].

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−4

−2

0

2

4

6

8

10

t

x 1(t), x

2(t)

0.3 0.4 0.5 0.6 0.70

0.2

0.4

0.6

t

x 1(t), x

2(t)

x

1(t)

x2(t)

Fig. 1. Global exponential stability ofsystem (3.1) with ∆tk = 0.013.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−50

0

50

100

150

200

t

x 1(t), x

2(t)

x1(t)

x2(t)

Fig. 2. Instability of system (3.1) withoutimpulses and hybrid term.


Example 3.2. Consider the following Machey-Glass [22] nonlinear delayed im-pulsive hybrid model:

x(t) = −0.68x(t) +1.7x(t− τ(t))

1 + x8(t− τ(t))+

3

k + 2xk, t ∈ [tk, tk+1),

x(t) =k + 1

2k2 + 1x(t−), t = tk, k ∈ N,

(3.3)

where 0 ≤ τ(t) ≤ τ < +∞. The corresponding system (3.3) without impulse andhybrid term is a chaotic system. Fig. 3 shows its simulation results for τ = 4 withthe initial function:

φ(t) =

0, t ∈ [−4, 0),

1.2, t = 0,(3.4)

It is easy to see that condition (H) holds for

f(t, x(t), x(t− τ(t))) =1.7x(t− τ(t))

1 + x8(t− τ(t))

with l1 = 0 and l2 = 1.7. By taking P = 13, λ3 = 24.64, λ4 = 0.0452, λ5 =0.1731, λ6 = 0.0769, then all the conditions of Theorem 2.2 are satisfied. By thecondition (iv) of Corollary 2.2, if ∆tk < 0.0933 the zero solution of system (3.3) isglobally exponentially stable for any time delay. The numerical simulation with∆tk = 0.092 and the initial function (3.4) is given in Fig. 4, the global exponentialconvergence rate is 0.0422.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.60.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x(t)

x(t−

4)

Fig. 3. Chaotic behaviors of system (3.3)without impulse and hybrid term.

0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

1.2

t

x(t)

4 5 6 7 8

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

t

x(t)

x(t)

Fig. 4. Global exponential stability ofsystem (3.3) with ∆tk = 0.092.

By Remark 2.3, let P = 13, τ = 4, Fig. 5 shows the relationship betweenλ and ∆tk in system (3.3). By Remark 2.3, in Theorem 2.2, the satisfaction ofboth (i) and (ii) is equivalent to ∆tk < h(λ). Hence, in Fig. 5, D can denote theglobally exponentially stable region.


0 0.5 1 1.5 2 2.5 30

0.02

0.04

0.06

0.08

0.1

λ

∆ t k

(λ*,∆ t*k)

h(λ)

D

Fig. 5. Globally exponentially region of system (3.3) for (λ,∆tk).

It can been seen from Example 3.2 that the impulses hybrid terms can con-tribute to global exponential stability for nonlinear delayed dynamical systemseven if the corresponding systems without impulses and hybrid terms may be un-stable or chaotic itself, which can be usually used as an effective control strategy tostabilize the underlying delayed dynamical systems in some practical applications.

4. Conclusions

In this paper, the global exponential stability criteria of linear and nonlinearimpulsive hybrid dynamical systems for any time delay are investigated. Byemploying the methods of Razumikhin technique and Lyapunov function, someglobal exponential stability criteria have been established. Our results have im-proved and generalized some of the known results existing in the literature. Twoexamples are given to illustrate the theoretical results. Furthermore, it is shownthat the impulses and hybrid term can stabilize an unstable or even chaotic dy-namical system, which is particularly meaningful for control and design of hybriddynamical systems in practice.

References

[1] Yu, H.G., Zhong, S.M., Ye, M., Dynamic analysis of an ecological modelwith impulsive control strategy and distributed time delay, Mathematics andComputers in Simulation, 80 (3) (2009), 619-632.

[2] Stamova, I.M., Stamov, G.T., Lyapunov-Razumikhin method for im-pulsive functional differential equations and applications to the populationdynamics, Journal of Computational and Applied Mathematics, (1-2) 130(2001), 163-171.

[3] Liu, X.Z., Wang, Q., The method of Lyapunov functional and exponentialstability of impulsive systems with time delay, Nonlinear Analysis, 66 (7)(2007), 1465-1484.


[4] Zhang, Y., Sun, J.T., Stability of impulsive linear differential equationswith time delay, IEEE Transactions on Circuits and Systems II: ExpressBriefs, 52(10) (2005), 701-705.

[5] Samuel, F.P., Gunasekar, T., Arjunan, M.M., Controllability re-sults for second order impulsive neutral functional integral differential inclu-sions with infinite delay, Italian Journal of Pure and Applied Mathematics,31 (2013), 319-332.

[6] Wu, X.T., Yan, L.T., Zhang, W.B., Chen, L. Exponential stability ofimpulsive stochastic delay differential systems, Discrete Dynamics in Natureand Society, 2012 (2012), 1-15.

[7] Wu, X.J., Du, W., Pan, G.A., Huang, W.T., The dynamical behaviorsof a Ivlev-type two-prey two-predator system with impulsive effect, IndianJournal of Pure and Applied Mathematics, 44 (1) (2013), 1-27.

[8] Fu, X.L., Li, X.D., Razumikhin-type theorems on exponential stability ofimpulsive infinite delay differential systems, Journal of Computational andApplied Mathematics, 224 (1) (2009), 1-10.

[9] Zhang, G.L., Song, M.H., Liu, M.Z., Exponential stability of impulsivedelay differential equations, Abstract and Applied Analysis, 2013 (2013),doi: 10.1155/2013/938027.

[10] Li, X.D., Fu, X.L., On the global exponential stability of impulsive func-tional differential equations with infinite delays or finite delays, Advances inDifference Equations, 19 (3) (2014), 442-447.

[11] Liu, B., Liu, X.Z., Liao, X.X., Stability and robustness of quasi-linearimpulsive hybrid systems, Journal of Mathematical Analysis and Applica-tions, 238 (2) (2003), 416-430.

[12] Hayakawa, T., Haddad, W.M., Volyanskyy, K.Y., Neural networkhybrid adaptive control for nonlinear uncertain impulsive dynamical systems,Nonlinear Analysis: Hybrid Systems, 2 (3) (2008), 862-874.

[13] Wang, P.G., Liu, X., New comparison principle and stability criteria forimpulsive hybrid systems on time scales, Nonlinear Analysis: Real WorldApplications, 7 (5) (2006), 1096-1103.

[14] Attia, S.A., Azhmyakov, V., Raisch, J., On an optimization problemfor a class of impulsive hybrid systems, Discrete Event Dynamic Systems,20 (2) (2010), 215-231.

[15] Guan, Z.H., Hill, D.J., Shen, X.M., On hybrid impulsive and switchingsystems and application to nonlinear control, IEEE Transactions on Auto-matic Control, 50 (7) (2005), 1058-1062.


[16] Taringoo, F., Caines, P.E., On the optimal control of impulsive hybridsystems on Riemannian manifolds, SIAM J. Control Optim., 51 (4) (2013),3127-3153.

[17] Adams, B.M., Davis, N., Epps, P., et al. Existence of solution forimpulsive hybrid differential equation, Mathematical Sciences, 6 (2012), 1-9.

[18] Chen, G.P., Li, J.M., Yang, Y., Modeling and stability analysis of hy-brid dynamical systems with extended differential Petri nets, InternationalJournal of Control, Automation and Systems, 10 (2) (2012), 238-248.

[19] Liu, X.Z., Shen, J.H., Stability theory of hybrid dynamical systems withtime delay, IEEE Transactions on Automatic Control, 51 (4) (2006), 620-625.

[20] Li, C.D., Ma, F., Feng, G., Hybrid impulsive and switching time-delaysystems, IET Control Theory and Applications, 3 (11) (2009), 1487-1498.

[21] Zhang, Y., Sun, J.T., Stability of impulsive linear hybrid systems withtime delay, Journal of Systems Science and Complexity, 23 (4) (2010), 738-747.

[22] Mackey, M.C., Glass, L., Oscillations and chaos in physiological controlsystems, Science, 197 (1977), 287-289.



FROM NEWTON TO KEPLER.One simple derivation of Kepler’s laws from Newton’s ones.

Frantisek Mosna

Department of MathematicsTechnical FacultyCzech University of Life Sciences, PragueCzech Republice-mail: [email protected]

Abstract. There is plenty of ways how to deduce Kepler’s laws of planetary motion

from Newton’s ones (law of universal gravitation and law of motion). We offer one of

them which is very simple and direct. It uses only mathematical tools and is suitable

for teaching purposes.

1. Kepler’s and Newton’s laws

Johannes Kepler (1571–1630) deduced his laws of planetary motion due to theexact measurement of planets performed by astronomer Tycho Brahe (1546–1601).The first and the second law were derived in Prague and published in work As-tronomia Nova in 1609. The third one was formulated in Linz and publishedHarmoniae Mundi in 1619.

Sixty five years later (in 1684), Isaac Newton (1643–1727) deduced his law ofuniversal gravitation from Kepler’s laws.

Plenty of literature deals with the relation between Kepler’s and Newton’slaws. The most of them use physical concepts, e.g. [1] and [2]. The geometricalway of derivation was discussed in Richard Feynmann lecture presented on 13thMarch 1964 at Caltech (its manuscript had been lost and it was found again in1992 at the office of author’s colleague Leighton) and it is very interesting [3].

In this text, we would like to present one way how to derive Kepler’s laws fromthe Newton’s law of universal gravitation and motion. The aim and advantage ofour deduction of especially the first one is its simplicity, directness, straightnessand using only mathematical tools without introducing other physical variables.It can be presented maybe even in secondary schools.

394 f. mosna

2. Formulation of the laws

Let us formulate this laws at first.

• Kepler’s laws of planetary motion:

(K1) The orbit of every planet is an elipse with the Sun at one of the two foci.

(K2) A line joining a planet and the Sun sweeps out equal areas during equalintervals of time, i.e. area velocity is constant.

(K3) QuotientT 3

a2is constant, where T is orbital period of any planet and a is

the major semi-axis of its orbit.

• Newton’s laws of universal gravitation and the second law of motion:

(NG) Any point mass M attracts every single other point mass m situated in thedistance r from M by a force

F = κmM

r2, where κ = 6.672 · 10−11Nm2/kg2

(acting in direction of radius-vector of both points).

(N2) The acceleration a of a body is parallel and directly proportional to the netforce F acting on the body,

F = am , where m is mass of a body.

3. Derivation of Kepler’s laws

At first, we will describe the situation in Figure 1. In the beginning, the point massm (planet) in the distance r0 from the point mass M (the Sun) has the velocityv0 in the direction perpendicular to the radius-vector of both points (M and m).

Figure 1: Situation and start condition

Let the point mass M be situated to origin of axes and let point mass mbe described by the position vector ~r(t) = (x(t), y(t), z(t)). According to (NG),

from newton to kepler 395

the force by which the point mass m is attracted to the point mass M can beexpressed by the formula

~F = −κ · mM|~r|3· ~r = −k · m~r

|~r|3, where k = κM. (1)

By (N2) it holds~F = m · ~r. (2)

From these two relations (1) and (2) we obtain equation

~r + k · ~r|~r|3

= 0. (3)

3.1. Trajectory is a plane curve

At first, we derive that the trajectory of point mass m is situated in some planecontaining the point mass M .

This fact will be proved if we show that the velocity ~r of the point m issituated in the same plane as radius-vector ~r of that point, i.e., when ~r × ~r is aconstant vector. Let us calculate the derivation of this cross product using relation(3)

d

dt(~r × ~r) = ~r × ~r + ~r × ~r︸︷︷︸

=0

= − k

|~r|3(~r × ~r) = 0.

That is why the cross product ~r × ~r is really constant. So, it is proved that themotion of a point in so called central-force field is a plane curve.

3.2. Trajectory is an ellipse (1. Kepler law)

We can assume for further calculation that the motion is situated in the planexy and we shall describe this motion by position vector ~r(t) = (x(t), y(t)) and~r(0) = (r0, 0) and ~r(0) = (0, v0) in consistency with initial conditions.

We transform the position vector to the polar coordinates r and ϕ

~r = (x, y) = (r cosϕ, r sinϕ).

We can differentiate twice

~r = (x, y) = r(cosϕ, sinϕ) + rϕ(− sinϕ, cosϕ),

~r = (x, y) = (r − rϕ2)(cosϕ, sinϕ) + (2rϕ+ rϕ)(− sinϕ, cosϕ)

and rewrite the initial conditions r(0) = 0 and ϕ(0) =v0

r0

.

This derivatives can be put into (3) and we obtain

(r − rϕ2 +k

r2)(cosϕ, r sinϕ) + (2rϕ+ rϕ)(− sinϕ, cosϕ) = 0.

396 f. mosna

From here, we have the equations

r − rϕ2 +k

r2= 0 and 2rϕ+ rϕ = 0. (4)

We multiply both sides of the second part of (4) by r and get

0 = 2rrϕ+ r2ϕ =d

dt(r2ϕ),

and so, r2ϕ is a constant. The initial conditions give us

ϕ =r0v0

r2. (5)

We put this result into the first part of (4)

r − r20v

20

r3+k

r2= 0 (6)

and we deal similarly with this equation (6). We multiply both sides of it by rand we obtain

0 = rr − r20v

20 ·

r

r3+ k · r

r2=

d

dt

(1

2r2 +

1

2r2

0v20 ·

1

r2− k · 1

r

).

This derivative is equal to zero and it again proves that

r2 + r20v

20 ·

1

r2− 2k · 1

r

is a constant. With respect to the initial conditions we get

r2 + r20v

20 ·

1

r2− 2k · 1

r= v2

0 − 2k · 1

r0

. (7)

We can express the derivative r from (7)

r =

√v2

0 − 2k · 1

r0

− r20v

20

r2+ 2k · 1

r. (8)

Now, we would like to express ϕ as a function of r, so we are looking for thefunction g such that ϕ = g(r) (see Figure 2).

The chain rule for derivatives gives us

dϕ

dt(t) =

dg

dr(r(t)) · dr

dt(t) or ϕ(t) =

dϕ

dr(r(t)) · r(t).

We put (5) and (8) into this relation and we obtain

r0v0

r2=dϕ

dr·

√v2

0 − 2k · 1

r0

− r20v

20

r2+ 2k · 1

r. (9)


Figure 2: Functions

We can rewrite the equation (9) into form

dϕ

dr=

r0v0

r√Ar2 +Br + C

, (10)

where

A = v20 − 2k · 1

r0

, B = 2k , C = −r20v

20 < 0

and the discriminantD = 4

(k − r0v

20

)2> 0.

We shall solve (10) by integration

ϕ =r0v0√−C

arcsinBr + 2C

r√D

+ c1 = arcsin2kr − 2r2

0v20

2r (k − r0v20)

+ c1.

The constant c1 = −π2

can be got from initial condition, hence

ϕ = arcsinr − 1

kr2

0v20

r(1− 1

kr0v2

0

) − π

2.

We continue the calculation

cosϕ = sin(ϕ+π

2) =

r − 1kr2

0v20

r(1− 1

kr0v2

0

)and

r =1kr2

0v20

1−(1− 1

kr0v2

0

)cosϕ

. (11)

This is an equation of conic section

r =p

1 + ε cosϕ,

where

p =1

kr2

0v20 ≥ 0 and ε =

1

kr0v

20 − 1 ≥ −1.

398 f. mosna

Parameter ε = 0 corresponds to the circle (and to the formula about the first

cosmic speed v1 =√

κMr0

), parameter ε = 1 corresponds to the parabola (and

to the relations concerning the second cosmic speed v2 =√

2κMr0

). The elliptic

trajectory remains for |ε| < 1, i.e., v0 < v2. So we got the 1. Kepler law fromhere.

3.3. Area velocity is constant (2. Kepler’s law)

In order to derive the 2. Kepler law, we must express the area, which is sweptout by radius-vector of point mass m between time t1 and t2 in polar coordinates,∆t = t2 − t1. We use again the expression of r by ϕ, i.e., r = h(ϕ), function h isinverse to the function g (see Figure 2).

Figure 3: Area of radius-vector

Let us calculate the area of ∆Ω. We use step by step the translation to thepolar coordinates (see Figure 3), the Fubini theorem and the substitution ϕ = ϕ(t)

|∆Ω| =

∫∫∆Ω

1 dxdy =

∫∫D

r drdϕ

=

∫ ϕ2

ϕ1

(∫ h(ϕ)

0

r dr

)dϕ =

∫ ϕ2

ϕ1

1

2h2(ϕ) dϕ

=

∫ t2

t1

1

2h2(ϕ(t)) · ϕ(t) dt =

∫ t2

t1

1

2r2(t) · ϕ(t) dt.

Now, we use relation (5) and get

|∆Ω| =∫ t2

t1

1

2v0r0 dt =

1

2v0r0(t2 − t1) =

1

2v0r0∆t, (12)

which gives us the 2. Kepler law about the constant area speed of radius-vectorof motion.


3.4. Relation between period and semi-major axes (3. Kepler’s law)

The 3. Kepler’s law is a simple consequence of obtained results. We shall use (12)for t1 = 0 and t2 = T , where T is a period of motion of point mass m, and we cancompare it with formula of ellipse area with semi-axes a and b

πab =1

2r0v0T. (13)

The relationsa =

p

1− ε2and b = a

√1− ε2 (14)

hold for parameters p, ε used for expression of ellipse in polar coordinates andsemi-axes a, b used in cartesian coordinates. From (14) we have b =

√ap and (13)

gives us πa√ap =

1

2r0v0T . Because p = 1

kv2

0r20, we have

πa

√a

kr0v0 =

1

2r0v0T

and from here we obtainT 2

a3=

4π2

k,

which gives the 3. Kepler’s law.

3.5. Mathematical apparatus

Finally, we summarize the mathematical apparatus used in our deductions:

1. properties of vectors, the dot (scalar, inner) product and cross (vector) pro-duct;

2. equation of ellipsex2

a2+y2

b2= 1 and formula of ellipse area P = abπ, where

a and b are semi-axes;

3. polar coordinates x = r cosϕ, y = r sinϕ;

4. equation of conic section in polar coordinates r(1 + ε cosϕ) = p, where pis parameter and ε is angular eccentricity, conic section is a hyperbola for|ε| > 1, parabola for |ε| = 1, ellipse for |ε| < 1, (circle for |ε| = 0); relationsb2 = a2(1− ε2), p = a(1− ε2), where a, b are semi-axes of ellipse;

5. chain rule for derivative, derivative of cross product;

6. integral ∫dx

x√Ax2 +Bx+ C

=1√−C

arcsinBx+ 2C

x√D

+ c1 ,

where C < 0, D = B2 − 4AC > 0;

400 f. mosna

7. double integral, its property, Fubini theorem and substitution theorem;

8. formula of ellipse area P = abπ.

References

[1] Arnold, V.I., Mathematical Methods of Classical Mechanics, Springer, NewYork, 1989.

[2] Sommerfeld, A., Mechanics, Academic Press, New York, 1952.

[3] Goodstein, D.L., Goodstein, J.R., Feynman’s lost lecture, Norton, NewYork, 1996, 1999.



HOPF MODULES IN THE WEAK YETTER-DRINFELDCATEGORIES

Yin Yanmin

Department of MathematicsShandong Jianzhu UniversityJinan, Shandong, 250101China

Abstract. Suppose that L is a weak Hopf algebra over the field k with a bijective

antipode and H is a weak Hopf algebra in the weak Yetter-Drinfeld category LLYD.

We prove that the fundamental theorem for right H-Hopf modules in LLYD.

Keywords: Weak Hopf algebra, Yetter-Drinfeld module, Fundamental theorem.

AMS 2010 Subject Classification: 16W30, O153.3.

1. Introduction

Weak Hopf algebras have been proposed by G. Bohm, F. Nill and K. Szlachanyias a generalization of ordinary Hopf algebras in the following sense: the definingaxioms are the same, but the multiplicativity of the counit and the comultiplica-tivity of the unit are replaced by weaker axioms. The initial motivation to studyweak Hopf algebras is their connection with the theory of algebra extension [1],and another important application of weak Hopf algebras is that they provide anatural framework for the study of dynamical twists in Hopf algebras [2].

Just like finite-dimensional Hopf algebras, finite-dimensional weak Hopf al-gebra also obey the mathematical beauty of giving rise to a self-dual notion: thedual of it can be canonically endowed with a weak Hopf algebra structure. Thenotion of a weak Yetter-Drinfeld category L

LYD over a weak Hopf algebra L hasbeen introduced by Bohm in [3], and further studied by Caenepeel in [4]. The pa-per [5] proves that if H is a finite-dimensional weak Hopf algebra in the categoryLLYD over a weak Hopf algebra L, then its linear dual H∗ is also a weak Hopfalgebra in L

LYD.In this paper, we prove the fundamental theorem for right H-Hopf modules

in LLYD. We also show that if H is dimensional, its dual H∗ has a right H-Hopf

module structure which is not analogous to usual one.

402 yin yanmin

2. Preliminaries

A weak Hopf algebra is a vector space L with the structure of an associativeunital algebra (L,m, µ) with multiplication m : L⊗ L −→ L and unit 1 ∈ L anda coassociative coalgebra (L,∆, ε) with comultiplication ∆ : L −→ L ⊗ L andcounit ε : L −→ k such that

(i) The comultiplication ∆ is a (not necessarily unit-preserving) homomorphismof algebras such that

(2.1) (∆⊗ id)∆(1) = (∆(1)⊗ 1)(1⊗∆(1)) = (1⊗∆(1))(∆(1)⊗ 1).

(ii) The counit satisfies the following identity

(2.2) ε(kgl) = ε(kg1)ε(g2l) = ε(kg2)ε(g1l), ∀k, g, l ∈ L.

(iii) There is a linear map SL : L −→ L called an antipode, such that, for alll ∈ L,

(2.3) m(id⊗ SL)∆(l) = (ε⊗ id)(∆(1)(l ⊗ 1)),

(2.4) m(SL ⊗ id)∆(l) = (id⊗ ε)((1⊗ l)∆(1)),

(2.5) SL(l) = SL(l1)l2SL(l3).

The linear map defined in (2.3) and (2.4) are called target and source counitalmaps and denoted by εt and εs respectively:

(2.6)εt(l) = ε(1(1)l)1(2) = ε(SL(l)1(1))1(2),

εs(l) = 1(1)ε(l1(2)) = 1(1)ε(1(2)SL(l)).

For all l ∈ L, we have

(2.7)l1 ⊗ εt(l2) = 1(1)l ⊗ 1(2),

εs(l1)⊗ l2 = 1(1) ⊗ l1(2),

(2.8)l1 ⊗ εs(l2) = l1(1) ⊗ SL(1(2)),

εt(l1)⊗ l2 = SL(1(1))⊗ 1(2)l.

We will briefly recall the necessary definitions and notions on the weak Hopfalgebras.

Definition 2.1. An algebra H is a left L-comodule algebra if H is a left L-comodule via x 7−→ σH(x) = x−1 ⊗ x0 such that

(1) σH(xy) = σH(x)σH(y) = x−1y−1 ⊗ x0y0,

(2) 1−1 ⊗ x10 = εs(x−1)⊗ x0 ∀x ∈ H.

hopf modules in the weak yetter-drinfeld categories 403

Applying (2), we have the following equality

1−1 ⊗ 10 = εs(1−1)⊗ 10 = SL(1(2))⊗ 1(1) −→ 1,

= SL(1(2))⊗ εt(1(1)) −→ 1,

= 1(1) ⊗ 1(2) −→ 1

Definition 2.2. An algebra H is a left L-module algebra if H is a left L-modulevia l ⊗ x 7−→ l −→ x such that

(1) l −→ xy = (l1 −→ x)(l2 −→ y),

(2) l −→ 1 = εt(l) −→ 1, ∀x, y ∈ H, l ∈ L,

the second equality is equivalent to εt(l) −→ x = (l −→ 1)x.

Definition 2.3. An algebra H is a left L-module coalgebra if H is a left L-modulevia l ⊗ x 7−→ l −→ x such that

(1) ∆(l −→ x) = (l −→ x)1 ⊗ (l −→ x)2 = (l1 −→ x1)⊗ (l2 −→ x2),

(2) εs(l) −→ x = x1ε(l −→ x2), ∀l ∈ L, x ∈ H,

the second equation is equivalent to

ε(lk −→ h) = ε(lk2)ε(k1 −→ h), ε(εs(l) −→ h) = ε(l −→ h), l, k ∈ L, h ∈ H.

Definition 2.4. An algebra H is a left L-comodule coalgebra if H is a leftL-comodule via x 7−→ σH(x) = x−1 ⊗ x0 such that

(1) x−1 ⊗ (x0)1 ⊗ (x0)2 = x−11 x−1

2 ⊗ x01 ⊗ x0

2,

(2) ε(x0)x−1 = ε(x0)εt(x−1) ∀x ∈ H.

3. Weak Hopf algebras in weak Yetter-Drinfeld category

Let L be a weak Hopf algebra with a bijective antipode SL. We recall that the weakYetter-Drinfeld category L

LYD is the braided monoidal categories whose objectsV are both left L-comodules and satisfy the following conditions:

(1) σV (v) = v−1 ⊗ v0 ∈ L⊗t V = 1(1)l ⊗ 1(2) −→ v|∀l ∈ L, v ∈ V ,(2) l1v

−1 ⊗ l2 −→ v0 = (l1 −→ v)−1l2 ⊗ (l1 −→ v)0i.e.,

(3) σV (l −→ v) = (l −→ v)−1 ⊗ (l −→ v)0 = l1v−1SL(l3)⊗ l2 −→ v0,

where the L-module action is denoted by l −→ v for l ∈ L, v ∈ V and the L-comodule structure map is denoted by σV : V −→ L⊗ V . We use the followingnotation:

σV (v) = v−1 ⊗ v0, (∆⊗ id)σV (v) = (id⊗ σV )σV (v) = v−2 ⊗ v−1 ⊗ v0).

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The braiding τ = τV, W : V ⊗t W −→ W ⊗t V in this category is given by

τ(1(1) −→ v ⊗ 1(2) −→ w) = v−1 −→ w ⊗ v0,

τ−1(1(1) −→ w ⊗ 1(2) −→ v) = v0 ⊗ S−1L (v−1) −→ w.

Let V ∈ LLYD. Then, for all v ∈ V , we have

εs(v−1)⊗ v0 = SL(1(2))⊗ 1(1) −→ v,

εt(v−1)⊗ v0 = SL(1(1))⊗ 1(2) −→ v, ∀v ∈ V.

In [5], Shen Bing-liang introduces the definition of weak Hopf algebra in theweak Yetter-Drinfeld category L

LYD. Moreover, they have showed that if H is afinite-dimensional weak Hopf algebra in L

LYD, then its dual H∗ is a weak Hopfalgebra in L

LYD.

Definition 3.1. Let L be a weak Hopf algebra with a bijective antipode SL. Anobject H ∈ L

LYD is called a weak bialgebra in this category if it is both an algebraand a coalgebra satisfying the following conditions:

(1) ∆(xy) = x1(x−12 −→ y1)⊗ x0

2y2,

ε(xyz) = ε(xy1)ε(y2z) = ε(x(y−11 −→ y2))ε(y

01z),

∆2(1) = 11 ⊗ 121′1 ⊗ 1′2 = 11 ⊗ (1−1

2 −→ 1′1)102 ⊗ 1′2.

(2) H is both a left L-module algebra, L-comodule algebra, L-module coal-gebra and L-comodule coalgebra.

(3) there exist an antipode S : H −→ H (here S is both left L-linear andL-colinear i.e., S is a morphism in the category of L

LYD ) satisfying

x1S(x2) = ε((x−1 −→ 11)x0)12,

S(x1)x2 = 11ε((1−12 −→ x)102),

S(x1)x2S(x3) = S(x), ∀x ∈ H.

Similar to the notation of weak Hopf algebra, we denote

εt(x) = ε((x−1 −→ 11)x0)12, εs(x) = 11ε((1

−12 −→ x)102).

According to the definitions of εt, εs we can obtain explicit expressions forthese coproducts

∆(εt(x)) = εt(x)11 ⊗ 12, ∆(εs(x)) = 11 ⊗ 12εs(x).

Furthermore, for x ∈ H,

ε(εt(x)) = ε((x−1 −→ 1)x0),

= ε((εt(x−1) −→ 1)x0),

= ε((SL(1(1)) −→ 1)(1(2) −→ x)),

= ε((εt(1(1)) −→ 1)(1(2) −→ x)),

= ε((1(1) −→ 1)(1(2) −→ x)),

= ε(x),


in a similar way we can compute ε(εs(x)) = ε(x). Applying Definition 3.1, oneobtains immediately the following identities

ε(xεt(y)) = ε(xy1S(y2)) = ε(xy1)ε(y2S(y3)) = ε(xy),

ε(εs(x)y) = ε(S(x1)x2y) = ε(S(x1)x2)ε(x3y) = ε(xy).

As S is both left L-linear and L-colinear, we can easily check that εt and εsare also both left L-linear and L-colinear. Moreover it is both an anti-algebramap and an anti-coalgebra map, that is

Sm = mτH,H(S ⊗ S), i.e., S(xy) = (x−1 −→ S(y))S(x0), x, y ∈ H,

∆S = (S ⊗ S)τH,H∆, i.e.,∆(S(x)) = (x−11 −→ S(x2))⊗ S(x0

1).

In this paper, we will always assume that the antipode S is bijective. Thecomposite-inverse S−1 satisfies

S−1m = m(S−1 ⊗ S−1)τ−1, i.e.,

S−1(xy) = S−1(y0)(S−1L (y−1) −→ S−1(x)),

∆S−1 = (S−1 ⊗ S−1)τ−1∆, i.e.,

∆(S−1(x)) = S−1(x02)⊗ S−1

L (x−12 ) −→ S−1(x1).

Proposition 3.2. Suppose H is a weak Hopf algebra in LLYD, the following iden-

tities holdεt S = S εS, εs S = S εt.

Proof. For x ∈ H we have

εt S(x) = [S(x)]1S([S(x)]2) = (x−11 −→ S(x2))S(S(x

01)),

= S(S(x1)x2) = S εS(x).

In a similar way, one can verify εs S = S εt.

As a preparation for the theorem below, we notice that Proposition 3.2 hascounterparts involving the antipode,

εt(x) = ε(S(x)11)12, εs(x) = 11ε(12S(x)).

As a matter of fact,

εt(x) = ε(εt(x)11)12 = ε(x1S(x2)11)12 = ε(εs(x1)S(x2)11)12 = ε(S(x)11)12.

The second equality can be proven analogously. Applying Proposition 3.2, onecan obtain

εs(x) = (S εt S−1)(x) = S(1(2)ε(S(S−1(x))1(1))) = S(1(2))ε(x1(1)),

εt(x) = (S εs S−1)(x) = S(1(1)ε(1(2)S(S−1(x))) = S(1(1))ε(1(2)x).

406 yin yanmin

In a similar way, we can verify the first equation.

Proposition 3.3. Suppose H is a weak Hopf algebra in LLYD. For all x ∈ H we

have the identities

x1 ⊗ εs(x2) = x11 ⊗ S(12), εt(x1)⊗ x2 = S(11)⊗ 12x.

Proof. Using the above definitions, one obtains

x1 ⊗ εs(x2) = x1 ⊗ S(12)ε(x211),

= x1(x−12 −→ 11′)ε(x

0212′11)S(12),

= x1(x−12 −→ 11)ε(x

0212)S(13),

= (h11)1ε((h11)2)⊗ S(12),

= h11 ⊗ S(12),

εt(x1)⊗ x2 = S(11)ε(12x1)⊗ x2,

= S(11)ε(1211′(1−12′ −→ x1))⊗ 102′x2,

= S(11)ε(12(1−13 −→ x1))⊗ 103x2,

= S(11)⊗ ε((12x)1)(12x)2,

= S(11)⊗ 12x.

4. Hopf modules in the Yetter-Drinfeld categories

Since a weak Hopf algebra H in the weak Yetter-Drinfeld categories LLYD is both

algebra and coalgebra, one can consider modules and comodules over H. As in thetheory of Hopf algebras, an H-Hopf module is an H-module which is also an H-comodule such that these two structures are compatible (the action ”commutes”with coaction):

Definition 4.1. Let H be a weak Hopf algebra in LLYD. A right H-Hopf module

M in LLYD such that it is both a right H-module and a right H-comodule via

ρM : M −→ M ⊗H, ρM(m) = m0 ⊗m1 and the following equations hold:

(1) ρM(mh) = m0(m−11 −→ h1)⊗m0

1h2,m ∈ M, h ∈ H,

(2) σM(mh) = m−1h−1 ⊗m0h0,m ∈ M, h ∈ H,

(3) m−1 ⊗ (m0)0 ⊗ (m0)1 = m−10 m−1

1 ⊗m00 ⊗m0

1m ∈ M,

(4) l −→ (mh) = (l1 −→ m)(l2 −→ h), l ∈ L, m ∈ M, h ∈ H,

(5) ρM(l −→ m) = (l1 −→ m0)(l2 −→ m1), l ∈ L, m ∈ M.

We remark that M⊗tH is a right H-module by (m⊗h)x = m(h−1 −→ x1)⊗h0x2

and a right H-comodule ρM⊗H(m⊗h) = m0⊗m−11 −→ h1⊗m0

1h2. The condition(1) means that the H-comodule structure ρM : M −→ M ⊗ H is H-linear, orequivalently the H-module structure map φM : M ⊗ H −→ M is H-colinear.


Also, (2) (resp. (4))⇐⇒ φM is L-colinear (resp. L-linear); (3)(resp. (5))⇐⇒ ρMis L-colinear (resp. L-linear).

Example 1. H itself is a right H-Hopf module (in LLYD) in the natural way. If V

is an object in LLYD, then so is V ⊗t H by l −→ (v ⊗ h) = (l1 −→ v)⊗ (l2 −→ h)

and σV⊗H(v ⊗ h) = v−1h−1 ⊗ v0h0. It is also both a right H-module and a rightH-comodule by (v ⊗ h)x = v ⊗ hx and ρV⊗H(v ⊗ h) = v ⊗ h1 ⊗ h2. One easilychecks that V ⊗t H is an H-Hopf module.

Lemma 4.2. If H is a weak Hopf algebra in LLYD and M a right H-Hopf module

in LLYD, we define M coH = m ∈ M |ρM(m) = m11 ⊗ 12 is a L-submodule, then

(1) M coH is a L-submodule.

(2) M coH is a L-subcomodule of M , so M coH ∈ LLYD.

Proof. (1) Let n ∈ M coH , then

ρM(l −→ n) = (l1 −→ n11)⊗ (l2 −→ 12),

= (l1 −→ n)(l2 −→ 11)⊗ (l3 −→ 12),

= [(l1 −→ n)⊗ 1]∆(εt(l2) −→ 1),

= ((1(1)l −→ n)⊗ 1)∆(1(2) −→ 1),

= [1(1) −→ (l −→ n)][1(2) −→ (1(1′) −→ 11)]⊗ 1(2′) −→ 12,

= (l −→ n)11 ⊗ 12.

Hence l −→ n ∈ M coH .

(2) Applying 1−1 ⊗ 10 = 1(1) ⊗ (1(2) −→ 1) and εt(l) −→ x = (l −→ 1H)x,we obtain

1−1 ⊗ (10)1 ⊗ (10)2 = 1(1) ⊗ (1(2) −→ 11)⊗ 12,

= 1(1) ⊗ (εt(1(2)) −→ 11)⊗ 12,

= 1(1) ⊗ (1(2) −→ 1)11 ⊗ 12,

= 1−1 ⊗ 1011 ⊗ 12.

For n ∈ M coH we do a calculation:

n−1 ⊗ (n0)0 ⊗ (n0)1 = n−10 n−1

1 ⊗ n00 ⊗ n0

1,

= n−11−11 1−1

2 ⊗ n0101 ⊗ 102,

= n−11−1 ⊗ n0(10)1 ⊗ (10)2,

= n−11−1 ⊗ n01011 ⊗ 12,

= n−1 ⊗ n011 ⊗ 12.

This implies that n−1 ⊗ n0 ∈ L⊗M coH , so M coH ∈ LLYD.

Theorem 4.3. If H is a weak Hopf algebra in LLYD and M a right H-Hopf module

in LLYD, M coH is defined as above. Then

408 yin yanmin

(1) Let P (m) = m0S(m1), m ∈ M , then P (m) ∈ M coH . If n ∈ M coH andh ∈ H, then ρM(nh) = nh1 ⊗ h2 and P (nh) = nεt(h).

(2) The map F : M coH ⊗t H −→ M, F (n⊗h) = nh is an isomorphism of Hopfmodules, the inverse map is given by G(m) = P (m0)m1.

Proof. (1) Since εt is a left L-comodule map we have σ(h1S(h2)) = h−1 ⊗ εt(h0).

Applying x1 ⊗ εs(x2) = x11 ⊗ S(12), we obtain

ρM(P (m)) = m0(m−11 m−1

2 −→ S(m3))⊗m01S(m

02),

= m0(m−11 −→ S(m2))⊗ εt(m

01),

= m0[S(m1)]1 ⊗ S−1 εs([S(m1)]2),

= m0S(m1)11 ⊗ S−1(S(12)),

= m0S(m1)11 ⊗ 12.

If n ∈ M coH and h ∈ H, then

ρ(nh) = n11(1−12 −→ h1)⊗ 102h2 = nh1 ⊗ h2,

P (nh) = nh1S(h2) = nεt(h).

(2) Since

F (l −→ (n⊗ h)) = F ((l1 −→ n)⊗ (l2 −→ h)),

= (l1 −→ n)(l2 −→ h),

= l −→ nh,

= l −→ F (n⊗ h),

then F is a left H-linear map. F is also left L-colinear by the following equality

σ(F (n⊗ h)) = σ(nh) = n1(1)h−1 ⊗ 1(2)h

0 = (id⊗ F )σ(n⊗ h),

clearly F is right H-linear. It is also right H-colinear by (1). Now we have

GF (n⊗ h) = P (nh1)⊗ h2 = nεt(h1)⊗ h2,

= n⊗ εt(h1)h2 = n⊗ S(11)12h,

= n⊗ h,

FG(m) = m0S(m1)m2 = m0εs(m1),

= [m0εs(m1)]0ε([m0εs(m1)]1),

= m0(m−11 −→ 11)ε(m

0112εs(m2)),

= m0ε(m1εs(m2)),

= m0ε(m111S(12)),

= m0ε(m1),

= m.


Example. Let H be a weak Hopf algebra in LLYD. M = H is defined as a right

H-Hopf module by ∆. Then MCoH = εt(H)|h ∈ H.

5. Application

By [5] we make H∗ into a weak Hopf algebra in LLYD. H∗ has the contragredient

left L-module structure, ie.,

(l −→ f)(h) = f(SL(l) −→ h), l ∈ L, f ∈ H∗, h ∈ H.

Also, since H is a finite-dimensional left L-comodule, H∗ has the transposed rightL-comodule structure and so it becomes a left L-comodule via σH∗ : H∗ −→L⊗H∗, σH∗(f) = f−1 ⊗ f 0, where

f 0(h)f−1 = f(h0)S−1L (h−1), h ∈ H.

Now assume that H is finite-dimensional, we will show that H∗ becomes a rightH-Hopf module in L

LYD. First, H∗ is a right H-module by

(fh)(x) = f(hx), f ∈ H∗, h ∈ H.

Second, H∗ is a right H-comodule using the identification θH : H∗ ⊗ H ∼=Hom(H,H), θH(f ⊗ h)(l) = f(h−1 −→ l)h0 as follows:

ρH∗ : H∗ −→ H∗ ⊗H ∼= Hom(H, H), ρH∗(f)(x) = f(x1)S(x2).

That is, ρH∗(f) = f0 ⊗ f1 means

(5.1) f(x1)S(x2) = ρH∗(f)(x) = θH(f0 ⊗ f1)(x) = f0(f−11 −→ x)f 0

1 , x ∈ H.

Proposition 5.1. H∗ is a right H-comodule by θH⊗H .

Proof. First, we check that H-comodule via above using θH⊗H . Applying σH to(5.1) we obtain

f−11 ⊗ f0(f

−21 −→ x)f 0

1 = x−12 ⊗ f(x1)S(x

02). (5.2)

Now for f ∈ H∗, x ∈ H, we have

θH⊗H((f0)0 ⊗ (f0)1 ⊗ f1)(x) = (f0)0((f0)−11 f−1

1 −→ x)(f0)01 ⊗ f 0

1 ,

= f0((f−11 −→ x)1)S((f

−11 −→ x)2)⊗ f 0

1 ,

= f0(f−21 −→ x1)(f

−11 −→ S(x2))⊗ f 0

1 ,

= (f−11 −→ S(x2))⊗ f0(f

−21 −→ x1)f

01

= (x−12 −→ S(x3))⊗ f(x1)S(x

02),

= ∆H(f(x1)S(x2)),

= ∆H(f0(f−11 −→ x)f 0

1 ),

= f0(f−11 −→ x)(f 0

1 )1 ⊗ (f 01 )2,

= f0((f1)−11 (f1)

−12 −→ x)(f1)

01 ⊗ (f1)

02,

= θH⊗H(f0 ⊗ (f1)1 ⊗ (f1)2)(x).

410 yin yanmin

It implies that (f0)0 ⊗ (f0)1 ⊗ f1 = f0 ⊗ (f1)1 ⊗ (f1)2.According to σV (l −→ v) = l1v

−1SL(l3)⊗l2 −→ v0, we calculate the followingequality

(1(1) −→ f0)((1(2) −→ f1)−1 −→ x)(1(2) −→ f1)

0

= f0((SL(1(1))1(2)f−11 SL(1(4))) −→ x)(1(3) −→ f 0

1 ),

= f0((εs(1(1))f−11 SL(1(3))) −→ x)(1(2) −→ f 0

1 ),

= f0((1(1′)f−11 SL(1(2))) −→ x)(1(1)1(2′) −→ f 0

1 ),

= f0((1(1)f−11 SL(1(2))) −→ x)(1(3) −→ f 0

1 ),

= f0((1 −→ f1)−1 −→ x)(1 −→ f1)

0,

= f0(f−11 −→ x)f 0

1 ,

so we obtain (1(1) −→ f0)⊗ (1(2) −→ f1) = f0 ⊗ f1.Applying ε to (5.1), we obtain

f(x) = f0(f−11 −→ x)ε(f 0

1 ),

= f0(εt(f−11 )ε(f 0

1 ) −→ x),

= f0(SL(1(1) −→ x))ε(1(2) −→ f1),

= (1(1) −→ f0)(x)ε(1(2) −→ f1),

= f0(x)ε(f1),

= (f0ε(f1))(x).

Hence (id⊗ ε)ρH∗(f) = f , thus H∗ becomes a right H-comodule.

Theorem 5.2. With the notation as above, then H∗ is a right H-Hopf module inLLYD. Moreover, (H∗)coH = f ∈ H∗|f(x1)x2 = f(11x)12, x ∈ H.

Proof. Now, we prove that H∗ is a right H-Hopf module. First, we will showthat (fh)0 ⊗ (fh)1 = f0(f

−11 −→ h1)⊗ f 0

1h2, since for x ∈ H,

θH (f0(f−11 −→ h1)⊗ f 0

1h2)(x)

= (f0(f−11 −→ h1))((f

01h2)

−1 −→ x)((f 01h2)

0),

= (f0(f−21 −→ h1))(f

−11 h−1

2 −→ x)f 01h

02,

= f0((f−21 −→ h1)(f

−11 h−1

2 −→ x))f 01h

02,

= f0(f−11 −→ (h1(h

−12 −→ x))f 0

1h02,

= f((h1(h−12 −→ x))1)S((h1(h

−12 −→ x))2)h

02,

= f(h1(h−22 h−2

3 −→ x1))S(h−12 h−1

3 −→ x2)S(h02)h

03,

= f(h1(h−22 −→ x1))S(h

−12 −→ x2)εs(h

02),

= f(h1((εs(h2))−2 −→ x1))S((εs(h2))

−1 −→ x2)(εs(h2))0,

= f(h11(1−22 −→ x1))(1

−12 −→ S(x2))S(1

02),

= f(h11(1−12 −→ x1))S(1

02x2),

= (f · h)(11(1−12 −→ x1))S(1

02x2),

= (f · h)(x1)S(x2),

= [(f · h)0 ⊗ [(f · h)1](x).


To verify that σH∗(fh) = f−1h−1 ⊗ f 0h0 for f ∈ H∗, h ∈ H, we compute forx ∈ H

f−1h−1f 0h0(x) = f−1h−1f 0(h0x),

= f(h0x0)S−1L (h−1x−1)h−2,

= f((1(1) −→ h)x0)S−1L (x−1)1(2)

= f((1(1′)1(1) −→ h)(1(2′) −→ x0))S−1L (x−1)1(2),

= f((1(1) −→ h)(εt(1(2)) −→ x0))S−1L (x−1)1(3),

= f((1(1) −→ h)(1(2) −→ 1)x0)S−1L (x−1)1(3),

= f((1(1′) −→ h)(1(2′) −→ (1(1) −→ 1))x0)S−1L (x−1)1(2),

= f(h(1(1) −→ 1)x0)S−1L (x−1)1(2),

= f(h(εt(1(1)) −→ x0))S−1L (x−1)1(2),

= f(h(S(1(1)) −→ x0))S−1L (x−1)1(2),

= f(h(S(S−1(1(2))) −→ x0)S−1L (x−1)S−1

L (1(1)),

= f(h(1(2) −→ x0))S−1L (1(1)x

−1),

= f(hx0)S−1L (x−1),

= (fh)(x0)S−1L (x−1),

= (fh)−1(fh)0(x).

Next, we want to check (l −→ fh) = (l1 −→ f)(l2 −→ h) for l ∈ L, h ∈ H,f ∈ H∗, since for x ∈ H

((l1 −→ f)(l2 −→ h))(x) = (l1 −→ f)((l2 −→ h)x),

= f(SL(l1) −→ ((l2 −→ h)x)),

= f((SL(l2)l3 −→ h)(SL(l1) −→ x)),

= f((εs(l2) −→ h)(SL(l1) −→ x)),

= f((SL(1(2)) −→ h)(SL(l1(1)) −→ x)),

= f((1(1) −→ h)(1(2) −→ (SL(l) −→ x))),

= f(h(SL(l) −→ x)),

= (fh)(SL(l) −→ x),

= (l −→ fh)(x).

We have f−1 ⊗ (f 0)0 ⊗ (f 0)1 = f−10 f−1

1 ⊗ f 00 ⊗ f 0

1 for f ∈ H∗, since for x ∈ H

f−10 f−1

1 ⊗ θH(f00 ⊗ f 0

1 )(x)

= f−10 f−2

1 ⊗ f 00 (f

−11 −→ x)f 0

1 ,

= f0((f−11 −→ x)0)S−1

L ((f−11 −→ x)−1)f−2

1 ⊗ f 01 ,

= f0(f−21 −→ x0)S−1

L (f−31 x−1S(f−1

1 ))f−41 ⊗ f 0

1 ,

= f0(f−21 −→ x0)f−1

1 S−1L (x−1)εt(f

−31 )⊗ f 0

1 ,

= f0((f−21 1(2) −→ x0)f−1

1 S−1L (1(1)x

−1)⊗ f 01 ,

= f0(f−21 −→ x0)f−1

1 S−1L (x−1)⊗ f 0

1 ,

412 yin yanmin

= f−11 S−1

L (x−1)⊗ f 01 f0(f

−21 −→ x0),

= (x0)−12 S−1

L (x−1)⊗ f((x0)1)S(((x0)2)

0),

= x−12 S−1

L (x−11 x−2

2 )⊗ f(x01)S(x

02),

= S−1L (x−1

1 εt(x−12 ))⊗ f(x0

1)S(x02),

= S−1L (x−1

1 SL(1(1)))⊗ f(x01)S(1(2) −→ x2),

= S−1L ((1(1) −→ x1)

−1)⊗ f((1(1) −→ x1)0)S(1(2) −→ x2),

= S−1L (x−1

1 )f(x01)⊗ S(x2),

= f 0(x1)f−1 ⊗ S(x2),

= f−1 ⊗ f 0(x1)S(x2),

= f−1 ⊗ (f 0)0((f0)−1

1 −→ x)(f 01 )

01,

= f−1 ⊗ θH((f0)0 ⊗ (f 0)1)(x).

Finally, we show that ρH∗(l −→ f) = l1 −→ f0 ⊗ l2 −→ f1. Since forl ∈ L, f ∈ H∗, x ∈ H

θH ((l1 −→ f0)⊗ (l2 −→ f1))(x)

= (l1 −→ f0)((l2 −→ f1)−1 −→ x)(l2 −→ f1)

0,

= (l1 −→ f0)(l2f−11 SL(l4) −→ x)(l3 −→ f 0

1 ),

= f0(SL(l1)l2f−11 SL(l4) −→ x)(l3 −→ f 0

1 ),

= f0(εs(l1)f−11 SL(l3) −→ x)(l2 −→ f 0

1 ),

= f0(1(1)f−11 SL(l2) −→ x)(l11(2) −→ f 0

1 ),

= f0(f−11 −→ (SL(l2) −→ x))(l1 −→ f 0

1 ),

= f((SL(l2) −→ x)1)(l1 −→ S((SL(l2) −→ x)2)),

= f(SL(l3) −→ x1)(l1 −→ S(SL(l2) −→ x2)),

= f(SL(l3) −→ x1)(l1SL(l2) −→ S(x2)),

= f(SL(l2) −→ x1)(εt(l1) −→ S(x2)),

= F (SL(1(2)l) −→ x1)(SL(1(1)) −→ S(x2)),

= f(SL(l) −→ (1(1) −→ x1))S(1(2) −→ x2),

= f(SL(l) −→ x1)S(x2),

= (l −→ f)(x1)S(x2),

= θH((l −→ f)0 ⊗ (l −→ f)1)(x).

From all above, H∗ is a right H-Hopf module in LLYD.

Applying Theorem 4.3, we can obtain the following result.

Corollary 5.3. H∗ is defined a right H-Hopf module in LLYD as above, then

H∗CoH ⊗t H ∼= H∗.

Corollary 5.4. Let L be a finite-dimensional Hopf algebra. Assume that H is afinite-dimensional Hopf algebra in the Yetter-Drinfeld category in L

LYD. H∗ isdefined a right H-Hopf module in L

LYD as above, then H∗CoH = Hom(H, k) =f ∈ H∗|f(x1)x2 = f(x)1H , x ∈ H.


Example. Recall that a groupoid G is a category in which every morphism is anisomorphism. In this section, we consider finite groupoids, i.e., groupois which afinite number of objects. The set of objects of G will be denoted by G0, and theset of morphisms by G1. The identity morphism on x ∈ G0 will also be denotedby x. For σ : x −→ y in G1, we write s(σ) = x and t(σ) = y, respectively for thesource and the target of σ. For every x ∈ G, Gx = σ ∈ G1|s(σ) = t(σ) = x is agroup.

Let G be a groupoid, and k a commutative ring. The groupoid algebra is

the direct product kG =⊕σ∈G1

kuσ, with multiplication defined by the formula

uσuτ = uστ , if t(τ) = s(σ) (0 if t(τ) = s(σ)). The unit element is 1 =∑x∈G0

ux. kG

is a weak Hopf algebra, with comultiplication, counit and antipode given by theformulas

∆(uσ = uσ ⊗ uσ, ε(uσ) = 1, S(uσ) = uσ−1

Using the formula

∆(1) =∑x∈G0

ux ⊗ ux,

suppose kG is free of finite rank as a k-module, hence (kG)∗ is also a weak Hopf

algebra. As a k-module, (kG)∗ =⊕σ∈G1

kvτ with ⟨vσ, uτ ⟩ = δσ, τ . Suppose (kG)∗

is defined a right H-Hopf module in LLYD as above. Then

((kG)∗)CoH =f ∈ (kG)∗|f(uσ)uσ =∑x∈G0

f(uxuσ)ux,

=f ∈ (kG)∗|f(uσ)uσ =∑x∈G0

f(uσ)ut(σ).

Acknowledgements. Research supported by Project of Shandong ProvinceHigher Educational Science and Technology Program (J12LI07) and the the doc-toral foundation of Shandong Jianzhu University (424111).

References

[1] Kadison, L., Nikshych, D., Frobenius Extensions and Weak Hopf Alge-bras, J. Algebra, 244 (2001), 312-342.

[2] Etingof, P., Nikshych, D., Dynamical Quantum Groups at Roots of 1,Duke Math. J., 108 (2001), 135-168.

[3] Bohm, G., Nill, F., Szlachanyi, K., Weak Hopf Algebras I. IntegralTheory and C∗-structure, J. Algebra, 221 (1999), 385-438.

414 yin yanmin

[4] Caenpeel, S., Wang Ding-duo, Yin Yan-min, Yetter-Drinfeld modulesover Weak Bialgebras, Ann. Univ. Ferrara, Sez. VII-Sc. Mat., 51 (2005), 69-98.

[5] Shen Bing-liang, Wang Shuan-hong,, Weak Hopf Algebra Duality inWeak Yetter-Drinfeld Categories and Applications, Internation ElectronicJournal of Algebra, 6 (2009), 74-94.

[6] Takeuchi, M., Hopf Modules in Yetter-Drinfeld-Drinfeld Categories,Comm. Algebra, 26 (9) (1998), 3057-3070.

[7] Bohm, G., Doi-Hopf Modules over Weak Hopf Algebras, Comm. Algebra, 28(2000), 4687-4689.



RIGHT ALTERNATIVE RINGS WITH x(yz)−y(xz) IN THE CENTER

K. Madhusudhan Reddy

School of Advanced SciencesVIT UniversityVellore, 632014, TamilnaduIndiae-mail: [email protected]

K. Suvarna

Department of MathematicsSri Krishnadevaraya UniversityAnantapur, 515002, Andhra PradeshIndiae-mail: [email protected]

Abstract. In [1] it was proved that if R is a prime right alternative ring of char. = 2, 3

with (R,R,U) ⊆ U or S(x2, x, y) = 0, then either U = C or R is strongly (−1, 1). In this

paper first we prove that if R is a prime right alternative ring with x(yz)− y(xz) ∈ U ,

then (R,R,U) ⊆ U . Using this we prove that either U = C or R is strongly (−1, 1).

Keywords and phrases: Right alternative ring, Char. = n, Strongly (−1, 1) Ring,

Center, Commutative Center.

AMS Subject Classification: 17D15.

1. Introduction

Throughout this paper R represents a right alternative ring of char. = 2. We de-note S(x, y, z) = (x, y, z)+(y, z, x)+(z, x, y). A right alternative ring satisfying theidentity [[R,R], R] = 0 is called a strongly (−1, 1) ring. The set C defined by C =c ∈ N(R)/[c, R] = 0 is called the center of R and Na(R) = v ∈ R/(x, x, v) = 0,∀x ∈ R is called the alternative nucleus of R. A ring R is said to be commutativecenter if U = u ∈ U |[u,R] = 0.

416 k. madhusudhan reddy, k. suvarna

In any right alternative ring we have the following identities:

(y, x, z) + (y, z, x) = 0,(1.1)

[xy, z] = x[y, z] + [x, z]y + 2(x, y, z) + (z, x, y),(1.2)

2S(x, y, z) = [[x, y], z] + [[y, z], x] + [[z, x], y],(1.3)

(z, x2, y) = (z, x, xy + yx),(1.4)

(z, x, xy) = (z, x, y)x,(1.5)

(wx, y, z) + (w, x, [y, z]) = w(x, y, z) + (w, y, z)x,(1.6)

([w, x], y, z)− [w, (x, y, z)] + [x, (w, y, z)] = (x,w, [y, z])−(w, x, [y, z]),(1.7)

((a, y, z), b, c) = ((a, b, c), y, z)− (a, b, (c, y, z))− (a, (b, y, z), c)+(a, b, c)[y, z]− (a, b, c[y, z]) + (a, b, [y, z])c.

(1.8)

For u ∈ U the following identities and relations hold in R.[1]

2[x, (z, y, u)] = [x, (u, y, z)] = (x, [y, z], u),(1.9)

([x, y], [z, w], u) = 0,(1.10)

(R, [R,R], U) ⊆ U,(1.11)

(a, a, (x, y, u)) = 0,(1.12)

((x, y, u), z, w) = 2(w, z, (x, y, u)),(1.13)

3(c, a, (a, y, u)) = ((a, a, c), y, u),(1.14)

3[x, (x, z, (a, b, u))] = [(x, x, z), (a, b, u)].(1.15)

In a right alternative ring of char. = 2, we have (R,R,U) ⊆ U if and only if

(1.16) (R, [R,R], U) = 0,

by (1.9). Also in any right alternative ring of char. = 2

(N(R) ∩ U) = C.(1.17)

First, we prove the following lemmas.

Lemma 1. If R is a right alternative ring with x(yz) − y(xz) ∈ U , then(R,R,U) ⊆ U .

Proof. By hypothesis, x(yz)− y(xz) ∈ U .

(xy)z − (x, y, z) + (y, x, z)− (yx)z ∈ U.

− (x, y, z) + (y, x, z) + [x, y]z ∈ U.(1.18)

right alternative rings with x(yz)− y(xz) in the center 417

By using semi-jacobian identity, we get

−[xz, y] + x[z, y] + (x, z, y) ∈ U.(1.19)

Now, by taking y = u ∈ U and using the definition of U , we get (x, z, u) ∈ U .Therefore, (R,R,U) ⊆ U .

Lemma 2. If R is a right alternative ring with x(yz)− y(xz) ∈ U , then the idealgenerated by (R,R,U) ∈ R is ⟨(R,R,U)⟩ = (R,R,U) +R(R,R,U).

Proof. Since x(yz)− y(xz) ∈ U , from Lemma 1, we have (R,R,U)) ⊆ U.Now, we obviously have (R,R,U)R = R(R,R,U) and

R(R(R,R,U)) ⊆ (R,R, (R,R,U)) +R2(R,R,U) ⊆ (R,R,U) +R(R,R,U).

This and (1.1) gives

(R(R,R,U))R ⊆ (R, (R,R,U), R) +R((R,R,U)R)⊆ (R,R, (R,R,U)) +R(R(R,R,U))⊆ (R,R,U) +R(R,R,U).

This proves Lemma 2.

Lemma 3. If R is a right alternative ring of char. = 2 with x(yz)− y(xz) ∈ U ,then K = x ∈ R/(x,R, U) = x(R,R,U) = 0 is an ideal of R such that

K⟨(R,R,U)⟩ = 0 and [[R,R], R] ⊆ K.

Proof. We let x ∈ K. Using U ⊆ Na(R) and (1.1), we first note that

0 = (x,R, U) = (R, x, U) = (x, U,R).

Since (R,R,U) ⊆ U , we have

(xR)(R,R,U) = x(R(R,R,U)) = x((R,R,U)R)= (x(R,R,U))R = 0

and(Rx)(R,R,U) = R(x(R,R,U)) = 0.

Now, (R,R,U) ⊆ U means 0 = (R, [x,R], U) = ([x,R], R, U) by using (1.16) andU ⊆ Na(R). Then this and (1.6) shows

(xR,R, U) = (Rx,R, U) ⊆ R(x,R, U)+(R,R,U)x+(R, x, [R,U ]) = x(R,R,U) = 0.

Thus it follows that K is an ideal of R. Using this, Lemma 2, and (R,R,U) ⊆ U ,we then see

K⟨(R,R,U)⟩ = K(R,R,U) +R(R,R,U) = (KR)(R,R,U) ⊆ K(R,R,U) = 0.

418 k. madhusudhan reddy, k. suvarna

That isK⟨(R,R,U)⟩ = 0.

Finally, we show that [[R,R], R] ⊆ K.First, by U ⊆ Na(R) and (1.16), we have

([R,R], R], R, U) = −(R, [[R,R], R], U) = 0.

Then, by using (1.2), (R,R,U) ⊆ U , (1.16) and (1.6), we see

[[R,R], R](R,R,U) ⊆ [[R,R](R,R,U), R] + [R,R][(R,R,U), R]+2([R,R], (R,R,U), R) + (R, [R,R], (R,R,U))= [[R,R](R,R,U), R]⊆ [([R,R]R,R,U), R] + [([R,R], R, U)R,R]+[([R,R], R, [R,U ]), R]= [([R,R], R, U)R,R] = 0,

by U ⊆ Na(R) and (1.16).This completes the proof of the lemma.

2. Main result

Theorem 1 If R is a prime right alternative ring of char. = 2 withx(yz)− y(xz) ∈ U , then either U = C or R is strongly (−1, 1).

Proof. From Lemma 1 we have (R,R,U) ⊆ U.Since (R,R,U) ⊆ U , from Lemma 3, we have

K⟨(R,R,U)⟩ = 0 and [[R,R], R] ⊆ K.

Since R is prime, either K = 0 or ⟨(R,R,U)⟩ = 0.If K = 0, then [[R,R], R] = 0.If ⟨(R,R,U)⟩ = 0, then U ⊆ U ∩N(R) = C by (1.17).Thus either U = C or R is strongly (−1, 1).This proves the theorem.

References

[1] Kleinfeld, E., Smith, H.F., On centers and Nuclei in prime right alter-native rings, Comm. Algebra, 22 (3) (1994), 829-855.



HYERS-ULAM STABILITY OF LINEAR DIFFERENTIALEQUATIONS OF SECOND ORDER WITH CONSTANTCOEFFICIENT

Jianming Xue

Oxbridge CollegeKunming University of Science and TechnologyKunming, Yunnan 650106P.R. Chinae-mail: [email protected]

Abstract. Y. Li and Y. Shen [12] have proved the Hyers-Ulam stability of differential

equation y′′(x)+αy′(x)+βy(x) = 0, in the condition that its characteristic equation has

two different positive roots. In this paper, we prove that the differential equation y′′(x)+

αy′(x) + βy(x) = 0 has the Hyers-Ulam stability, no matter whether its characteristic

roots are real or complex. Therefore the results obtained in this paper improve and

extend the ones of [12].

Keywords: Hyers–Ulam stability; differential equations; characteristic equation.

2010 Mathematics Subject Classification: 34K20, 39A10.

1. Introduction

In 1940, Ulam gave a wide ranging talk before the Mathematics Club of theUniversity of Wisconsin, in which he discussed a number of important unsolvedproblems (see [1] ). Among those was the question concerning the stability ofhomomorphisms: Let G1be a group and G2 be a metric group with a metric d(·, ·).Given any δ > 0, does there exist an ε > 0 such that if a function h : G1 → G2

satisfies the inequality

d(h(xy), h(x)h(y)) < ε for all x, y ∈ G1,

then there exists a homomorphism H : G1 → G2 with d(h(x), H(x)) < δ for allx ∈ G1?

The problem for the case of approximately additive mappings was solved byHyers [2] when G1 and G2 are Banach spaces and the result of Hyers was genera-lized by Rassias (see [3]). Since then, the stability problems of functional equa-tions have been extensively investigated by several mathematicians (see [3]-[5]).

420 j. xue

C. Alsina and R. Ger were the first authors who investigated the Hyers-Ulam sta-bility of differential equations. In 1998, they proved in [6] the following: Assumethat a differential equation f : I → R is a solution of the differential inequality

|y′(t)− y(t)| ≤ ε,

where I is an open subinterval of R. Then there exists a solution f0 : I → R ofthe differential equation y′(t) = y(t) such that |f(t)− f0(t)| ≤ ε, for all t ∈ I.

The result of Hyers-Ulam stability for first-order linear differential equationshas been generalized by many researchers (see [7]-[11]). Jung [11] proved thegeneralized Hyers-Ulam stability of differential equations of the form

ty′(t) + αy(t) + βtrx0 = 0

and also applied this result to the investigation of the Hyers-Ulam stability of thedifferential equation

t2y′′(t) + αty′(t) + βy(t) = 0.

Recently, Y. Li and Y. Shen [12] discussed the Hyers-Ulam stability of lineardifferential equations of second order y′′(x)+αy′(x)+βy(x) = 0, in the conditionthat the characteristic equation λ2 + αλ+ β = 0 has two positive roots.

The aim of this paper is to investigate the Hyers-Ulam stability of the fol-lowing linear differential equations of second order.

(1.1) y′′(x) + αy′(x) + βy(x) = 0

and

(1.2) y′′(x) + αy′(x) + βy(x) = f(x),

where y : [a, b] → C is a twice continuously differentiable function, f : [a, b] → Cis continuous function and α, β ∈ R.

First of all, we give the definition of the Hyers-Ulam stability.

Definition 1.1. We say that equation (1.1) has the Hyers-Ulam stability if thereexists a constant K ≥ 0 with the following property, for every ε > 0, if

||y′′(x) + αy′(x) + βy(x)|| ≤ ε,

then there exists some u : [a, b] → C which is a twice continuously differentiablefunction satisfying u′′(x) + αu′(x) + βu(x) = 0 such that

||y(x)− u(x)|| ≤ Kε.

We call such K a Hyers-Ulam stability constant for equation (1.1).

hyers-ulam stability of linear differential equations ... 421

2. Main results

In the following theorem, we prove the Hyers-Ulam stability of the differentialequation (1.1).

Theorem 2.1. Equation (1.1) has the Hyers-Ulam stability, where y : [a, b] → Cis a twice continuously differentiable function.

Proof. Let ε > 0 and y : [a, b] → C be a twice continuously differentiable functionsuch that ||y′′ + αy′ + βy|| ≤ ε.

We will show that there exists a constant K independent of ε and y such that||y − u|| ≤ Kε for some twice continuously differentiable function u : [a, b] → Csatisfying

u′′ + αu′ + βu = 0.

Let λ1 and λ2 be the roots of the characteristic equation

λ2 + αλ+ β = 0.

Define g(x) = y′(x)− λ1y(x), then

||g′(x)− λ2g(x)|| = ||y′′(x)− λ1y′(x)− λ2y

′(x) + λ1λ2y(x)||= ||y′′(x) + αy′(x) + βy(x)|| ≤ ε.

Let Z(x) = e−λ2(x−a)g(x), for each x ∈ [a, b], then

(2.1)

||Z(x)− Z(t)|| = ||e−λ2(x−a)g(x)− e−λ2(t−a)g(t)||

=∥∥∥ x∫

t

d

dv(e−λ2(v−a)g(v))dv

∥∥∥=

∥∥∥ x∫t

e−λ2(v−a)[g′(v)− λ2g(v)]dv∥∥∥

≤ ε∣∣∣ x∫t

e||λ2||(v−a)dv∣∣∣ ≤ e||λ2||(b−a)(b− a)ε,

for any x, t ∈ [a, b].For any x ∈ [a, b], it follows from (2.1) that

||g(x)− eλ2(x−b)g(b)|| = ||eλ2(x−a)(Z(x)− Z(b))|| ≤ e2||λ2||(b−a)(b− a)ε.

Let g1(x) = eλ2(x−b)g(b), then g1(x) satisfies

(2.2) g′1(x)− λ2g1(x) = 0

and||g(x)− g1(x)|| ≤ e2||λ2||(b−a)(b− a)ε.

422 j. xue

Since g(x) = y′(x)− λ1y(x), we have that

||y′(x)− λ1y(x)− g1(x)|| ≤ e2||λ2||(b−a)(b− a)ε.

Let W (x) = e−λ1(x−a)y(x) −x∫a

e−λ1(v−a)g1(v)dv, for each x ∈ [a, b]. By an

argument similar to the above, we can show that there exists

u(x) = eλ1(x−b)y(b)− eλ1(x−a)

b∫x

e−λ1(v−a)g1(v)dv

such that||y − u|| ≤ e2(||λ2||+||λ1||)(b−a)(b− a)2ε

and the twice continuously differentiable function u : [a, b] → C satisfying

(2.3) u′(x)− λ1u(x) = g1(x).

Finally, it follows from (2.2) and (2.3) that u′′ + αu′ + βu = 0. Thus, theproof is completed.

Theorem 2.2. Assume that f : [a, b] → C is continuous function such that f(x)is integrable on [a, c] for each c ∈ [a, b]. If a twice continuously differentiablefunction y : [a, b] → C satisfies the differential inequality

(2.4) ||y′′(x) + αy′(x) + βy(x)− f(x)|| ≤ ε,

for all x ∈ [a, b], then (2.4) has the Hyers-Ulam stability.

Proof. Similar to the proof of Theorem 2.1. Let λ1 and λ2 be the roots ofcharacteristic equation λ2 + αλ+ β = 0.

Define g(x) = y′(x)− λ1y(x), we have

||g′(x)− λ2g(x)− f(x)|| = ||y′′(x)− λ1y′(x)− λ2y

′(x) + λ1λ2y(x)− f(x)||= ||y′′(x) + αy′(x) + βy(x)− f(x)|| ≤ ε.

Let g1(x) = eλ2(x−b)g(b)− eλ2(x−a)b∫x

e−λ2(v−a)f(v)dv, then g1(x) satisfies

g′1(x)− λ2g1(x)− f(x) = 0

and||g(x)− g1(x)|| ≤ e2||λ2||(b−a)(b− a)ε.

Since g(x) = y′(x)− λ1y(x), we have

||y′(x)− λ1y(x)− g1(x)|| ≤ e2||λ2||(b−a)(b− a)ε.

hyers-ulam stability of linear differential equations ... 423

By an argument similar to Theorem 2.1, we can show that there exists

u(x) = eλ1(x−b)y(b)− eλ1(x−a)

b∫x

e−λ1(v−a)g1(v)dv

such that||y − u|| ≤ e2(||λ2||+||λ1||)(b−a)(b− a)2ε

and the twice continuously differentiable function u : [a, b] → C satisfies

u′′ + αu′ + βu− f(x) = 0.

Thus, the proof is completed.

Remark 2.3. The roots of characteristic equation λ2 + αλ + β = 0 of equation(1.1) can be real or complex.

Acknowledgements. This research was supported by Scientific Research Fundof Yunnan Provincial Education Department (No. 2013C157).

References

[1] Ulam, S.M., Problems in Modern Mathematics, John Wiley & Sons, NewYork, NY, USA, 1964.

[2] Hyers, D.H., On the stability of the linear functional equation, Proc. Nat.Acad. Sci. U.S.A., 27 (1941), 222-224.

[3] Rassias, Th.M., On the stability of linear mapping in Banach spaces, Proc.Amer. Math. Soc., 72 (1978), 297-300.

[4] Jun, K.-W., Lee, Y.-H., A generalization of the Hyers-Ulam-Rassias sta-bility of Jensen’s equation, J. Math. Anal. Appl., 238 (1999), 305-315.

[5] Park, C.-G., On the stability of the linear mapping in Banach modules,J. Math. Anal. Appl., 275 (2002), 711-720.

[6] Alsina, C., Ger, R., On some inequalities and stability results related tothe exponential function, J. Inequal. Appl., 2 (1998), 373-380.

[7] Takahasi, S.E., Miura, T., Miyajima, S., On the Hyers-Ulam stabilityof the Banach space-valued differential equation y′ = λy, Bull. Korean Math.Soc., 39 (2002), 309-315.

[8] Miura, T., Miyajima, S., Takahasi, S.-E., A characterization of Hyers-Ulam stability of first order linear differential operators, J. Math. Anal. Appl.,286 (2003), 136-146.

424 j. xue

[9] Takahasi, S.-E., Takagi, H., Miura, T., Miyajima, S., The Hyers-Ulam stability constants of first order linear differential operators, J. Math.Anal. Appl., 296 (2004), 403-409.

[10] Jung, S.-M., Hyers-Ulam stability of linear differential equations of firstorder (II), Appl. Math. Lett., 19 (2006), 854-858.

[11] Jung, S.-M., Hyers-Ulam stability of linear differential equations of firstorder (III), J. Math. Anal. Appl., 311 (2005), 139-146.

[12] Li, Y., Shen, Y., Hyers-Ulam stability of linear differential equations ofsecend order, Appl. Math. Lett., 23 (2010), 306-309.



A THEOREM ABOUT FINITE GROUPS

WITH SPECIAL CONJUGACY CLASSES1

Xianglin Du

Chongqing Three Gorges University

Wanzhou, Chongqing, 404100

P.R. China

e-mail: [email protected]

Yuming Feng

1. School of Electronic Information Engineering

Southwest University

Chongqing, 400715

P.R. China

2. School of mathematics and statistics

Chongqing Three-Gorges University


P.R. China


http://math1.sanxiau.edu.cn/ics/News/Show.asp?id=277

Jinkui Liu

Chongqing Three Gorges University


P.R. China


Abstract. Let G be a finite group. G has the property that for any conjugacy class

length, G has exactly two conjugacy classes having such length. The paper classifies all

possible stucture of the finite group G under a condiction that G′ is nilpotent.

Keywords: finite groups, conjugacy class length, character.

2010 MR Subject Classification: 20D10, 20D60.

1Supported by Scientific and Technological Research Program of Chongqing Municipal Edu-

cation Commission (Grant Nos. KJ1401006, KJ1401019) and the Fundamental Research Funds

for the Central Universities.

426 x. du, y. feng, j. liu

1. Introduction

In 1973, F. Mark ([1]) began to investigate the “S3-conjecture”, where S3

is the only finite group whose conjugacy classes all have different length. Many

authors also studied this problem in special cases ([3], [4]). In 1994, J.P. Zhang

([5]) proved the conjecture is true if the group is a finite solvable group. In 2004,

C.M. Boner and M.B. Ward ([2]) investigated the similar problem that if G is

a finite group with exactly two conjugacy classes of the same length, and G′ is

nilpotent, then G ∼= Z2, D10, A4. This paper continues this kind of problems. We

will prove that if G is a finite group with the property P ∗(P ∗ will be defined later),

and G′ is nilpotent, then G ∼= Z2, G ∼= Z2 × S3 or G ∼= Z3 o Z4.

For the sake of convenience, we first give some notations. Let x ∈ G, we

denote by o(x) the order of x, xG denotes the conjugacy class containing x, |xG|denotes the length of the conjugacy class of x, A char G means that A is a

character subgroup of G. Let Irr(G) be the set of all irreducible characters of G,

Irr∗(G) be the set of nonlinear irreducible characters of G. We call a finite group

G having property P ∗ if G has property that for any conjugacy class length, G

has exactly two conjugacy classes having such length. We assume that all groups

in this paper are finite groups. All other symbols used in this paper are standard

and the readers can consult [7] for more information.

We begin our work with the following lemmas.

2. Preliminaries

Lemma 2.1. ([5]) Suppose that G is a finite solvable group. If the conjugacy

classes of G all have different length, then G ∼= S3.

In the following proof, we suppose that G is a finite group with property P ∗,

and G′ is nilpotent.

Lemma 2.2. Z(G) ∼= Z2 and G/G′ is an Abelian 2-group.

Proof. Because G contains exactly two conjugacy classes of length 1, and the

conjugacy classes length of any elements of Z(G) is 1, we have that Z(G) ∼= Z2.

Let z ∈ Z(G), z = 1, x ∈ G, and o(x) is odd, then the length of conjugacy classes

of x and zx are equal. o(x) is odd implies CG(x) = CG(x2), which concludes that

the conjugacy length of x and x2 are equal too. Since G has property P ∗, two

elements among x, x2, zx must be conjugate. The order of x and zx are different

implies that x is conjugate to x2. Therefore, there exists g ∈ G, such that x2 = xg.

a theorem about finite groups with special conjugacy classes 427

So we have that x = x−1xg = [x, g] ∈ G′, from which we know that G/G′ is an

Abelian 2-group.

Lemma 2.3. Let P2 ∈ Syl2(G). If P2 is non-abelian, then G′ = Z2H,P ′2 = Z2,

where Z2 = Z(G), H is a 2′-Hall normal subgroup of G.

Proof. By Lemma 2.2, G/G′ is an Abelian 2-group. Since G′ is nilpotent, we can

set G′ = L×H, where L ≤ P2. Then P ′2 ≤ L ≤ P2. Since L char G′G, we have

that LG and L∩Z(P2) = ∅. Also, since H char G′G, HG and the elements

of L can commute with the elements of H. Thus Z(G) = Z2 = L ∩ Z(P2) ≤ L.

Now, we prove L = Z2. Otherwise, since LG, L consists of conjugacy classes

of G, |L| = |Z2|+|aG|+...,+|bG|, where a, b ∈ L. Let |aG| be the minimal length of

the conjugacy classes of non-unit elements of L. Clearly, the G-conjugacy classes

length of L is power of 2. If |aG| ≥ 4,we have a contradiction immediately since

|Z2| = 2. Therefore, |aG| = 2. But G has another conjugacy class of length 2, let

it be xG, that is |xG| = 2. If x ∈ L, then |xG|+ |aG| = 4. Since G has the property

P ∗, the other conjugacy classes length of L is at least 4, with the same reason

we have a contradiction. If x /∈ L, it is easy to know that |xG| = |(zx)G| and|aG| = |(za)G|, where z ∈ Z(G). G has property P ∗ implies that x is conjugate

to zx, and a is conjugate to za. Because |aG| = 2, we can conclude that for

any element g of G, ag = az or ag = a. With the same reason we have that

xg = zx, or xg = x. Furthermore, it is easy to prove for any g ∈ G, (ax)g = ax,

or (ax)g = zax. So |(ax)G| = 2 or ax ∈ Z(G). Obviously, ax /∈ Z(G). We assert

that ax is not conjugate to a. Otherwise, there exists g ∈ G such that a = (ax)g,

then a = (ax)g = ax or a = (ax)g = zax, both of them are impossible. In the

same way, we can prove that ax is not conjugate to x. Therefore, (ax)G is the

third conjugacy class with length 2 except aG, xG, it contradicts to the fact that

G has property P ∗.

Therefore, |L| = 2, and P ′2 = L = Z2.

Lemma 2.4. P2 is an Abelian subgroup.

Proof. Let |P2| = 2n, if P2 is not an Abelian subgroup, by Lemma 2.3, G′ = Z2H,

P ′2 = Z2. Let N is a minimal normal subgroup of G, and N ≤ H. Since H ≤ G′,

and G′ is nilpotent, N is a p-group and N ≤ Z(H). Let M = P2N . We con-

sider a quotient group M = M/Z2 = P2N ≤ G = G/Z2, where P2 = P2/Z2,

N = NZ2/Z2. Clearly, N is a minimal normal subgroup of M . If M = M/Z2

is Abelian, then for any x ∈ P2, y ∈ N , such that y−1yx ∈ Z2 ∩ N = 1 since

N G. Therefore xy = yx for any x ∈ P2, y ∈ N . Since N ≤ Z(H), we have

N ≤ Z(G) = Z2, a contradiction. Therefore, M = M/Z2 is non- Abelian.


P2 is Abelian and N is a minimal normal subgroup of M impliy that M ′ = N .

Let χ ∈ Irr(M), χ(1) > 1. Since N is an Abelian normal subgroup of M , we

have χ(1) | |M : N | = 2n−1(Ref [6]). Also, since P2 is Abelian, for any a ∈ P2,

(|aM |, χ(1)) = 1. Therefore, χ(a) = 0, or ρ(a) = ωI(Ref[6]), where ρ is an

irreducible representation of M which affords χ, I is a χ(1)×χ(1) identity matrix.

If ρ(a) = ωI, since M ′ = N , for any g ∈ N , a−1ag = h ∈ N . Therefore,

ρ(a) = ρ(a)ρ(h), and ρ(h) = I follows. So we have that h ∈ kerχ.

(i) if h = 1, then 1 < N ∩ kerχ M . N is a minimal normal subgroup of M ,

implies that N ≤ kerχ. It means that χ may be viewed as a irreducible character

of M/N ∼= P2. But P2 is Abelian, hence χ is one degree irreducible character, it

contradicts to χ(1) > 1.

(ii) if h = 1, then a ∈ CG(N). Let CG(N) = K × H( K ≤ P2), then K char

CG(N) G, we have K G. So KZ2 G. The elements of H and KZ2 can

commute since H is 2′-Hall normal subgroup of G. Thus any x ∈ KZ2, |xG| ispower of 2. By class equation, 2s = |KZ2| = 2+2n1+ ...,+2ni , n1 ≤ n2 ≤ ...,≤ ni.

if n1 ≥ 2, we have a contradiction immediately. If n1 = 1, then KZ2 contains

an element u with |uG| = 2. So G contains another element v with |vG| = 2

since G has property P ∗. It is easy to see that H ≤ CG(v) and v ∈ KZ2. Hence

2s = |KZ2| = 2 + 2 + 2 + 2n3 + ...,+2ni , 2 ≤ n3 ≤ ...,≤ ni, a contradiction.

Therefore, for any a ∈ P2, a = 1, χ(a) = 0.

Now, we restrict χ on P2, we have

(χ, χ)P2=

1

|P2|∑a∈P2

χ(a)χ(a−1) =χ2(1)

|P2|.

Therefore, χ2(1) = |P2|(χ, χ)P2= 2n−1(χ, χ)P2

. Let χ = n1θ1+n2θ2+...+ntθt,

where θi ∈ Irr(P2) and ni ≥ 1. Since θi(1) = 1, χ(1) = n1 + n2 + ... + nt, by

orthogonality relation of irreducible character, (χ, χ)P2= n2

1 + n22 + ... + n2

t ≥n1 + n2 + ... + nt = χ(1), that is χ2(1) = 2n−1(χ, χ)P2

≥ 2n−1χ(1). Thus, we can

conclude that χ(1) ≥ 2n−1. Since χ(1) is power of 2, we have (χ, χ)P2is power of 2.

M ′ = N means that the number of one degree irreducible characters of M ′ is

|M/M ′| = |P2| = 2n−1. Let |N | = pm, then

2n−1pm = |M | = |M/M ′|+∑

χ∈Irr∗(M)

χ2(1) = 2n−1 + 2n−1∑

χ∈Irr∗(M)

(χ, χ)P2.

Therefore, pm = 1 +∑

χ∈Irr∗(M)

(χ, χ)P2. Since (χ, χ)P2

≥ χ(1) ≥ 2n−1 and

(χ, χ)P2is power of 2, pm = 1 + 2n−1

∑i

2ni . Hence 2n−1 | pm − 1.

a theorem about finite groups with special conjugacy classes 429

We have known before that N G, and N ≤ Z(H). So for any x ∈ N , |xG|is power of 2.

Clearly, |xG| = |(zx)G|, they are not conjugate because o(x) = o(zx). Since

x ∈ N and zx /∈ N and G has property P ∗, the distinct conjugacy classes of

elements of N have distinct length.

So pm = |N | = 1 + |xG1 | + |xG

2 | + ... + |xGs | = 1 + 2m1 + 2m2 + ... + 2ms ,

m1 < m2 < ... < ms. Because 2n−1 | pm − 1 and |P2| = |P2||Z2| = 2n, we have

s = 1, m1 = n− 1.That is pm − 1 = 2n−1. But N = 1∪ xG, it implies that P2 acts

fixed-point-free on N by conjugacy. By [7] 7.24, P2 is a cyclic group or generalized

quaternion group. Since P2 is Abelian, P2 = P2/Z(G) is a a cyclic group, which

implies that P2 is Abelian. It contradicts to our hypothesis of P2 non-Abelian.

In fact, we have proven that P2 is an Abelian subgroup.

3. Main theorem

Theorem 3.1. Suppose that G is a finite group with property P ∗, and G′ is

nilpotent, then G ∼= Z2, G ∼= Z2 × S3, or G ∼= Z3 o Z4.

Proof. Since G has property P ∗, Z(G) = Z2 = ⟨z⟩, o(z) = 2. Clearly, G ∼= Z2, if

G is Abelian.

IfG is not Abelian, let∑

1=xG|x ∈ G, zx ∈ xG,∑

2=xG|x ∈ G, zx /∈ xG.Then G =

∑1 ∪

∑2. Clearly, xG ∈

∑2 if and only if (zx)G ∈

∑2. Let

∑2 =∑

21 ∪∑

22, where∑

21 = xG1 , x

G2 , ..., x

Gl ,

∑22 = (zx1)

G, (zx2)G, ..., (zxl)

G.Because |xG

i | = |(zxi)G|, i = 1, 2, ..., l, and G has property P ∗, the length of

conjugacy classes of∑

21 are distinct.

Firstly, we assert that∑

1 = ∅. Otherwise, let xG ∈∑

1. By definition of∑

1,

zx ∈ xG, so there exists g ∈ G such that zx = xg. We decompose x as production

of a 2-element a and a 2′- element b, x = ab = ba. Let o(a) = 2s, o(b) = t,

(2, t) = 1. Since (zab)t = ((ab)g)t = zat = (at)g, if let y = at, then zy ∈ yG. Since

a is a 2-element and t is odd, y is a 2-element. Let P2 ∈ Syl2(G), y ∈ P2. Since

P2 is Abelian, |yG| is a odd number. On the other hand, since zy ∈ yG, Z(G)

acts on conjugacy class yG. Obviously, the action is faithful. Hence 2 | |yG|, it isa contradiction. Thus

∑1 = ∅.

Now, considering G = G/Z(G), we will prove that every conjugacy classes in

G/Z(G) has distinct length.

Let x ∈ G/Z(G), since∑

1 = ∅, zx /∈ xG. It is easy to see that CG(x) ≤ CG(x).

Let g ∈ CG(x), then we have xg = x, or xg = xz. By∑

1 = ∅, we can conclude that


xg = x, that is g ∈ CG(x). Therefore, CG(x) = CG(x). We have known that the

conjugacy classes of∑

21 has distinct length. So if let∑∗

21 = (x)G |xG ∈∑

21,then the conjugacy classes of

∑∗21 has distinct length too. Also since G/Z(G) =∑∗

21 = (x)G | xG ∈∑

2, by Lemma 2.1, G/Z(G) ∼= S3. Therefore, G is a group

with order 12. But all finite groups of order 12 are the following:

Z12, Z2 × Z2 × Z3, A4, G ∼= Z2 × S3, G ∼= Z3 o Z4.

It is easy to check that, G ∼= Z2 × S3 or G ∼= Z3 o Z4. The proof is completed.

References

[1] Markel, F.M., Groups with many conjugate elements, J. Alg., 26 (1973),

69-74.

[2] Hayashi, M., On a generalization of F.M. Markel’s theorem, Hokkaido

Math. J., 4 (1975), 278-280.

[3] Ward, M.B., Finite groups in which no two distinct conjugacy classes have

the same order, Arch. Math., 54 (1990), 111-116.

[4] Zhang, J.P., Finite groups with many conjugate elements, J. Alg., 170

(1994), 608-624.

[5] Boner, C.M., Ward, M.B., Finite groups with two conjugacy classes of

the same order, Rocky Journal of Mountain Mathematics, 31 (2) (2001),

401-416.

[6] Isaacs, I.M., Character theory of finite groups, Academic Press, New York

San Francisco, London, 1976.

[7] Kurzweil, H., Endliche Gruppen, Berlin, Heidelberg, Springer-Veflag,

1977.



ENERGY OF AN INTUITIONISTIC FUZZY GRAPH

B. Praba

SSN College of EngineeringKalavakkam, Chennai – 603110Indiae-mail: [email protected]

V.M. ChandrasekaranG. Deepa1

VIT UniversityVellore – 632014Indiae-mails: [email protected]

[email protected]

Abstract. In this paper, the concept of energy of fuzzy graph is extended to the

energy of an intuitionistic fuzzy graph. We have defined the adjacency matrix of an

intuitionistic fuzzy graph and the energy of an intuitionistic fuzzy graph is defined in

terms of its adjacency matrix. These concepts are illustrated with real time example.

The lower and upper bound for the energy of an intuitionistic fuzzy graph are also

derived.

Keywords: fuzzy set, intuitionistic fuzzy set, fuzzy graph, energy of fuzzy graph.

1. Introduction

Fuzzy set was introduced by Zadeh [18] whose basic component is only a member-ship function. The generalization of Zadeh’s fuzzy set, called intuitionistic fuzzyset was introduced by Atanassov [2] which is characterized by a membership func-tion and a non-membership function. In Zadeh’s fuzzy set, the sum of membershipdegree and a non-membership degree is equal to one. In Atanassov intuitionisticfuzzy set, the sum of membership degree and a non-membership degree does notexceed one.

The foundation for graph theory was laid in 1735 by Leonhard Euler when hesolved the ”Konigsberg bridges” problem. Many real life problems can be repre-sented by graph. In computer science, graphs are used to represent networks ofcommunications, data organization, computational devices, the flow of computa-tion, etc. The link structure of a website could be represented by a directed graphin which the vertices are the web pages available at the website and a directed


432 b. praba, v.m. chandrasekaran, g. deepa

edge from page A to page B exists if and only if A contains a link to B [13].A similar approach can be taken to problems in travel, biology, computer chipdesign and many other fields. Hence graph theory is widely used in solving realtime problems. But when the system is large and complex it is difficult to extractthe exact information about the system using the classical graph theory. In suchcases fuzzy graph is used to analyze the system.

The first definition of fuzzy graphs was proposed by Kafmann [10] in 1973,from the Zadeh’s fuzzy relations [18] [19] [20]. But Rosenfeld [14] introducedanother elaborated definition including fuzzy vertex and fuzzy edges and seve-ral fuzzy analogs of graph theoretic concepts such as paths, cycles, connected-ness and etc. The first definition of intuitionistic fuzzy graphs was proposed byAtanassov [3]. I. Gutman [6] in 1978 introduced the concept of ”graph energy”as the sum of the absolute values of the eigen values of the adjacency matrix ofthe graph. Certain bounds on energy are discussed in [4], [12] and [7]. Energy ofdifferent graphs including regular [8], non-regular [9], circulant [16] and randomgraphs [17] is also under study. Energy is defined for signed graphs in [5] and forweighted graphs in [15]. The energy of graph is extended to the energy of fuzzygraph in [1]. In this paper, we extend the concept of energy of fuzzy graph tothe energy of an intuitionistic fuzzy graph. This paper is organized as follows.In Section 2, we give all the basic definitions related to fuzzy sets, intuitionisticfuzzy sets, energy of graph and energy of fuzzy graph. In Section 3, we definethe energy of an intuitionistic fuzzy graph. The lower and upper bounds for theenergy of an intuitionistic fuzzy graph are also derived. In Section 4, we illustratethese concepts by taking the website of http://www.pantechsolutions.net/ . InSection 5, we give the conclusion.

2. Preliminaries

2.1. Energy of graph

Definition 2.1. A graph G is a pair of set (V,E), denoted by G = (V,E), whereV is a set of vertices and E is a set of edges. Each edge in E is a pair of verticesin V . Each edge is associated with a set consisting of either one or two verticescalled its endpoints.

Definition 2.2. A graph in which the edges are unordered vertex pair is calledan undirected graph. A graph in which the edges are ordered vertex pair is calleda directed graph. Hence if there is an edge from vi to vj in G then (vi, vj) ∈ E.

Definition 2.3. An edge whose endpoints are the same is called a loop. A graphwithout loops and parallel edges is called a simple graph. Two vertices that areconnected by an edge are called adjacent. The adjacency matrix A = [aij] for agraph G = (V,E) is a matrix with n rows and n columns, n = |V | and its entries

defined by aij =

1, if (vi, vj) ∈ E,

0, otherwise.

energy of an intuitionistic fuzzy graph 433

Definition 2.4. The spectrum of a matrix is defined as a set of its eigenvalues.Let G = (V,E) be a simple graph with n vertices and m edges. Let A = [aij] bethe adjacency matrix of G. As in a simple graph, there can be at most 1 edgebetween 2 vertices. So, the entries in A are either 0 or 1. The diagonal elementsare zero since there are no loops. A is symmetric and so the spectrum of A is real.An eigenvalue of A are called eigenvalues of G[4] and the spectrum of A is calledthe spectrum of G. Energy of a simple graph G = (V,E) with adjacency matrixA is defined as the sum of absolute values of eigenvalues of A. It is denoted byE (G). That is,

E(G) =n∑

i=1

|λi|,

where λi is an eigenvalue of A, i = 1, 2..., n. Since A is a symmetric matrix whichdiagonal elements are zero,

n∑i=1

λi = 0.

By comparing the coefficients of λn−2 in the characteristic polynomial

n∏i=1

(λ− λi) = |A− λI| ,

we get ∑1≤i<j≤n

λiλj = −m.

Theorem 2.1. Let G be a simple graph with |V | = n vertices and m edges and

A be the adjacency matrix of G then

√2m+ n(n− 1)|A| 2n ≤ E(G) ≤

√2mn.

2.2. Fuzzy set

Definition 2.5. Let X be a nonempty set. A fuzzy set A in X is defined asA = (x, µA(x)) /x ∈ X, which is characterized by a membership functionµA (x) : X → [0, 1] and a fuzzy set satisfying the following conditionµA(x) + γA(x) = 1, where γA(x) = 1 − µA(x) is the nonmembership function(see [18]).

2.3. Intuitionistic fuzzy set

Definition 2.6. Let X be a nonempty set. An intuitionistic fuzzy set A in X isdefined as A = (x, µA(x), γA(x))/x ∈ X which is characterized by a membershipfunction µA(x) : X → [0, 1] and the nonmembership function γA(x) : X → [0, 1]and satisfying the following condition

(i) 0 ≤ µA(x) + γA(x) ≤ 1,∀x ∈ X,

(ii) 0 ≤ µA(x), γA(x), πA(x) ≤ 1, ∀x ∈ X,

(iii) πA(x) = 1− µA(x)− γA(x),


where πA(x) is called the intuitionistic fuzzy index of element x in A, the valuedenotes a measure of nondeterminacy. Obviously, if πA(x) = 0, then the intuitio-nistic fuzzy set A is just a Zadeh’s fuzzy set (see [2]).

2.4. Energy of fuzzy graph [1]

Definition 2.7. Let V be a nonempty set. A fuzzy subset of V is a functionσ : V → [0, 1]. σ is called the membership function and σ (v) is called themembership of v where v ∈ V . Let V1 and V2 be two nonempty sets and σ1 andσ2 be two fuzzy subsets of V1 and V2 respectively. Define a fuzzy subset µ ofV1 × V2 as µ (vi, vj) ≤ min σ1 (vi) , σ2 (vj). Then, µ is called a fuzzy relation,from σ1 to σ2. Suppose σ1 (x) = 1, ∀x ∈ V1 and σ2 (y) = 1,∀y ∈ V2. Then, µ iscalled a fuzzy relation from V1 into V2. µ (vi, vj) is interpreted as the strength ofrelation between vi and vj. Suppose V1 = V2 = V and σ1 = σ2 = σ. Then, µis called a fuzzy relation on σ. Suppose V1 = V2 = V and σ1 (x) = 1, ∀x ∈ V1,and σ2 (y) = 1, ∀y ∈ V2. Then µ is called a fuzzy relation on V . From the abovedefinitions, it follows that binary relations on crisp sets are particular cases offuzzy relations. Let V be a nonempty set and σ be a fuzzy subset of V . Let µbe a fuzzy relation on σ. µ is said to be symmetric, if µ (vi, vj) = µ (vj, vi) forvi, vj ∈ V . A fuzzy relation can also be expressed by a matrix called fuzzy relationmatrix M = [mij], where mij = µ (vi, vj).

A fuzzy graph G = (V, σ, µ) is a nonempty set V together with a pair offunction (σ, µ) where σ is a fuzzy subset of V and µ is a fuzzy relation on σ. LetM = [mij] be a fuzzy relation matrix defined on µ. This represents the strengthof the relation between the vertices. Hence we have the following definitions.

Definition 2.8. [1] The adjacency matrix A of a fuzzy graph G = (V, σ, µ) is ann× n matrix where n = |V | defined as A = [aij], where aij = µ (vi, vj). Note thatA becomes the usual adjacency matrix when all the nonzero membership valuesare 1, i.e., when the fuzzy graph becomes a crisp graph.

Definition 2.9. [1] Let G = (V, σ, µ) be a fuzzy graph and A be its adjacencymatrix. The eigenvalues of A are called eigenvalues of G. The spectrum of A iscalled the spectrum of G. It is denoted by Spec G.

Definition 2.10. [1] Let G = (V, σ, µ) be a fuzzy graph and A be its adjacencymatrix. Energy of G is defined as the sum of the absolute values of the eigenvaluesof A.

Theorem 2.2. [1] Let G = (V, σ, µ) be a fuzzy graph with |V | = n and µ∗ =e1, e2, ..., em. If mi = µ (ei) is the strength of the relation associated with the ith

edge, then √√√√2m∑i=1

m2i + n(n− 1)|A| 2n ≤ E(G) ≤

√√√√2

(m∑i=1

m2i

)n.


3. Energy of an intuitionistic fuzzy graph

In this section, we define the energy of an intuitionistic fuzzy directed graphwithout loops. The link structure of a website could be represented by a di-rected intuitionistic fuzzy graph. The links are considered as vertices and thepath between the links are considered as edges. The weightage of the each edgeare considered as the number of visitors (membership value), the number of nonvisitors (non - membership value) and drop off (intuitionistic fuzzy index) amongthat link structure. The lower and upper bounds for the energy of an intuitionisticfuzzy graph are also obtained.

Definition 3.1. An intuitionistic fuzzy graph is defined as G = (V,E, µ, γ) whereV is the set of vertices and E is the set of edges. µ is a fuzzy membership functiondefined on V × V and γ is a fuzzy non - membership function defined on V × V .We denote µ (vi, vj) by µij and γ (vi, vj) by γij such that (i) 0 ≤ µij + γij ≤ 1(ii) 0 ≤ µij, γij, πij ≤ 1, where πij = 1 − µij − γij. Hence (V × V, µ, γ) is anintuitionistic fuzzy set.

Example 3.1.

Figure 1: G1. An intuitionistic fuzzy graph

Definition 3.2. An intuitionistic fuzzy adjacency matrix of an intuitionistic fuzzygraph is defined as the adjacency matrix of the corresponding intuitionistic fuzzygraph. That is for an intuitionistic fuzzy graph G = (V,E, µ, γ), an intuitionisticfuzzy adjacency matrix is defined by A (IG) = [aij] where aij = (µij, γij). Notethat µij represents the strength of the relationship between vi and vj and γijrepresents the strength of the non-relationship between vi and vj.


Example 3.2. For an intuitionistic fuzzy graph in Fig. 1, the adjacency matrix is

A (IG) =

0 (0.6, 0.2) 0 (0.3, 0.6)

(0.4, 0.3) 0 (0.5, 0.1) 0(0.5, 0.2) (0.7, 0.1) 0 (0.6, 0.3)(0.9, 0.1) (0.8, 0.1) (0.3, 0.5) 0

Definition 3.3. The adjacency matrix of an intuitionistic fuzzy graph can be writ-ten as two matrices one containing the entries as membership values and the othercontaining the entries as non-membership values, i.e., A (IG) = ((µij) , (γij)) ,where

A (µij) =

0 0.6 0 0.30.4 0 0.5 00.5 0.7 0 0.60.9 0.8 0.3 0

and A (γij) =

0 0.2 0 0.60.3 0 0.1 00.2 0.1 0 0.30.1 0.1 0.5 0

Definition 3.4. The eigenvalue of an intuitionistic fuzzy adjacency matrix A (IG)is defined as (X, Y ), where X is the set eigenvalues of A (µij) and Y is the set ofeigenvalues of A (γij).

Definition 3.5. The spectrum of an intuitionistic fuzzy adjacency matrix A (IG)is the defined as (X,Y ), where X is the set eigenvalues of A (µij) and Y is the setof eigenvalues of A (γij).

Definition 3.6. The energy of an intuitionistic fuzzy graph G = (V,E, µ, γ) isdefined as (∑

λi∈X

|λi|,∑δi∈Y

|δi|

)

where∑λi∈X

|λi| is defined as an energy of the membership matrix denoted by

E (µij(G)) and∑δi∈Y

|δi| is defined as an energy of the nonmembership matrix

denoted by E (γij(G)).

Example 3.3. For an intuitionistic fuzzy graph in Fig. 1,Spec(µij (G1)) = 1.2406,−0.7153,−0.2627 + 0.2332i,−0.2627− 0.2332iSpec(γij (G1)) = 0.6441,−0.0148,−0.3146 + 0.1629i,−0.3146− 0.1629iE (µij (G1)) = 1.2406 + 0.7153 + 0.3513 + 0.3513 = 2.6585

E (γij (G1)) = 0.6441 + 0.0148 + 0.3543 + 0.3543 = 1.3675

Theorem 3.1. Let G be an intuitionistic fuzzy directed graph (without loops)with |V | = n and |E| = m and A(IG) = ((µij), (γij)) be an intuitionistic fuzzyadjacency matrix of G then


(i)

√2∑

1≤i<j≤n

µijµji + n(n− 1)|A| 2n ≤ E (µij(G)) ≤√2n

∑1≤i<j≤n

µijµji,

where |A| is the determinant of A(µij), and

(ii)

√2∑

1≤i<j≤n

γijγji + n(n− 1)|B| 2n ≤ E (γij(G)) ≤√2n

∑1≤i<j≤n

γijγji,

where |B| is the determinant of A(γij).

Proof. Upper bound. Apply Cauchy’s Schwartz inequality to the n numbers1, ..., 1 and |λ1| , |λ2| , ..., |λn|,

n∑i=1

|λi| ≤√n

√√√√ n∑i=1

|λi|2(1)

(n∑

i=1

λi

)2

=n∑

i=1

|λi|2 + 2∑

1≤i<j≤n

λiλj(2)

By comparing the coefficients of λn−2 in the characteristic polynomialn∏

i=1

(λ− λi) = |A− λI| ,

we get

(3)∑

1≤i<j≤n

λiλj = −∑

1≤i<j≤n

µijµji

Substituting (3) in (2), we get

(4)n∑

i=1

|λi|2 = 2∑

1≤i<j≤n

µijµji

Substituting (4) in (1), we getn∑

i=1

|λi| ≤√n

√2∑

1≤i<j≤n

µijµji =

√2n

∑1≤i<j≤n

µijµji

E (µij(G)) ≤√

2n∑

1≤i<j≤n

µijµji

Lower bound

(E (µij (G)))2 =

(n∑

i=1

|λi|

)2

=n∑

i=1

|λi|2 + 2∑

1≤i<j≤n

|λiλj|

= 2∑

1≤i<j≤n

µijµji +2n(n− 1)

2AM |λiλj|


AM |λiλj| ≥ GM |λiλj| , 1 ≤ i < j ≤ n

E (µij (G)) ≥√

2∑

1≤i<j≤n

µijµji + n (n− 1)GM |λiλj|

GM |λiλj| =

( ∏1≤i<j≤n

|λiλj|

) 2n(n−1)

=

(n∏

i=1

|λi|n−1

) 2n(n−1)

=

(n∏

i=1

|λi|

) 2n

= |A|2n

E (µij (G)) ≥√

2∑

1≤i<j≤n

µijµji + n (n− 1) |A|2n

Therefore,√2∑

1≤i<j≤n

µijµji + n (n− 1) |A|2n ≤ E (µij (G)) ≤

√2n

∑1≤i<j≤n

µijµji

Similarly, we can prove√2∑

1≤i<j≤n

γijγji + n (n− 1) |B|2n ≤ E (γij (G)) ≤

√2n

∑1≤i<j≤n

γijγji

Example 3.4. (Illustration to Theorem 3.1) For an intuitionistic fuzzy graphin Fig. 1,

E (µij (G1)) = 2.6585 its Lower bound = 2.4599 and Upper bound = 2.8844 andE (γij (G1)) = 1.3675 its Lower bound = 0.9878 and Upper bound = 1.4967.

4. Numerical examples

In this section, we explain the concept for the energy of an intuitionistic fuzzygraph and lower and upper bounds for the energy of an intuitionistic fuzzy graphthrough a real time example.

We have taken the website http://www.pantechsolutions.net/. This websiteis modeled as an intuitionistic fuzzy graph by considering the navigation of thecustomer. An intuitionistic fuzzy graph of this site for four different time periodsis taken for each of these periods the energy of an intuitionistic fuzzy graph andits lower and upper bounds are calculated. The energy is also represented intermsof bar diagram. We have taken the four links 1. microcontroller-boards, 2./log-inhtml, 3./ and 4./ project kits for our calculation.


Example 4.1. In the website http://www.pantechsolutions.net/ we consider fourlinks 1. microcontroller-boards, 2./log-in html, 3./ and 4./ project kits for theperiod July 16, 2013 to August 15, 2013.


For an intuitionistic fuzzy graph in Fig. 2,

Spec(µij (G2)) = −0.2000, 0.0000, 0.2000, 0.0000Spec(γij (G2)) = 0.9927,−0.1841,−0.4043 + 0.2306i,−0.4043− 0.2306iE (µij (G2)) = 0.2000 + 0.0000 + 0.2000 + 0.0000 = 0.4 its Lower bound = 0.2828and Upper bound = 0.5657

E (γij (G2)) = 0.9927+0.1841+0.4655+0.4655 = 2.1078 its Lower bound = 1.9047and Upper bound = 2.2271.

Example 4.2. In the same website (mentioned above in example 4.1.), we con-sider the same four links for the period August 16, 2013 to September 15, 2013.



For an intuitionistic fuzzy graph in Fig.3,

Spec(µij (G3)) = −0.2000, 0.0000, 0.2000, 0.0000Spec(γij (G3)) = 0.8964,−0.1893,−0.3536 + 0.2114i,−0.3536− 0.2114iE (µij (G3)) = 0.2000 + 0.0000 + 0.2000 + 0.0000 = 0.4 its Lower bound = 0.2828and Upper bound = 0.5657


Example 4.3. In the same website (mentioned above in example 4.1.), we con-sider the same four links for the period September 16, 2013 to October 15, 2013.


For an intuitionistic fuzzy graph in Fig.4,

Spec(µij (G4)) = −0.1732, 0.1732, 0.0000, 0.0000Spec(γij (G4)) = 0.9418,−0.2000,−0.3709 + 0.1053i,−0.3709− 0.1053iE (µij (G4)) = 0.1732+0.1732+0.0000+0.0000 = 0.3464 its Lower bound = 0.2449and Upper bound = 0.4899


Example 4.4. In the same website (mentioned above in Example 4.1.),we consider the same four links for the period of October 16, 2013 to Novem-ber 15, 2013.



For an intuitionistic fuzzy graph in Fig. 5,

Spec(µij (G5)) = 0.0000, 0.1879,−0.1532,−0.0347Spec(γij (G5)) = 0.9940,−0.1302,−0.4000,−0.4638E (µij (G5)) = 0.0000+0.1879+0.1532+0.0347 = 0.3758 its Lower bound = 0.2449and Upper bound = 0.4899


The following bar diagrams represents the energy of four links for the abovefour periods corresponding the membership and non-membership values respec-tively.

Table 1: Table for energy of membership values


From the above bar diagram the energy for the period July 16 to August 15 andAugust 16 to September 15 are high and same comparing the other period.

Table 2: Table for energy of non membership values

From the above bar diagram the energy for the period July 16 to August 15 ishigh comparing the other period.

5. Conclusion

In this paper, we defined the energy of an intuitionistic fuzzy graph interms of itsadjacency matrix. The lower and upper bounds for the energy of an intuitionisticfuzzy graph are derived. These concepts are illustrated with real time example.

Acknowledgments. The authors thank the managements for their constantsupport towards the successful completion of this work.

References

[1] Anjali, N., Sunil Mathew, Energy of a fuzzy graph, Annals of FuzzyMathematics and Informatics, 2013.

[2] Atanassov, K., Intuitionistic fuzzy sets, Fuzzy sets and systems, 20 (1986),87-96.

[3] Atanassov, K., Intuitionistic Fuzzy Sets: Theory and Applications,Springer-Verlag, Heidelberg, 1999.


[4] Brualdi, R.A., Energy of a graph, Notes to AIM Workshop on spectra offamilies of atrices described by graphs, digraphs, and sign patterns, 2006.

[5] Germina, K.A., Shahul Hameed, K., Thomas Zaslavsky, On pro-ducts and line graphs of signed graphs, their eigen values and energy, LinearAlgebra Appl., 435 (2011), 2432-2450.

[6] Gutman, I., The energy of a graph, Ber. Math. Statist. Sekt. Forschungs-zentram Graz., 103 (1978), 1-22.

[7] Gutman, I., The energy of a graph: old and new results, in Algebraic Com-binatorics and Applications, A. Betten, A. Kohner, R. Laue, and A. Wasser-mann (eds.), Springer, Berlin, 2001, 196-211.

[8] Gutman, I., Zare Firoozabadi, S., de la Pe na, J.A., Rada, J.,On the energy of regular graphs, MATCH Commun. Math. Comput. Chem.,57 (2007), 435-442.

[9] Indula, Gl., Vijayakumar, A., Energies of some non-regular graphs,J. Math. Chem., 42 (2007), 377-386.

[10] Kauffman, A., Introduction a la theorie des sous ensembles Flous, Massonet cie., vol. 1. (1973).

[11] Lee, K.H., First Course on Fuzzy theory and Applications, Springer-Verlag,Berlin, 2005.

[12] Liu, H., Lu, M., Tian, F., Some upper bounds for the energy of graphs,J. Math. Chem., 42 (2007), 377-386.

[13] Praba, B., Sujatha, R., Application of Fuzzy Markov model for web-testing, Ganita Sandesh, vol. 21 (2) (2007), 111-120.

[14] Rosenfeld, A., Fuzzy graphs, in L.A. Zadeh, K. cS. Fu, K. Tanaka andM. Shimura (eds.), Fuzzy sets and their applications to cognitive and deci-sion process, Academic Press, New York, 1975, 75-95.

[15] Shao, J., Gong, F., Du, Z., Extremal energies of weighted trees andforests with fixed total weight sum, MATCH Commun. Chem., 66 (2011),879-890.

[16] Shparlinski, I., On the energy of some circulant graphs, Linear Algebraand Appl., 414 (2006), 378-382.

[17] Wenxue Du, Xueliang Li, Yiyang Li, Various energies of randomgraphs, MATCH. Commun. Math. Comput. Chem., 64 (2010), 251-260.

[18] Zadeh, L.A., Fuzzy sets, Information and Control, (8) (1965), 338-353.


[19] Zadeh, L.A., Similarity relations and fuzzy ordering, Informatoins scien-ces, 3 (1971), 177-200.

[20] Zadeh, L.A., Is there a need for fuzzy logic?, Information sciences, 178(2008), 2751-2779.



ON UPPER AND LOWER ALMOST CONTRA-ω-CONTINUOUSMULTIFUNCTIONS

C. CarpinteroDepartment of MathematicsUniversidad De OrienteNucleo De Sucre CumanaVenezuela and Facultad de Ciencias BasicasUniversidad del AtlanticoBarranquillaColombiae-mail: [email protected]

N. RajeshDepartment of MathematicsRajah Serfoji Govt. CollegeThanjavur-613005TamilnaduIndiae-mail: nrajesh [email protected]

E. RosasDepartment of MathematicsUniversidad De OrienteNucleo De Sucre CumanaVenezuela and Facultad de Ciencias BasicasUniversidad del AtlanticoBarranquillaColombiae-mail: [email protected]

S. SaranyasriDepartment of MathematicsDepartment of MathematicsM.R.K. Institute of TechnologyKattumannarkoil, Cuddalore – 608 301TamilnaduIndiae-mail: srisaranya [email protected]

Abstract. In this paper, we introduce and study the almost contra-ω-continuous mul-

tifunctions between topological spaces and obtain some characterizations and properties

of such multifunctions.

Keywords: regular open set, ω-open set, almost contra-ω-continuous multifunctions.

2010 Mathematics Subject Classification: 54C60, 54C08

446 c. carpintero, n. rajesh, e. rosas, s. saranyasri

1. Introduction

Generalized open sets play a very important role in General Topology and theyare now the research topics of many topologists worldwide. Indeed a significanttheme in General Topology and Real analysis concerns the various modified formsof continuity, separation axioms, etc., by utilizing generalized open sets. Oneof the most well known notions and also an inspiration source is the notion ofω-open [10] sets introduced by H.Z. Hdeib in 1982 and used by Al-Zoubi andAl-Nashef [1] in 2003. Various types of functions play a significant role in thetheory of classical point set topology. A great number of papers dealing with suchfunctions have appeared, and a good many of them have been extended to thesetting of multifunction [2], [6],[7], [9], [17], [18]. In this paper, we introduce andstudy almost contra-ω-continuous multifunctions between topological spaces andobtain some characterizations of such multifunctions.

2 Preliminaries

Throughout this paper, (X, τ) and (Y, σ) (or simply X and Y ) always meantopological spaces in which no separation axioms are assumed unless explicitlystated. For a subset A of (X, τ), Cl(A) and Int(A) denote the closure of Awith respect to τ and the interior of A with respect to τ , respectively. Recently,as generalization of closed sets, the notion of ω-closed sets were introduced andstudied by Hdeib [10]. A point x ∈ X is called a condensation point of A if foreach U ∈ τ with x ∈ U , the set U ∩ A is uncountable. A is said to be ω-closed[10] if it contains all its condensation points. The complement of an ω-closedset is said to be ω-open. It is well known that a subset W of a space (X, τ)is ω-open if and only if for each x ∈ W , there exists U ∈ τ such that x ∈ Uand U\W is countable. The family of all ω-open subsets of a topological space(X, τ) denoted by ωO(X, τ), forms a topology on X finer than τ and the familyof all ω-closed subsets of a topological space (X, τ) is denoted by ωC(X, τ). Theω-closure and the ω-interior, that can be defined in the same way as Cl(A) andInt(A), respectively, will be denoted by ωCl(A) and ω Int(A), respectively. We setωO(X, x) = A : A ∈ ωO(X) and x ∈ A and ωC(X, x) = A : A ∈ ωC(X) andx ∈ A. A subset A is said to be regular open [20] (resp. semiopen [13], preopen[14], α-open [15], semi-preopen [3]) if A = Int(Cl(A)) (resp. A ⊂ Cl(Int(A)),A ⊂ Int(Cl(A)), A ⊂ Int(Cl(Int(A))), A ⊂ Cl(Int(Cl(A)))). The complement of aregular open (resp. semiopen, preopen, semi-preopen) set is called a regular closed(resp. semiclosed, preclosed, semi-preclosed). The intersection of all semiclosed(resp. preclosed, α-closed, semi-preclosed) subsets of (X, τ) containing A ⊂ X iscalled the semiclosure (resp. preclosure, α-closure, semi-preclosure) of A and isdenoted by sCl(A) (resp. pCl(A), αCl(A), spCl(A)). The θ-semiclosure [12] ofA, denoted by sClθ(A), is defined to be the set of all x ∈ X such that A ∩ Cl(U)= ∅ for every semiopen set U containing x. A subset A is called θ-semiclosed [12]if and only if A = sClθ(A). The complement of a θ-semiclosed set is called a θ-semiopen set [12]. The family of all regular open (resp. regular closed, semiopen,

on upper and lower almost contra-ω-continuous ... 447

semiclosed, α-open, semi-preopen, semi-preclosed) sets of (X, τ) is denoted byRO(X) (resp. RC(X), SO(X), SC(X), αO(X), SPO(X), SPC(X)). By amultifunction F : (X, τ) → (Y, σ), following [4], we shall denote the upper andlower inverse of a set B of Y by F+(B) and F−(B), respectively, that is, F+(B)= x ∈ X : F (x) ⊂ B and F−(B) = x ∈ X : F (x) ∩ B = ∅. In particular,F−(Y ) = x ∈ X : y ∈ F (x) for each point y ∈ Y and for each A ⊂ X, F (A)= ∪x∈AF (x). Then F is said to be surjection if F (X) = Y and injection if x = yimplies F (x) ∩ F (y) = ∅.

Definition 2.1 A multifunction F : (X, τ) → (Y, σ) is said to be:

1. upper almost ω-continuous [5] if for each point x ∈ X and each open set Vcontaining F (x), there exists U ∈ ωO(X, x) such that U ⊂ F+(Int(Cl(V )));

2. lower almost ω-continuous [5] if for each point x ∈ X and each open setV such that F (x) ∩ V = ∅, there exists U ∈ ωO(X, x) such that U ⊂F−(Int(Cl(V ))).

3. upper contra-ω-continuous [6] if for each point x ∈ X and each closed set Vcontaining F (x), there exists U ∈ ωO(X, x) such that U ⊂ F+(V );

4. lower contra-ω-continuous [6] if for each point x ∈ X and each closed set Vsuch that F (x) ∩ V = ∅, there exists U ∈ ωO(X, x) such that U ⊂ F−(V );

5. upper weakly ω-continuous [7] if for each x ∈ X and each open set V of Ysuch that x ∈ F+(V ), there exists U ∈ ωO(X, x) such that U ⊂ F+(Cl(V ));

6. lower weakly ω-continuous [7] if for each x ∈ X and each open set V of Ysuch that x ∈ F−(V ), there exists U ∈ ωO(X, x) such that U ⊂ F−(Cl(V )).

Definition 2.2 A subset K of a space X is said to be S-closed [21] (resp. ω-compact [2]) relative to X if every cover of K by regular closed (resp. ω-open)sets of X has a finite subcover. A space X is said to be S-closed (resp. ω-compact)if X is S-closed (resp. ω-compact) relative to X.

Lemma 2.3 [1] Let A and B be subsets of a space (X, τ).

1. If A ∈ ωO(X) and B ∈ τ , then A ∩B ∈ ωO(B);

2. If A ∈ ωO(B) and B ∈ ωO(X), then A ∈ ωO(X).

Lemma 2.4 [17] For a multifunction F : (X, τ) → (Y, σ), the following holds:

1. G+F (A×B) = A ∩ F+(B);

2. G−F (A×B) = A ∩ F−(B),

for any subset A of X and B of Y , where GF : X → X × Y is defined asGF (x) = x × F (x) for every x ∈ X.


Definition 2.5 [2] A function f : (X, τ) → (Y, σ) is said to be almost contra-ω-continuous if f−1(W ) ∈ ωO(X) for every W ∈ RC(Y ).

3. On upper and lower almost contra-ω-continuous multifunctions

Definition 3.1 A multifunction F : (X, τ) → (Y, σ) is said to be:

1. upper almost contra-ω-continuous if for each point x ∈ X and each regu-lar closed set V with x ∈ F+(V ), there exists U ∈ ωO(X, x) such thatU ⊂ F+(V );

2. lower almost contra-ω-continuous if for each point x ∈ X and each regu-lar closed set V with x ∈ F−(V ), there exists U ∈ ωO(X, x) such thatU ⊂ F−(V ).

Theorem 3.2 If F : (X, τ) → (Y, σ) is an upper (lower) almost contra-ω-conti-nuous multifunction, then it is upper (lower) weakly ω-continuous.

Proof. Let x ∈ X and V be an open subset of Y with F (x) ⊂ V . This implies thatCl(V ) is a regular closed set with F (x) ⊂ Cl(V ). Since F is upper almost contra-ω-continuous, there exists U ∈ ωO(X, x) such that U ⊂ F+(Cl(V )). Hence, F isupper weakly ω-continuous.

The following example shows that the converse of the above Theorem 3.2 isnot true in general.

Example 3.3 Let X = ℜ with the topologies τ = ∅,ℜ,ℜ − Q and σ =∅,ℜ,ℜ − Q. Define F : (ℜ, τ) → (ℜ, σ) as follows: F (x) = x. Then Fis upper weakly-ω-continuous multifunction but is not upper almost contra-ω-continuous multifunction.

Corollary 3.4 If F : (X, τ) → (Y, σ) is almost contra-ω-continuous, then it isweakly ω-continuous.

Theorem 3.5 If F : (X, τ) → (Y, σ) is an upper (lower) contra-ω-continuousmultifunction, then it is upper (lower) almost contra ω-continuous multifunction.

Proof. The proof is obvious.

The following example shows that the converse of the above Theorem 3.5 isnot true in general.

Example 3.6 Let X = ℜ with the topology τ = ∅,ℜ,ℜ−Q. And Y = a, b, cwith the topology σ = ∅, Y, a, b, a, b, b, c. Take a fixed number e ∈ Q,and define F : (ℜ, τ) → (Y, σ) as follows:

F (x) =

b if x ∈ Q− ec if x ∈ (ℜ−Q) ∪ e.

Then F is upper almost contra-ω-continuous multifunction but is not uppercontra-ω-continuous multifunction.


Corollary 3.7 [2] If F : (X, τ) → (Y, σ) is contra-ω-continuous, then F is almostcontra-ω-continuous.

Theorem 3.8 For a multifunction F : (X, τ) → (Y, σ), the following statementsare equivalent:

1. F is upper almost contra-ω-continuous;

2. F+(A) ∈ ωO(X) for every regular closed A of Y ;

3. F−(U) ∈ ωC(X) for every regular open subset U of Y ;

4. F−(Int(Cl(A))) ∈ ωC(X) for every open subset A of Y ;

5. F+(Cl(Int(A))) ∈ ωO(X) for every closed subset A of Y ;

6. for each x ∈ X and for each V ∈ SO(Y ) with F (x) ⊂ V , there existsU ∈ ωO(X, x) such that F (U) ⊂ Cl(V );

7. F+(V ) ⊂ ω Int(F+(Cl(V ))) for every V ∈ SO(Y ).

Proof. (1) ⇔ (2): Let A ∈ RC(Y ) and x ∈ F+(A). Since F is upper almostcontra-ω-continuous, there exists U ∈ ωO(X, x) such that U ⊂ F+(A). Thus,F+(A) ∈ ωO(X). The converse is obvious.

(2) ⇔ (3) and (4) ⇔ (5): It follows from the fact that F+(Y \A) = X\F−(A) forevery subset A of Y .

(3) ⇔ (4): Let A be an open subset of Y . Since IntCl(A)) is regular open, thenF−(Int(Cl(A))) is ω-closed. The converse is obvious.

(5) ⇔ (2): It is similar to that of (3) ⇔ (4).

(6) ⇒ (7): Let V ∈ SO(Y ) and x ∈ F+(V ). Then F (x) ⊂ V . By (6), there existsU ∈ ωO(X, x) such that F (U) ⊂ Cl(V ). This implies that x ∈ U ⊂ F+(Cl(V )).Hence, x ∈ ω Int(F+(Cl(V ))) and F+(V ) ⊂ ω Int(F+(Cl(V ))).

(7) ⇒ (2): Let A ∈ RC(Y ). Since A ∈ SO(Y ), then F+(A) ⊂ ω Int(F+(Cl(A)));hence F+(A) ∈ ωO(X).

(2) ⇒ (6): Let x ∈ X and V ∈ SO(Y ) with F (x) ⊂ V . Since Cl(V ) ∈ RC(Y ),there exists A ∈ ωO(X, x) such that A ⊂ F+(Cl(V )). Hence F (A) ⊂ Cl(V ).


1. F is lower almost contra-ω-continuous;

2. F−(A) ∈ ωO(X) for every regular closed A of Y ;

3. F+(U) ∈ ωC(X) for every regular open subset U of Y ;

4. F+(Int(Cl(A))) ∈ ωC(X) for every open subset A of Y ;


5. F−(Cl(Int(A))) ∈ ωO(X) for every closed subset A of Y ;

6. for each x ∈ X and for each V ∈ SO(Y ) with F (x) ∩ V = ∅, there existsU ∈ ωO(X, x) such that F (u) ∩ Cl(V ) = ∅ for each u ∈ U ;

7. F−(V ) ⊂ ω Int(F−(Cl(V ))) for every V ∈ SO(Y ).

Proof. The proof is similar to that of Theorem 3.8.

Definition 3.10 [8] Let U be a subset of a topological space (X, τ). The set∩V ∈ RO(X) : U ⊂ V is called the r-kernel of U and is denoted by r-Ker(U).

Lemma 3.11 [8] The following properties hold for subsets U, V of a space X:

1. x ∈ r-Ker(U) if and only if U ∩ V = ∅ for any regular closed set V con-taining x.

2. U ⊂ r-Ker(U) and U = r-Ker(U) if U is regular open in X.

3. If U ⊂ V then r-Ker(U) ⊂ r-Ker(V ).

Corollary 3.12 [2] For a function f : (X, τ) → (Y, σ), the following statementsare equivalent:

1. f is almost contra-ω-continuous;

2. f−1(F ) ∈ ωO(X) for every F ∈ RC(Y );

3. for each x ∈ X and each F ∈ RC(Y, f(x)), there exists U ∈ ωO(X, x) suchthat f(U) ⊂ F ;

4. for each x ∈ X and each U ∈ RO(Y, f(x)), there exist V ∈ ωC(X, x) suchthat f(V ) ⊂ U ;

5. f−1(Int(Cl(G))) ∈ ωC(X) for every open subset G of Y ;

6. f−1(Cl(Int(F ))) ∈ ωO(X) for every closed subset F of Y ;

7. f(ωCl(A)) ⊂ rKer(f(A)) for every subset A of X;

8. ωCl(f−1(B)) ⊂ f−1(rKer(B)) for every subset B of Y .



2. F−(A) ∈ ωO(X) for every θ-semiopen A of Y ;

3. F+(U) ∈ ωC(X) for every θ-semiclosed subset U of Y ;

4. ωCl(F+(Int(Cl(B)))) ⊂ F+(sCl(B) for every subset B of Y ;


5. ωCl(F+(B)) ⊂ F+(sClθ(B)) for every subset B of Y ;

6. F (ωCl(A)) ⊂ sClθ(F (A)) for every subset A of X.

Proof. (1) ⇒ (2): Let G be any θ-semiopen set of Y . There exists a family ofregular closed sets Kα : α ∈ ∆ such that G = ∪Kα : α ∈ ∆. It follows fromTheorem 3.9 (ii) that F−(G) = ∪F−(Kα) : α ∈ ∆ is ω-open.

(2) ⇒ (3): This is obvious.

(3) ⇒ (4): Let B be any subset of Y . Then Int(Cl(B)) is regular open and itis θ-semiclosed in Y . Therefore, we have that F+(Int(Cl(B))) is ω-closed andωCl(F+(Int(Cl(B)))) = F+(Int(Cl(B))) ⊂ F+(sCl(B)).

(4) ⇒ (5): Let B be any subset of Y . For any regular open set V with B ⊂ V ,we have ωCl(F+(B)) ⊂ Cl(F+(V )) = ωCl(F+(Int(Cl(V )))) ⊂ F+(sCl(V )) =F+(V ). Therefore, ωCl(F+(B)) ⊂ F+(∩V ∈ RO(Y ) : B ⊂ V ) = F+(sClθ(B)).

(5) ⇒ (1): Let V be any semiopen set of Y . Then we have X\ω Int(F−(Cl(V ))) =ωCl(F+(Y \Cl(V ))) ⊂ F+(sClθ(Y \Cl(V ))) = F+(Y \Cl(V )) = X\F−(Cl(V )).Therefore, we obtain F−(V ) ⊂ F−(Cl(V )) ⊂ ω Int(F−(Cl(V ))). By Theorem 3.9(7), F is lower almost contra-ω-continuous.

(5) ⇒ (6): Let A be a subset of X and B = F (A). Then A ⊂ F+(B) andωCl(A) ⊂ ωCl(F+(B)) ⊂ F+(sClθ(B)). Therefore, we have F (ωCl(A)) ⊂F (F+(sClθ(B))) ⊂ sClθ(B) = sClθ(F (A)).

(6) ⇒ (5): Let B be any subset of Y . Then we have F (ωCl(F+(B))) ⊂sClθ(F (F+(B))) ⊂ sClθ(B); hence ωCl(F+(B)) ⊂ F+(sClθ(B)).

Corollary 3.14 For a function f : (X, τ) → (Y, σ), the following properties areequivalent:

1. f is almost contra-ω-continuous;

2. f−1(V ) ∈ ωO(X) for each θ-semiopen set V of Y ;

3. f−1(F ) ∈ ωC(X) for each θ-semiclosed set F of Y .

4. for each x ∈ X and each U ∈ SO(Y, f(x)), there exist V ∈ ωO(X, x) suchthat f(V ) ⊂ Cl(U);

5. f−1(V ) ⊂ ω Int(f−1(Cl(V ))) for every V ∈ SO(Y ).

6. f(ωCl(A)) ⊂ sClθ(f(A)) for every subset A of X;

7. ωCl(f−1(B)) ⊂ f−1(sClθ(B)) for every subset B of Y .

8. ωCl(f−1(V )) ⊂ f−1(sClθ(V )) for every open subset V of Y .

9. ωCl(f−1(V )) ⊂ f−1(sCl(V )) for every open subset V of Y .

10. ωCl(f−1(V )) ⊂ f−1(Int(Cl(V ))) for every open subset V of Y .




2. ωCl(F−(Int(K))) ⊂ F−1(K) for every semiclosed set K of Y ;

3. ωCl(F−(Int(sCl(B)))) ⊂ F−(sCl(B)) for every B ⊆ Y ;

4. F+(s Int(B)) ⊂ ω Int(F+(Cl(s Int(B)))) for every B ⊆ Y .

Proof. (1) ⇒ (2): Let K be a semiclosed set of Y . Then Y \K is semiopen.By Theorem 3.8 (7), it follows that F+(Y \K) ⊂ ω Int(F+(Y \ Int(K))). HenceX\F−(K) ⊂ ω Int(F+(Y \ Int(K)))=ω Int(X\F−(Int(K)))=X\ωCl(F−(Int(K))).Hence, ωCl(F−(Int(K))) ⊂ F−1(K).

(2) ⇒ (3): Let B be any subset of Y . Then sCl(B) is semiclosed in Y and henceωCl(F−(Int(sCl(B)))) ⊂ F−(sCl(B)).

(3) ⇒ (4): Let B be any subset of Y . Then we have

X\F+(s Int(B)) = F−(sCl(Y \B)) ⊃ ωCl(F−(Int(sCl(Y \B))))

= ωCl(F−(Int(Y \s Int(B)))) = ωCl(F−(Y \Cl(s Int(B))))

= ωCl(X\F+(Cl(s Int(B)))) = X\ω Int(F+(Cl(s Int(B)))).

Hence, F+(s Int(B)) ⊂ ω Int(F+(Cl(s Int(B)))).

(4) ⇒ (1): Let V be any semiopen set of Y . Then V = s Int(V ) and henceF+(V ) ⊂ ω Int(F+(Cl(V ))). By Theorem 3.8 (7), F is upper almost contra-ω-continuous.



2. ωCl(F+(Int(K))) ⊂ F+(K) for every semiclosed set K of Y ;

3. ωCl(F+(Int(sCl(B)))) ⊂ F+(sCl(B)) for every B ⊆ Y ;

4. F−(s Int(B)) ⊂ ω Int(F−(Cl(s Int(B)))) for every B ⊆ Y .

Proof. The proof is similar to that of Theorem 3.15.Recall that a topological space is said to be extremely disconnected if the

closure of every open set is open in the space.

Theorem 3.17 Let (Y, σ) be an extremely disconnected space. Then a multifunc-tion F : (X, τ) → (Y, σ) is upper almost contra-ω-continuous if and only if it isupper almost ω-continuous.


Proof. Let x ∈ X and V be any regular open set of Y containing F (x). Since(Y, σ) is extremely disconnected, V is regular closed and hence semiopen. ByTheorem 3.8, there exists U ∈ ωO(X, x) such that F (U) ⊂ Cl(V ) = V . ThenF is upper almost ω-continuous. Conversely, let K be any regular closed subsetof Y . Since (Y, σ) is extremely disconnected, K is also regular open and byTheorem 3.4 of [5], F+(K) is ω-open. By Theorem 3.8, F is upper almost contra-ω-continuous.

Theorem 3.18 Let (Y, σ) be an extremely disconnected space. Then a multifunc-tion F : (X, τ) → (Y, σ) is lower almost contra-ω-continuous if and only if F islower almost ω-continuous.


Theorem 3.19 The following statements are equivalent for a multifunctionF : (X, τ) → (Y, σ):

1. F is upper (lower) almost contra-ω-continuous;

2. F+(Cl(V ))(F−(Cl(V ))) is ω-open in X for every V ∈ SPO(Y );

3. F+(Cl(V ))(F−(Cl(V ))) is ω-open in X for every V ∈ SO(Y );

4. F−(Int(Cl(V )))(F+(Int(Cl(V )))) is ω-closed in X for every V ∈ PO(Y ).

Proof. (1) ⇒ (2): Suppose that V is any semi-preopen set of Y . SinceCl(V ) ∈ RC(Y ), by Theorem 3.8, F−(Cl(V )) is ω-open in X.

(2) ⇒ (3): This is obvious, since SO(Y ) ⊂ SPO(Y ).

(3) ⇒ (4): Let V ∈ PO(Y ). Then Y \ Int(Cl(V )) is regular closed and henceit is semiopen. Then, we have X\F−(Int(Cl(V ))) = F+(Y \ Int(Cl(V ))) =F+(Cl(Y \ Int(Cl(V )))) ∈ ωO(X). Hence F−(Int(Cl(V ))) ∈ ωC(X).

(4) ⇒ (1): If V is any regular open set of Y . Then V ∈ PO(Y ) and henceF−(V ) = F−(Int(Cl(V ))) is ω-closed in X. Therefore, F is upper almost contra-ω-continuous.

The proof of the second case is similar.

Lemma 3.20 [16] For a subset V of a topological space (Y, σ), the following prop-erties hold:

1. αCl(V ) = Cl(V ) for every V ∈ SPO(Y );

2. pCl(V ) = Cl(V ) for every V ∈ SO(Y );

3. sCl(V ) = Int(Cl(V )) for every V ∈ PO(Y ).

Corollary 3.21 The following statements are equivalent for a multifunctionF : (X, τ) → (Y, σ):


1. F is upper (lower) almost contra-ω-continuous;

2. F+(αCl(V ))(F−(αCl(V ))) is ω-open in X for every V ∈ SPO(Y );

3. F+(pCl(V ))(F−(pCl(V ))) is ω-open in X for every V ∈ SO(Y );

4. F−(sCl(V ))(F+(sCl(V ))) is ω-closed in X for every V ∈ PO(Y ).

Proof. This is an immediate consequence of Theorem 3.19 and Lemma 3.20.



2. ωCl(F−(V )) ⊂ F−(Int(Cl(V )) for every open subset V of Y ;

3. ωCl(F−(V )) ⊂ F−(sCl(V )) for every open subset V of Y .

Proof. (2) ⇒ (1): Let V ∈ RO(Y ). Then ωCl(F−(V )) ⊂ F−(Int(Cl(V )) =F−(V ). This implies that F−(A) is ω-closed and hence F is upper almost contra-ω-continuous.

(1) ⇒ (2): Let V be an open set. We have Int(Cl(V )) ∈ RO(Y ). By (1),F−(Int(Cl(V )) is ω-closed. Since V ⊂ Int(Cl(A)), then F−(A) ⊂ F−(Int(Cl(A))).Thus, ωCl(F−(V )) ⊂ F−(sCl(V )).

(2) ⇔ (3): It follows from Lemma 3.20.



2. ωCl(F+(V )) ⊂ F+(Int(Cl(V )) for every open subset V of Y ;

3. ωCl(F+(V )) ⊂ F+(sCl(V )) for every open subset V of Y .


Theorem 3.24 Let F : (X, τ) → (Y, σ) be any multifunction. If ωCl(F−(V )) ⊂F−(r-Ker(V )) for every subset V of Y , then F is upper almost contra-ω-con-tinuous.

Proof. Let V ∈ RO(Y ). By Lemma 3.11, ωCl(F−(V )) ⊂ F−(r-Ker(V )) =F−(V ). This implies that ωCl(F−(V )) = F−(V ); hence F−(V ) is ω-closed.By Theorem 3.8, F is upper almost contra-ω-continuous.

Theorem 3.25 Let F : (X, τ) → (Y, σ) be any multifunction. If F (ωCl(V )) ⊂ r-Ker(F (V )) for every subset V of Y , then F is lower almost contra-ω-continuous.


Proof. Let H be a regular open set of X. Then F (ωCl(F+(H))) ⊂ r-Ker(H)and ωCl(F+(H)) ⊂ F+(r-Ker(H)). By Lemma 3.11,

ωCl(F+(H)) ⊂ F+(r-Ker(H)) = F+(H).

We have ωCl(F+(H)) = F+(H). This implies that F+(H) is ω-closed in X.By Theorem 3.9, F is lower almost contra-ω-continuous.

Definition 3.26 A topological space (X, τ) is said to ω-T2 [2], if for each pair ofdistinct points x and y in X, there exist disjoint ω-open sets U and V in X suchthat x ∈ U and y ∈ V .

Lemma 3.27 [19] If A and B are disjoint compact subsets of an Urysohn spaceX, there exist open sets U and V of X such that A ⊂ U , B ⊂ V and Cl(U) ∩Cl(V ) = ∅.

Theorem 3.28 If F : (X, τ) → (Y, σ) is an upper almost contra-ω-continuousinjective multifunction into an Urysohn space Y and F (x) is compact for eachx ∈ X, then X is ω-T2.

Proof. For any distinct points x1, x2 ∈ X, we have F (x1) ∩ F (x2) = ∅, sinceF is injective and F (x1) and F (x2) are disjoint compact sets, by Lemma 3.27,there exist open sets V1 and V2 such that F (x1) ⊂ V1, F (x2) ⊂ V2 and Cl(V1) ∩Cl(V2) = ∅. Since Cl(V1) and Cl(V2) are regular closed sets and F is upper almostcontra-ω-continuous, there exist U1 ∈ ωO(X, x1) and U2 ∈ ωO(X, x2) such thatF (U1) ⊂ Cl(V1), F (U2) ⊂ Cl(V2); hence U1 ∩ U2 = ∅ and X is ω-T2.

Definition 3.29 A subset A of a topological space (X, τ) is said to be ω-densein X if ωCl(A) = X.

Definition 3.30 A multifunction F : (X, τ) → (Y, σ) is called upper weaklycontinuous [19], if for each open set V containing F (x) and for each x ∈ X, thereexists an open set U containing x such that F (U) ⊂ Cl(V ).

Theorem 3.31 Let X be a topological space and Y an Urysohn space. If thefollowing four conditions are satisfied:

1. F : (X, τ) → (Y, σ) is an upper weakly continuous multifunction,

2. G : X → Y is an upper almost contra-ω-continuous multifunction,

3. F (x) and G(x) are compact sets of Y for each x ∈ X,

4. A = x ∈ X : F (x) ∩G(x) = ∅,

then A is ω-closed. Moreover if F (x) ∩ G(x) = ∅ for each point x ∈ X in aω-dense set D, then F (x) ∩G(x) = ∅ for each x ∈ X.


Proof. Suppose that x ∈ X\A. Then we have F (x) ∩G(x) = ∅. Since F (x) andG(x) are disjoint compact sets of an Urysohn space, by Lemma 3.27, there existopen sets V and W such that F (x) ⊂ V and G(x) ⊂ W and Cl(V ) ∩Cl(W ) = ∅.Since F is upper weakly continuous and F (x) ⊂ V , there exists an open set U1

containing x such that F (U1) ⊂ Cl(V ). Since Cl(W ) is regular closed, G(x) ⊂ Wand G is upper almost contra-ω-continuous, there exists U2 ∈ ωO(X, x) such thatG(U2) ⊂ Cl(W ). Let U = U1 ∩ U2, then U is ω-open and U ∩ A = ∅. Therefore,x ∈ X\ωCl(A) and hence A is ω-closed. On the other hand, if F (x) ∩ G(x) = ∅on an ω-dense set D of X, then we have X = ωCl(D) ⊂ ωCl(A) = A. Therefore,we obtain F (x) ∩G(x) = ∅ for each x ∈ X.

Definition 3.32 A subset A of a space (X, τ) is said to be:

1. α-regular [11], if for each a ∈ A and any open set U containing a, thereexists an open set V of X such that a ∈ V ⊂ Cl(V ) ⊂ U ;

2. α-paracompact [11], if every X-open cover A has an X-open refinementwhich covers A and is locally finite for each point of X.

Lemma 3.33 [11] If A is an α-regular and α-paracompact subset of a space Xand U is an open neighborhood of A, then there exists an open set V of X suchthat A ⊂ V ⊂ Cl(V ) ⊂ U .

For a multifunction F : (X, τ) → (Y, σ), the multifunction Cl(F ) : X → Yis defined by Cl(F )(x) = Cl(F (x)) for each point x ∈ X. Similarly, we denotesCl(F ), pCl(F ), αCl(F ), spCl(F ), ωCl(F ).

Lemma 3.34 [18] If F : (X, τ) → (Y, σ) is a multifunction such that F (x) isα-paracompact α-regular for each x ∈ X, then for each open set V of Y , G+(V )= F+(V ) and for each closed set K of Y , G−(K) = F−(K), where G denotesCl(F ), sCl(F ), pCl(F ), αCl(F ), spCl(F ), ωCl(F ).

Lemma 3.35 [18] If F : (X, τ) → (Y, σ) is a multifunction, then for each openset V of Y , G−(V ) = F−(V ) and for each closed set K of Y , G+(K) = F+(K),where G denotes Cl(F ), sCl(F ), pCl(F ), αCl(F ), spCl(F ), ωCl(F ).

Theorem 3.36 A multifunction F : (X, τ) → (Y, σ) is upper almost contra-ω-continuous if and only if G is upper almost contra-ω-continuous.

Proof. Let K be a regular closed set of Y . By Theorem 3.8 and Lemma 3.35,G+(K) = F+(K) is a ω-open set of X. Hence, G is upper almost contra-ω-continuous. Conversely, Let K be a regular closed set of Y . By Theorem 3.8 andLemma 3.35, F+(K) = G+(K) is a ω-open set of X. Hence, G is upper almostcontra-ω-continuous.

Theorem 3.37 Let F : (X, τ) → (Y, σ) be a multifunction such that F (x) isα-regular α-paracompact for each x ∈ X. Then F is lower almost contra-ω-continuous if and only if G is lower almost contra-ω-continuous.


Proof. Let K be a regular closed set of Y . By Theorem 3.9 and Lemma 3.34,G−(K) = F−(K) is an ω-open set of X. Hence, G is lower almost contra-ω-continuous. Conversely, Let K be a regular closed set of Y . By Theorem 3.9 andLemma 3.34, F−(K) = G−(K) is an ω-open set of X. Hence, G is lower almostcontra-ω-continuous.

Theorem 3.38 Let F : (X, τ) → (Y, σ) be a multifunction and U be an open sub-set of X. If F is a lower (upper) almost contra-ω-continuous, then F |U : U → Yis lower (upper) almost contra-ω-continuous.

Proof. Let V be any regular closed set of Y . Let x ∈ U and x ∈ (F |U)−(V ).Since F is a lower almost contra-ω-continuous multifunction, then there existsG ∈ ωO(X, x) such that G ⊂ F−(V ). Then by Lemma 2.3, x ∈ G ∩ U ∈ ωO(U)and G ∩ U ⊂ (F |U)−(V ). This shows that F |U is a lower almost contra-ω-conti-nuous multifunction. The proof of the second case is similar.

Theorem 3.39 Let Ui : i ∈ ∆ be an open cover of a space X. A multifunc-tion F : (X, τ) → (Y, σ) is upper almost contra-ω-continuous if and only if therestriction F |Ui

: Ui → Y is upper almost contra-ω-continuous for each i ∈ ∆.

Proof. Suppose that F is an upper almost contra-ω-continuous multifunction.Let i ∈ ∆, x ∈ Ui and V be a regular closed set of Y containing F |Ui

(x). Since Fis an upper almost contra-ω-continuous multifunction and F (x) = F |Ui

(x), thereexistsG ∈ ωO(X, x) such that F (G) ⊂ V . Set U = G∩Ui, then x ∈ U ∈ ωO(Ui, x)and F |Ui

(U) = F (U) ⊂ V . Therefore, F |Uiis upper almost contra-ω-continuous.

Conversely, let x ∈ X and V ∈ RC(Y ) containing F (x). There exists i ∈ ∆ suchthat x ∈ Ui. Since F |Ui

is upper almost contra-ω-continuous and F (x) = F |Ui(x),

there exists U ∈ ωO(Ui, x) such that F |Ui(U) ⊂ V . Then we have U ∈ ωO(X, x)

and F (U) ⊂ V . Therefore, F is upper almost contra-ω-continuous.

Theorem 3.40 Let Ui : i ∈ ∆ be an open cover of a space X. A multifunc-tion F : (X, τ) → (Y, σ) is lower almost contra-ω-continuous if and only if therestriction F |Ui

: Ui → Y is lower almost contra-ω-continuous for each i ∈ ∆.

Proof. The proof is similar to that of Theorem 3.39 and is thus omitted.

For a multifunction F : (X, τ) → (Y, σ), the graph multifunctionGF : X → Yis defined as follows: GF (x) = x × F (x) for every x ∈ X.

Theorem 3.41 If GF : X → X × Y is an upper almost contra-ω-continuousmultifunction, then F : (X, τ) → (Y, σ) is an upper almost contra-ω-continuousmultifunction.

Proof. Let x ∈ X and K ∈ RC(Y ) with F (x) ⊂ K. Since X × K is regularclosed in X × Y and GF (x) ⊂ X ×K, then there exists U ∈ ωO(X, x) such thatGF (U) ⊂ X × K. By Lemma 2.4, U ⊂ G+

F (X × K) = F+(K) and F (U) ⊂ K.Thus, F is an upper almost contra-ω-continuous multifunction.


Theorem 3.42 If GF : X → X × Y is a lower almost contra-ω-continuous mul-tifunction, then F : (X, τ) → (Y, σ) is a lower almost contra-ω-continuous multi-function.

Proof. Let x ∈ X and K ∈ RC(Y ) with x ∈ F−(K). Since X × K is regularclosed inX×Y andGF (x)∩(X×K)=(x×F (x))∩(X×K)=x×(F (x)∩K) = ∅.Since GF is lower almost contra-ω-continuous, then there exists U ∈ ωO(X, x)such that U ⊂ G−

F (X × K). Then U ⊂ F−(K). Hence, F is a lower almostcontra-ω-continuous.

Theorem 3.43 Let F : (X, τ) → (Y, σ) be an upper almost contra-ω-continuoussurjective multifunction and F (x) is a S-closed relative to Y for each x ∈ X. IfA is a ω-compact relative to X, then F (A) is a S-closed relative to Y .

Proof. Let Vi : i ∈ ∆ be any cover of F (A) by regular closed sets of Y . Foreach x ∈ A, there exists a finite subset ∆(x) of ∆ such that

F (x) ⊂ ∪Vi : i ∈ ∆(x).

Put V (x) = ∪Vi : i ∈ ∆(x). Then F (x) ⊂ V (x) and there exists U(x) ∈ωO(X, x) such that F (U(x)) ⊂ V (x). Since U(x) : x ∈ A is a cover of A byω-open sets in X, there exists a finite number of points of A, say, x1, x2,....xn suchthat A ⊂ ∪U(xi) : 1 = 1, 2, ....n. Therefore, we obtain

F (A) ⊂ F

(n∪

i=1

U(xi)

)n∪

i=1

F (U(xi))n∪

i=1

V (xi) ⊂n∪

i=1

∪i∈∆(xi)

Vi.

This shows that F (A) is a S-closed relative to Y .

Theorem 3.44 Let X and Xi be topological spaces for i ∈ I. If F : X →∏i∈I

Xi

is an upper (lower) almost contra-ω-continuous multifunction, then Pi F is anupper (lower) almost contra-ω-continuous multifunction for each i ∈ I, where

Pi :∏i∈I

Xi → Xi is the projection for each i ∈ I.

Proof. Let Hi be a regular closed subset of Xj. We have

(Pj F )+(Hj) = F+(P+j (Hj)) = F+

(Hj ×

∏i=j

Xi

).

Since F is an upper almost contra-ω-continuous multifunction, F+

(Hj ×

∏i=j

Xi

)is ω-open in X. Hence, Pi F is an upper (lower) almost contra-ω-continuous.


Theorem 3.45 Let Xi and Yi be topological spaces and Fi : Xi → Yi be a mul-

tifunction for each i ∈ I. If F :∏i∈I

Xi →∏i∈I

Yi, defined by F (xi) =∏i∈I

Fi(xi),

is upper (lower) almost contra-ω-continuous multifunction, then Fi is upper (lower)almost contra-ω-continuous multifunction for each i ∈ I.

Proof. Let Hi ⊂ Yi be a regular closed subset. Since F is upper almost contra-ω-continuous multifunction, F+(Hi × Π

i =jYj) = F+

i (Hi)× Πi=j

Xj is an ω-open set.

Thus, F+i (Hi) is an ω-open set; hence F is upper almost contra-ω-continuous

multifunction.

References

[1] Al-Zoubi, K., Al-Nashef, B., The topology of ω-open subsets, Al-Manarah, (9) (2003), 169-179.

[2] Al-Omari, A., Noorani, M.S.M., Contra-ω-continuous and almost ω-continuous functions, Int. J. Math. Math. Sci. (9) (2007), 169-179.

[3] Andrijevic, D., Semi-preopen sets, Mat. Vesnik, 38 (1986), 24-32.

[4] Banzaru, T., Multifunctions and M-product spaces, Bull. Stin. Tech. Inst.Politech. Timisoara, Ser. Mat. Fiz. Mer. Teor. Apl., 17 (31) (1972), 17-23.

[5] Carpintero, C., Rajesh, N., Rosas, E., Saranyasri, S., Some proper-ties of Upper/lower almost ω-continuous functions, Sci. Stud. Res. Ser. Math.Inform., por aparecer en 2014.

[6] Carpintero, C., Rajesh, N., Rosas, E., Saranyasri, S., On upper andlower contra-ω-continuous multifunctions, to appear in Novi Sad J. Math.,(2014).

[7] Carpintero, C., Rajesh, N., Rosas, E., Saranyasri, S., Upper andlower weakly ω-continuous functions (submitted).

[8] Ekici, E., On contra R-continuity and a weak form, Indian J. Math., 46(2-3) (2004), 267-281.

[9] Ekici, E., Jafari, S., Popa V., On almost contra-continuous multifunc-tions, Lobachesvskii J. Math., 30 (2) (2009), 124-131.

[10] Hdeib, H.Z., ω-closed mappings, Revista Colombiana Mat., 16 (1982),65-78.

[11] Kovacevic, Subsets and paracompactness, Univ. u. Novom Sadu, Zb.Rad. Prirod. Mat. Fac. Ser. Mat., 14 (1984), 79-87.


[12] Joseph, J.E., Kwack, M.H., On S-closed spaces, Proc. Amer. Math. Soc.,80 (1980), 341-348.

[13] Levine, N., Semiopen sets and semicontinuity in topological spaces, Amer.Math. Monthly, 70 (1963), 36-41.

[14] Mashhour, A.S., Abd El-Monsef, M.E., El-Deep, S.N., On precon-tinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt, 53(1982), 47–53.

[15] Njastad, O., On some classes of nearly open sets, Pacific J. Math., 15(1965), 961-970.

[16] Noiri, T., On almost continuous functions, Indian J. Pure Appl. Math., 20(1989), 571-576.

[17] Noiri, T., Almost weakly continuous multifunctions, Demonstratio Math.,26 (1993), 363-380.

[18] Noiri, T., Popa, V., A unified theory of weak continuity for multifunctions,Stud. Cerc. St. Ser. Mat. Univ. Bacau, 16 (2006), 167-200.

[19] Simithson, R.E., Almost and weak continuity for multifunctions, Bull. Cal-cutta Math. Soc., 70 (1978), 383-390.

[20] Stone, M., Applications of the theory of boolean rings to general topology,Trans. Amer. Math. Soc., 41 (1937), 374-381.

[21] Thompson, T., S-closed spaces, Proc. Amer. Math. Soc., 60 (1976), 335-338.



IRREDUCIBLE IDEALS IN RINGS

B. Venkateswarlu

Department of MathematicsGIT, GITAM UniversityVisakhapatnam 530 045, A.P.India.e-mail: [email protected]

R. Vasu Babu

Department of MathematicsShri Vishnu Engineering College for WomenBhimavaram, W.G. Dist., A.P.Indiae-mail: [email protected]

Tigist Embiale

Department of MathematicsGondar UniversityGondarEthiopiae-mail: [email protected]

Abstract. It is well known that the ideals of any ring form an algebraic lattice under the

set inclusion ordering, in which the finitely generated ideals are precisely the compact

elements. Strongly irreducible ideals of a ring were studied by W.J. Heinzer, L.J. Ratliff

and D.E. Rush [3] and α−irreducible and α−strongly irreducible ideals of a ring were

characterized by X. Lu and H. Qin [8]. In this paper, we extend these results for elements

of a general algebraic lattice and obtain the results on ideals of rings and on submodules

of an R− module as consequences of our general results. Also, we characterize algebraic

lattices satisfying the ascending chain condition.

Keywords: α−irreducible, α−strongly irreducible, compact element, algebraic lattice,

ascending chain condition, module over a ring.

2000 Mathematics Subject Classification: 06D10, 06B23.


The concepts of irreducible elements and prime elements are crucial in the studyof the structure theory of general algebraic systems, in particular, in that of dis-tributive lattices. For example, the prime ideals play a vital role in the pioneeringwork of M.H. Stone [5] on the representation theory of distributive lattices. Theprime ideals of a lattice are precisely the prime elements in the lattice of its ideals.In a general lattice, prime elements are also called strongly irreducible elements.

462 b. venkateswarlu, r. vasu babu, t. embiale

First, we recall certain elementary concepts and notations from Theory ofLattices [1]. A partially ordered set (L,≤) is called a lattice (complete lattice)if every two element subset (respectively, every arbitrary subset) of L has bothinfimum and supremum in L; For any subset A of L, we write inf A or

∧A or

∧a∈A

a

for the infimum (greatest lower bound) of A and supA or∨A or

∨a∈A

a for the

supremum ( least upper bound ) of A. If A is a finite set a1, a2, · · · , an, then we

writen∧

i=1

ai or a1∧a2∧· · ·∧an for the inf A andn∨

i=1

ai or a1∨a2∨· · ·∨an for the supA.

If (L,≤) is a lattice, then a∧b = infa, b and a∨b = supa, b give us two binaryoperations ∧ and ∨ on L which are both associative, commutative and idempotentand satisfy the absorption laws (a∧(a∨b) = a = a∨(a∧b)). Conversely if ∧ and ∨are binary operations on a non empty set L which satisfy all the above propertiesand if the partial order ≤ on L is defined by a = a ∧ b ⇔ a ≤ b ⇔ a ∨ b = b,then (L,≤) is a lattice in which a ∧ b and a ∨ b are respectively the infimum andsupremum of any a, b.

A lattice (L,≤) is called bounded if it has smallest element 0 (that is, 0 ≤ afor all a ∈ L) and the largest element 1 (that is, a ≤ 1 for all a ∈ L). A completelattice is necessarily bounded. Logically, the infimum and supremum of the emptyset, if they exist, are the largest element 1 and smallest element 0 respectively.An element a of a complete lattice L is called compact if, for any X ⊆ L, a ≤supX =⇒ a ≤ supF for some finite subset F of X. A complete lattice L is calledan algebraic lattice if every element of L is the supremum of a set of compactelements in L. An element m in a partially ordered set (L,≤) is called maximalif m ≤ a ∈ L implies m = a.

2. Algebraic Lattices with ACC

Let us recall that a partially ordered set (P,≤) is said to satisfy the ascendingchain condition (ACC) if any increasing sequence in P terminates at a finite stage;that is, if a1 ≤ a2 ≤ · · · ≤ an ≤ · · · is a sequence in P such that an ≤ an+1 for all n,then there exists n0 such that an0 = an0+k for all k > 0. It is well known that (P,≤)satisfies ACC if and only if every non empty subset of P has a maximal element.In the following, we characterize algebraic lattices satisfying the ascending chaincondition.

Theorem 2.1. Let (L,≤) be an algebraic lattice. Then (L,≤) satisfies the ACCif and only if every element of L is compact.

Proof. Suppose that (L,≤) satisfies ACC. Let a ∈ L. We can assume that a = 0,since the smallest element 0 is always compact. Let X be a subset of L such thata ≤ supX. Since a = 0, X is non empty. Consider the set

A = supF | F is a finite subset of X.

A is a non empty subset of L. Since L satisfies ACC, A has a maximal element.Let m be a maximal element in A. Then, m = supF, for some finite subset F

irreducible ideals in rings 463

of X. For any x ∈ X, we have m = supF ≤ sup(F ∪ x) ∈ A and, by themaximal element of m,m = sup(F ∪x), so that x ≤ m for all x ∈ X. Therefore,m = supX and a ≤ m = supF and F is a finite subset of X. Thus, a is compact.

Conversely, suppose that every element of L is compact. Let a1 ≤ a2 ≤· · · ≤ an ≤ · · · be an increasing sequence in L and a = sup

n∈Z+

an. Then a is

compact and hence a = an1 ∨ an2 ∨ · · · ∨ anr for some n1, n2, · · · , nr ∈ Z+. Ifn = maxn1, n2, · · · , nr, then a = an1 ∨ an2 ∨ · · · ∨ anr = an and hence ak ≤ anfor all k. Therefore an = an+k for all k ∈ Z+. Thus (L,≤) satisfies the ascendingchain condition.

Recall that an ideal I of a commutative ring R with unity is called finitelygenerated if there exist a1, a2, · · · , an ∈ R such that

I = r1a1 + r2a2 + · · ·+ rnan | ri ∈ R.

Also, R is said to be a Noetherian ring if the lattice of ideals ofR satisfies ascendingchain condition. It is well known that the ideals of any ring form an algebraiclattice, under the set inclusion order, in which the compact elements are preciselythe compact elements. We record the following well known result as a consequenceof the above theorem.

Corollary 2.2. The following are equivalent to each other for any ring R.

(1) R is a Noetherian ring.

(2) Every ideal of R is compact in the lattice of ideals of R.

(3) Every ideal of R is finitely generated.

(4) Any nonempty class of ideals has a maximal member.

On the same lines as discussed above, the submodule of an R−module (whereR is any ring) form an algebraic lattice under the set inclusion ordering in whichthe finitely generated R−submodules are precisely the compact elements. AnR−module is said to be Noetherian if the lattice of R−submodules satisfies theascending chain condition.

Corollary 2.3. The following are equivalent to each other for any module M overa ring R.

(1) M is a Noetherian R− module.

(2) Every submodule of M is compact in the lattice of submodules of M .

(3) Every submodule of M is finitely generated.

(4) Any non empty class of submodules of M has a maximal member.

It is known that a complete lattice is algebraic if and only if it is isomorphicto the lattice of subuniverses of an (universal) algebra. In view of this, we havethe following

Corollary 2.4. The following are equivalent to each other for any universal al-gebra A.


(1) The lattice of subuniverses of A satisfies ACC.

(2) Every subuniverse of A is compact in the lattice of subuniverses of A.

(3) Any subuniverse of A is finitely generated.

(4) Any class of subuniverses of A has a maximal member.

3. α−Irreducibility

In this section, we recall the notions of irreducible element and strongly irreducibleelement in a general complete lattice and discuss certain important properties ofthese in order to facilitate characterizations of these among ideals of a ring.

Definition 3.1. Let (L,∧,∨) be a bounded lattice and 1 = p ∈ L.

(1) p is said to be irreducible if, for any a and b ∈ L,

p = a ∧ b =⇒ p = a or p = b.

(2) p is said to be strongly irreducible or prime if, for any a and b ∈ L,

a ∧ b ≤ p =⇒ a ≤ p or b ≤ p.

Clearly, every prime element in any bounded lattice is irreducible and theconverse is not true. However, if the lattice is distributive, then every irreducibleelement is prime. In the following, we write | A | for the cardinality of a set A.

Definition 3.2. Let (L,≤) be a complete lattice, 1 = p ∈ L and α a cardinalnumber.

(1) p is said to be α−irreducible if, for any A ⊆ L with | A |≤ α,

p = inf A =⇒ p ∈ A.

(2) p is said to be α−strongly irreducible or α−prime if, for any A ⊆ L with| A |≤ α,

inf A ≤ p =⇒ a ≤ p for some a ∈ A.

An element p is irreducible (prime) if and only if p is α−irreducible (respec-tively α−prime) for all positive integers α. Also, clearly every α− prime element isα− irreducible and the converse is not true; for, consider the set N of non negativeintegers together with the partial order defined by ‘ a ≤ b ⇐⇒ b divides a ′. Then(N,≤) is a complete lattice in which, for any subset A, infimum and supremumof A are precisely the LCM and GCD of A respectively. Here 2 is α− irreduciblefor all cardinal numbers α. But 2 is not α−prime, since the infimum of the set ofall odd primes is 0 and 0 < 2.

However, if the lattice L satisfies the infinite join distributivity (that is,x∨(inf A) = infx∨a | a ∈ A for all x ∈ L and A ⊆ L), then every α−irreducibleelement in L is α−prime. Note that the complete lattice (N,≤) discussed abovedoes not satisfy the infinite join distributivity. The following two theorems areproved in 2.2 and 2.7 of [6].

irreducible ideals in rings 465

Theorem 3.3. An element p = 1 in an algebraic lattice L is α−irreducible forevery cardinal number α if and only if there exists a compact non zero elementc ∈ L such that p is a maximal element in the set x ∈ L | c x.

Theorem 3.4. An element p = 1 in an algebraic lattice L is α−prime for everycardinal number α if and only if there exists a compact non zero element c ∈ Lsuch that p is the unique maximal element in the set x ∈ L | c x.

Also, it is proved in 2.5 of [6] that any element of an algebraic lattice L is the infi-mum of a set of elements in L which are α−irreducible for all cardinal numbers α.

4. Irreducible ideals of a ring

It is well known that ideals of a ring R form a complete lattice under the setinclusion order, in which the infimum of any class Iα∈∆ of ideals of R issimply the set intersection, while the supremum is the ideal generated

∪α∈∆

Iα.

Let us denote the set of all ideals of a ring R by I (R).

Theorem 4.1. Let I be a proper ideal of a ring R. Then I is α−irreducible inthe lattice I (R) for every cardinal number α if and only if there exists a finitenon empty subset F of R such that I is a maximal member in the set of all idealsof R not containing F .

Proof. First, observe that the lattice I (R) of ideals of R is an algebraic latticein which the compact elements are precisely the finitely generated ideals of R.Also, for any ideal J of R and for any subset F or R, we have F ⊆ J if and onlyif < F >⊆ J, where < F > is the ideal generated by F in R. Now, the theoremfollows from Theorem 3.3.

Note that an ideal I of a ring R with unity is a proper ideal if and only ifI does not contain the unity element. This, together with the above theorem,implies the following.

Corollary 4.2. Let R be a ring with unity. Then, any maximal ideal of R isα−irrducible in I (R) for all cardinal numbers α.

A proof analogous to that of Theorem 4.1 together with Theorem 3.4 yieldthe following.

Theorem 4.3. A proper ideal I of a ring R is α−prime in I (R) for all cardinalnumbers α if and only if there exists a non empty finite subset F of R such thatI is the unique maximal member in the set of all ideals of R not containing F.

Recall that a ring with unity is called a local ring if it has a unique maximalideal. The following is an immediate consequence of the above theorem.

Corollary 4.4. In a local ring, the maximal ideal is α− prime for all cardinalnumbers α.


Example 4.5. Consider the ring Z of integers. It is well known that any ideal ofZ is of the form nZ for some non negative integer n. Z is a Noetherian ring andevery ideal of Z is compact in the lattice I (Z) of ideals of Z. Since

0 = ∩pZ | p is a prime number,

0 is not α−irreducible for any infinite cardinal α, However, 0 is irreducible,since

0 = nZ ∩mZ =⇒ n = 0 or m = 0.

It can be proved that I (Z) is a distributive lattice and that a non zero ideal nZ isirreducible (and hence prime) if and only if n = pr for some prime number p anda positive integer r. Also, any ideal of Z is not α−prime, for an infinite cardinalα; for, let nZ be a non zero proper ideal of Z. Then n > 1. Consider the set

X = pZ | p is a prime number not dividing n.

Then infX = 0 ⊆ nZ and pZ " nZ for all pZ ∈ X. Thus nZ is not α−prime.On the other hand, nZ is α−irreducible for all cardinals α if and only if n = pr

for some prime number p and a positive integer r.

Theorems 4.1 and 4.3 can be extended to submodules of an R−module M,since a submodule of M is compact in the lattice of submodules of M if and onlyif it is finitely generated over R.

Acknowledgement. The authors thank Professor U.M. Swamy for his help inpreparing this paper.

References

[1] Birkhoff, G., Lattice theory, AMS Colloquium Publications, Amer. Math.Soc., 1967.

[2] Birkhoff,G., Frink,O., Representations of lattices by sets, Trans. Amer.Mth.Soc., 64 (1948), 299-316.

[3] Heinzer, W.J., Ratliff Jr., L.J., Rush, D.E., Strongly irreducibleideals of a commutative ring, J.of Pure and Appl. Algebra, 166 (2002), 267-275.

[4] Stanely Burris, Sankappanavar, H.P., A Course in universal algebra,Springer-Verlag, New York, 1980.

[5] Stone, M.H., Topological representation of distributive lattices and Brouw-erian logics, Casopis Pet. Mat. Fys., 67 (1937), 125.

[6] Swamy, U.M., Venkateswarlu, B., Irreducible Elements in AlgebraicLattices, International Journal of Algebra and Computations, vol. 8, no. 8(2010), 969-975.

[7] Venkateswarlu, B., Swamy, U.M., A characterization of irreducibleelements in Algebraic lattices, accepted for publication in Afrika Mathematika(DOI:10.1007/s13370-014-0231-5).

[8] Xinnmin Lu, Hourong Qin, α-Irreducible and α-Strongly IrreducibleIdeals of ring, Southeast Asian Bulletin of Mathematics, 32 (2008) 299-303.

Accepted: 6.04.2014


INCLUSION RESULTS ON SUBCLASSES OF STARLIKEAND CONVEX FUNCTIONS ASSOCIATEDWITH STRUVE FUNCTIONS

T. Janani

G. Murugusundaramoorthy1

School of Advanced SciencesVIT UniversityVellore – 632014Indiaemails: [email protected]

[email protected]

Abstract. The present investigation our goal is to determine necessary and sufficient

condition for Struve functions belonging to the classes J ∗λ (α, β) and G∗

λ(α, β).

Keywords: Starlike functions, Convex functions, Uniformly Starlike functions, Uni-

formly Convex functions, Hadamard product, Bessel function, Struve function.

2000 Mathematics Subject Classification: 30C45.

1. Introduction

Let A be the class of analytic functions in the unit disk

U = z ∈ C : |z| < 1

of the form

(1) f(z) = z +∞∑n=2

anzn z ∈ U.

As usual, we denote by S the subclass of A consisting of functions which arenormalized by f(0) = 0 = f ′(0) − 1 and also univalent in U. Denote by T thesubclass of A consisting of functions whose non-zero coefficients from second on,is given by


468 t. janani, g. murugusundaramoorthy

(2) f(z) = z −∞∑n=2

anzn,

Also, for functions f ∈ A given by (1) and g ∈ A given by

g(z) = z +∞∑n=2

bnzn,

we define the Hadamard product (or convolution) of f and g by

(f ∗ g)(z) = z +∞∑n=2

anbnzn, z ∈ U.

A function f ∈ A is said to be starlike of order α (0 ≤ α < 1), if and only if

ℜ(

zf ′(z)f(z)

)> α (z ∈ U). This function class is denoted by S∗(α). We also write

S∗(0) ≡ S∗, where S∗ denotes the class of functions f ∈ A that f(U) is starlikewith respect to the origin. A function f ∈ A is said to be convex of order α

(0 ≤ α < 1) if and only if ℜ(1 + zf ′′(z)

f ′(z)

)> α (z ∈ U). This class is denoted by

K(α). Further, K(0) = K, the well-known standard class of convex functions. Itis an established fact that f ∈ K(α) ⇐⇒ zf ′ ∈ S∗(α).

It is well known that the special functions (series) play an important rolein geometric function theory, especially in the solution by de Branges [6] of thefamous Bieberbach conjecture. There is an extensive literature dealing with geo-metric properties of different types of special functions, especially for the genera-lized, Gaussian hypergeometric functions [5], [7], [12] and the Bessel functions [1],[2], [3], [8].

We recall here the Struve function of order p (see [10], [15]), denoted by Hp

is given by

(3) Hp(z) =∞∑n=0

(−1)n

Γ(n+ 32) Γ(p+ n+ 3

2)

(z2

)2n+p+1

,∀z ∈ C

which is the particular solution of the second order non-homogeneous differentialequation

(4) z2ω′′(z) + zω′(z) + (z2 − p2)ω(z) =4(z/2)p+1

√πΓ(p+ 1

2)

where p is unrestricted real(or complex) number. The solution of the non-homogeneousdifferential equation

(5) z2ω′′(z) + zω′(z)− (z2 + p2)ω(z) =4(z/2)p+1

√πΓ(p+ 1

2)

is called the modified Struve function of order p and is defined by the formula

inclusion results on subclasses of starlike ... 469

Lp(z) = −ie−ipπ/2Hp(iz) =∞∑n=0

1

Γ(n+ 32) Γ(p+ n+ 3

2)

(z2

)2n+p+1

,∀z ∈ C.

Let the second order non-homogeneous linear differential equation [15] (also see[10] and references cited therein),

(6) z2ω′′(z) + bzω′(z) + [cz2 − p2 + (1− b)p]ω(z) =4(z/2)p+1

√πΓ(p+ b

2)

where b, p, c ∈ C which is natural generalization of Struve equation. It is ofinterest to note that when b = c = 1, then we get the Struve function (3) and forc = −1, b = 1 the modified Struve function (5). This permit us to study Struveand modified Struve functions. Now, denote by wp,b,c(z) the generalized Struvefunction of order p given by

wp,b,c(z) =∞∑n=0

(−1)n(c)n

Γ(n+ 32) Γ(p+ n+ b+2

2)

(z2

)2n+p+1

,∀z ∈ C

which is the particular solution of the differential equation (6).Although the se-ries defined above is convergent everywhere, the function ωp,b,c is generally notunivalent in U. Now, consider the function up,b,c defined by the transformation

up,b,c(z) = 2p√πΓ

(p+

b+ 2

2

)z

−p−12 ωp,b,c (

√z),

√1 = 1.

By using well known Pochhammer symbol (or the shifted factorial) defined, interms of the familiar Gamma function, by

(a)n =Γ(a+ n)

Γ(a)=

1 (n = 0),

a(a+ 1)(a+ 2) · · · (a+ n− 1) (n ∈ N = 1, 2, 3, . . .)

we can express up,b,c(z) as

up,b,c(z) =∞∑n=0

(−c/4)n

(m)n (3/2)nzn

= b0 + b1z + b2z2 + ...+ bnz

n + ...,

where m =(p+ b+2

2

)= 0,−1,−2, . . . . This function is analytic on C and satisfies

the second-order inhomogeneous linear differential equation

4z2u′′(z) + 2(2p+ b+ 3)zu′(z) + (cz + 2p+ b)u(z) = 2p+ b.

For convenience, throughout in the sequel, we use the following notations

wp,b,c(z) = wp(z),

up,b,c(z) = up(z),

m = p+b+ 2

2


and for if c < 0,m > 0(m = 0,−1,−2, . . . ) let,

zup(z) = z +∞∑n=2

(−c/4)n−1

(m)n−1 (3/2)n−1

zn = z +∞∑n=2

bn−1zn

and

Ψ(z) = z(2− up(z)) = z −∞∑n=2

(−c/4)n−1

(m)n−1 (3/2)n−1

zn(7)

In this paper, we introduce two new subclasses of S namely Jλ(α, β) andGλ(α, β) to discuss some inclusion properties.

For some α (0 ≤ α < 1), λ (0 ≤ λ ≤ 1), β > 0 and functions of the form (1),we let Jλ(α, β) be the subclass of S of satisfying the analytic criteria

ℜ(

zf ′(z)

(1− λ)z + λf(z)− α

)> β

∣∣∣∣ zf ′(z)

(1− λ)z + λf(z)− 1

∣∣∣∣and Gλ(α, β) the subclass of S of satisfying the analytic criteria

ℜ(

zf ′(z) + z2f ′′(z)

(1− λ)z + λzf ′(z)− α

)> β

∣∣∣∣ zf ′(z) + z2f ′′(z)

(1− λ)z + λzf ′(z)− 1

∣∣∣∣ .Also denote J ∗

λ (α, β) = Jλ(α, β)∩ T and G∗λ(α, β) = Gλ(α, β)∩ T , the subclasses

of T .

Example 1 [4] For some α(0 ≤ α < 1), β > 0 and choosing λ = 1 and functionsof the form (2), we let T SP(α, β) be the subclass of S of satisfying the analyticcriteria

ℜ(zf ′(z)

f(z)− α

)> β

∣∣∣∣zf ′(z)

f(z)− 1

∣∣∣∣and UCT (α, β) the subclass of S of satisfying the analytic criteria

ℜ(1 +

zf ′′(z)

f ′(z)− α

)> β

∣∣∣∣zf ′′(z)

f ′(z)

∣∣∣∣ .Note that T SP(α, 0) ≡ T ∗(α, 0) and UCT (α, 0) ≡ C(α)[11], further

T SP(0, β) ≡ T Sp(β) and UCT (0, β) ≡ UCT (β)[13]

Example 2 For some α(0 ≤ α < 1), β > 0 and choosing λ = 0 and functions ofthe form (2), we let

(i) USD(α, β) the subclass of S of satisfying the analytic criteria

ℜ (f ′(z)− α) > β |f ′(z)− 1|

and

(ii) UCD(α, β) the subclass of S of satisfying the analytic criteria

ℜ ((zf ′(z))′ − α) > β |(zf ′(z))′ − 1| .


Suitably specializing the parameters we get the various subclasses studied in[9] and see the references cited therein.(also see [4], [14])

Recently, Yagmur and Orhan [15] (see [10]) have determined various suffi-cient conditions for the parameters p, b and c such that the functions up,b,c(z) orz → zup,b,c(z) to be univalent, starlike, convex and close to convex in the openunit disk. Motivated by results on connections between various subclasses of ana-lytic univalent functions by using hypergeometric functions (see [5], [7], [12]) andby work of Baricz [1], [2], [3]. In this paper,we we obtain sufficient condition forfunction h(z), given by

(8)

hµ(z) = (1− µ)zup(z) + µzup′(z)

= z +∞∑n=2

(1 + nµ− µ)(−c/4)n−1

(m)n−1 (3/2)n−1

zn.

where 0 ≤ µ ≤ 1 in the present investigation our goal is to determine sufficientcondition for function hµ(z) belonging to the classes Jλ(α, β) and Gλ(α, β).

2. Main results and their consequences

We recall the following necessary and sufficient conditions for the functionsf ∈ J ∗

λ (α, β), f ∈ G∗λ(α, β) and the subclasses stated in the Examples 1 and

2 which are relevant for our study.

Lemma 1 A function f(z) of the form (1) is in

(i) the class Jλ(α, β) if

(9)∞∑n=2

[n(1 + β)− λ(α + β)]|an| ≤ 1− α.

(ii) the class Gλ(α, β) if

(10)∞∑n=2

n[n(1 + β)− λ(α + β)]|an| ≤ 1− α.

The above sufficient conditions are also necessary for functions f of theform (2).


(i) the class T SP(α, β) if and only if

∞∑n=2

[n(1 + β)− (α+ β)]|an| ≤ 1− α.

(ii) the class UCT (α, β) if and only if

∞∑n=2

n[n(1 + β)− (α+ β)]|an| ≤ 1− α.



(i) the class USD(α, β) if and only if

∞∑n=2

n(1 + β)|an| ≤ 1− α.

(ii) the class UCD(α, β) if and only if

∞∑n=2

n2(1 + β)|an| ≤ 1− α.

Theorem 1 If c < 0,m > 0(m = 0,−1,−2, ... then hµ(z) ∈ Jλ(α, β) if

(11)

µ(1 + β)u′′p(1) + [(2µ+ 1)(1 + β)− µλ(α + β)]u′

p(1)

+ [(1 + β)− λ(α + β)]up(1)

≤ 2− α(1 + λ) + β(1− λ).

Proof. Since zup(z) = z +∞∑n=2

(−c/4)n−1

(m)n−1 (3/2)n−1

zn,

up(1)− 1 =∞∑n=2

(−c/4)n−1

(m)n−1 (3/2)n−1

,(12)

and differentiating zup(z) with respect to z and taking z = 1 we have

zu′p(z) + up(z) = 1 +

∞∑n=2

n(−c/4)n−1

(m)n−1 (3/2)n−1

zn−1

u′p(1) + up(1)− 1 =

∞∑n=2

n(−c/4)n−1

(m)n−1 (3/2)n−1

.

Further, differentiating zu′p(z) + up(z) with respect to z and taking z = 1, we get

zu′′p(z) + 2u′

p(z) =∞∑n=2

n(n− 1)(−c/4)n−1

(m)n−1 (3/2)n−1

zn−2

u′′p(1) + 2u′

p(1) =∞∑n=2

n(n− 1)(−c/4)n−1

(m)n−1 (3/2)n−1

.(13)

Since hµ(z) ∈ Jλ(α, β), by virtue of Lemma 1 and (9) it suffices to show that

(14)∞∑n=2

(1 + nµ− µ)[n(1 + β)− λ(α + β)]

((−c/4)n−1

(m)n−1 (3/2)n−1

)≤ 1− α.


Now, let

S(n, λ, β, α) =∞∑n=2

(1 + nµ− µ)[n(1 + β)− λ(α + β)]

((−c/4)n−1

(m)n−1 (3/2)n−1

)

S(n, λ, β, α) = µ(1 + β)∞∑n=2

n2

((−c/4)n−1

(m)n−1 (3/2)n−1

)

+ [(1 + β)(1− µ)− λµ(α + β)]∞∑n=2

n

((−c/4)n−1

(m)n−1 (3/2)n−1

)

− λ(α + β)(1− µ)∞∑n=2

((−c/4)n−1

(m)n−1 (3/2)n−1

).

Writing n2 = n(n− 1) + n, we get

S(n, λ, β, α) = µ(1 + β)∞∑n=2

n(n− 1)

((−c/4)n−1

(m)n−1 (3/2)n−1

)+ [(1 + β)− λµ(α+ β)]

∞∑n=2

n

((−c/4)n−1

(m)n−1 (3/2)n−1

)− λ(α+ β)(1− µ)

∞∑n=2

((−c/4)n−1

(m)n−1 (3/2)n−1

)From (12), (13), 13 and taking z = 1, we get

S(n, λ, β, α) ≤ µ(1 + β)u′′p(1) + [(1 + β)− λµ(α + β)](u′

p(1) + up(1)− 1)

−λ(α + β)(1− µ)(up(1)− 1)

= µ(1 + β)u′′p(1) + [(2µ+ 1)(1 + β)− λµ(α + β)]u′

p(1)

+[(1 + β)− λ(α + β)(up(1)− 1)

But this expression is bounded above by 1− α if (11) holds.

Thus, the proof is complete.

Theorem 2 If c < 0,m > 0(m = 0,−1,−2, ... then zup(z) ∈ Jλ(α, β) if

(15) (1 + β)u′p(1) + [(1 + β)− λ(α+ β)]up(1) ≤ 2− α(1 + λ) + β(1− λ).

Proof. By virtue of Lemma 1 of (9), it suffices to show that

∞∑n=2

[n(1 + β)− λ(α + β)]

((−c/4)n−1

(m)n−1 (3/2)n−1

)≤ 1− α.


Since h0(z) = zup(z), hence by taking µ = 0 in (14) we get the above inequa-lity. Hence by taking µ = 0 in the Theorem 1, we get the desired result givenin 15.

Theorem 3 If c < 0,m > 0(m = 0,−1,−2, ... then zup(z) ∈ Gλ(α, β) if

(16)

(1 + β)u′′p(1) + [3(1 + β)− λ(α + β)]u′

p(1)

+ [(1 + β)− λ(α + β)]up(1)

≤ 2− β(λ− 1)− α(λ+ 1).

Proof. By virtue of Lemma 1 of (10), it suffices to show that

∞∑n=2

n[n(1 + β)− λ(α + β)]

((−c/4)n−1

(m)n−1 (3/2)n−1

)≤ 1− α.

By definition zup(z) ∈ Gλ(α, β) ⇔ zu′p(z) ∈ Jλ(α, β). That is by taking µ = 1 we

have h1(z) = zu′p(z) ∈ Jλ(α, β), hence by taking µ = 1 in the Theorem 1, we get

the desired result given in 16.

Remark 1 The above conditions (11) and (16) are also necessary for functionsΨ(z) given by (7) and of the form

h∗µ(z) = (1− µ)Ψ(z) + µΨ′(z)

= z −∞∑n=2

(1 + nµ− µ)(−c/4)n−1

(m)n−1 (3/2)n−1

zn

is in the classes J ∗λ (α, β) and G∗

λ(α, β) respectively.

Further, by taking λ = 0(or)λ = 1 in Theorems 2 and 3, we state the followingcorollaries without proof.

Corollary 1 If c < 0,m > 0(m = 0,−1,−2, ...) then z(2− up(z)),

(i) is in T SP(α, β) if and only if

(1 + β)u′p(1) + (1− α)up(1) ≤ 2(1− α).

(ii) is in UCT (α, β) if and only if

(1 + β)u′′p(1) + (3− 2β − α)u′

p(1) + (1− α)up(1) ≤ 2(1− α).


Corollary 2 If c < 0,m > 0(m = 0,−1,−2, ...) then z(2− up(z)),

(i) is in USD(α, β) if and only if

(1 + β)[u′p(1) + up(1)] ≤ 2− α + β.

(ii) is in UCD(α, β) if and only if

(1 + β)[u′′p(1) + 3u′

p(1) + up(1)] ≤ 2− α + β.

References

[1] Baricz, A., Geometric properties of generalized Bessel functions, Publ.Math. Debrecen, 73 (1-2) (2008), 155–178.

[2] Baricz, A., Geometric properties of generalized Bessel functions of complexorder, Mathematica, 48 (71) (1) (2006), 13–18.

[3] Baricz, A., Generalized Bessel functions of the first kind, Lecture Notes inMath., vol. 1994 , Springer-Verlag, 2010.

[4] Bharati,R., Parvatham, R., Swaminathan, A., On subclasses ofuniformly convex functions and corresponding class of starlike functions,Tamkang J. Math., 26 (1) (1997), 17–32.

[5] Cho, N.E., Woo, S.Y., Owa, S., Uniform convexity properties for hyper-geometric functions, Fract. Cal. Appl. Anal., 5 (3) (2002), 303–313.

[6] de Branges, L., A proof of the Bierberbach conjucture, Acta. Math., 154(1985), 137–152.

[7] Merkes, E., Scott, B.T., Starlike hypergeometric functions, Proc. Amer.Math. Soc., 12 (1961), 885–888.

[8] Mondal, S.R., Swaminathan, A., Geometric properties of GeneralizedBessel functions, Bull. Malays. Math. Sci. Soc., 35 (1) (2012), 179–194.

[9] Murugusundaramoorthy, G., Magesh, N., On certain subclasses ofanalytic functions associated with hypergeometric functions, Appl. Math. Let-ters, 24,(2011), 494–500.

[10] Orhan, H., Yagmur, N., Geometric properties of generalized Struve func-tions, in The International Congress in Honour of Professor Hari M. Srivas-tava, Bursa, Turkey, August, 2012.

[11] Silverman, H., Univalent functions with negative coefficients, Proc. Amer.Math. Soc., 51 (1975), 109–116.


[12] Silverman, H., Starlike and convexity properties for hypergeometric func-tions, J. Math. Anal. Appl., 172 (1993), 574–581.

[13] Subramanian, K.G., Murugusundaramoorthy, G., Balasubrah-manyam, P., Silverman, H., Subclasses of uniformly convex and uniformlystarlike functions, Math. Japonica, 42(3), (1995), 517–522.

[14] Subramanian, K.G., Sudharsan, T.V., Balasubrahmanyam, P.,Silverman, H., Classes of uniformly starlike functions, Publ. Math. De-brecen., 53 (3–4) (1998), 309–315.

[15] Yagmur, N., Orhan, H., Starlikeness and convexity of generalized Struvefunctions, Abstract and Appl. Anal., (2013), Article ID 954513, 6 pages.

Accepted: 6.04.2014


THE MODIFIED (w/g)-EXPANSION METHODAND ITS APPLICATIONS FOR SOLVINGTHE MODIFIED GENERALIZED VAKHNENKO EQUATION

Elsayed M.E. Zayed

Department of MathematicsFaculty of ScienceZagazig UniversityZagazigEgypte-mail: [email protected]

Ahmed H. Arnous

Engineering mathematics and Physics DepartmentHigher Institute of EngineeringEl ShoroukEgypte-mail: [email protected]

Abstract. The modified (w/g)-expansion method for finding traveling wave solutions

of nonlinear evolution equations is presented in this paper, which can be thought of as

the generalization of the well-known (G′/G)-expansion method given recently by Wang

et al. When the w and g are taken special choices, some familiar expansion methods

can be obtained. Based on these interesting results, we further give two new forms of

expansions via the modified (g′/g2)-expansion method and the modified (g′) expansion

method. In order to well illustrate the effectiveness of these two modified expansion

methods, they are applied to a modified generalized Vakhnenko equation.

Keywords: the modified (w/g)-expansion method; modified (g′/g2)-expansion method;

modified (g′)-expansion method; nonlinear evolution equations; traveling wave solutions;

a modified generalized Vakhnenko equation.

AMS Subject Classifications: 35K99, 35P05, 35P99.

1. Introduction

The investigation of the exact traveling wave solutions of nonlinear evolutionequations plays an important role in the study of soliton theory. In recent years, avariety of powerful methods have been applied for constructing exact traveling andsolitary wave solutions of nonlinear evolution equations, such as the tanh-functionexpansion method [1]–[3], Jacobi elliptic function method [4], [5], Exp-function

478 e.m.e. zayed, a.h. arnous

method [6], [7], the F-function expansion method [8], the inverse scattering method[9], Hirota bilinear transformation [10], the Backlund transform method [11], the(G′/G)-expansion method [12], [13], the sine-cosine method [6], [14], the modifiedsimple equation method [15]–[18], the multiple exp-function method [19], [20],the transformed rational function method [21]–[23], the local fractional variationiteration method [24], the local fractional series expansion method [25], the (w/g)-expansion method [26] and so on.

The objective of this paper is to introduce the modified (w/g)-expansion me-thod for finding exact solutions for a modified generalized Vakhnenko equationin mathematical physics. The proposed method is based on the assumption thatthese exact solutions can be expressed by a polynomial in (w/g)i, i = 0, 1, ...,mand that w, g satisfy the following auxiliary equation

(1.1)

(w

g

)′

= a+ b

(w

g

)+ c

(w

g

)2

,

namely

(1.2) w′ g − wg′ = ag2 + bwg + cw2,

where a, b, c are arbitrary constants, while ′ = d/dξ and ξ = k(x − V t), k and Vare constants. The degree m of this polynomial can be obtained by consideringthe homogeneous balance between the highest-order derivatives and the nonlinearterms appearing in the given nonlinear evolution equations. The coefficients ofthis polynomial can be obtained by solving a set of algebraic equations resultedfrom the process of using the proposed method. The rest of this paper is organizedas follows. In Section 2, we describe a new modified (w/g)-expansion method. InSection 3, we apply this method to solve the modified generalized Vakhnenkoequation. In Section 4, Physical explanations of some obtained solutions areobtained. In Section 5, some conclusions are given.

2. Description of a new modified (w/g)- expansion method

For a given nonlinear evolution equation

(2.1) P (u, ut, ux, uxt, utt, uxx, ...) = 0,

where P is a polynomial in the function u(x, t) and its partial derivatives in whichthe nonlinear terms are involving. We use the wave transformation

(2.2) u(x, t) = u(ξ), ξ = k(x− V t),

where k, V are constants to reduce equation (2.1) into the following ODE:

(2.3) Q(u(r), u(r+1), ...

)= 0,

the modified (w/g)-expansion method and its applications... 479

where u(r) = dr

dξr, r ≥ 0, and r is the least order derivatives in the given equation.

Setting u(r) = υ(ξ), where υ(ξ) is a new function of ξ, then equation (2.3) reducesto the following ODE:

(2.4) F (υ, υ′, υ′′, ...) = 0,

where ′ = d/dξ. We further introduce the formal solution of equation (2.4) in thefollowing ansatz:

(2.5) u(r)(ξ) = υ(ξ) =m∑i=0

αi

(w

g

)i

,

where w and g satisfy equation (1.2) and αi(i = 0, 1, ...,m) are constants to bedetermined later, while r ≥ 0. To determine υ(ξ) explicitly, we take the followingfour steps:

Step 1. Determine the positive integer m in equation (2.5) by balancing thehighest-order derivatives and the nonlinear terms in equation (2.4).

Step 2. Substitute (2.5) into (2.4) and collect all terms with the same powersof (w/g)i, (i = 0, 1, ...,m), together, thus the left-hand side of equation (2.4) isconverted into a polynomial in (w/g)i. Then set each coefficient of this polynomialto zero, to derive a set of algebraic equations for αi, k, V.

Step 3. Solve these algebraic equations by the use of Maple or Mathematica tofind the values of αi, k, V.

Step 4. Use the results obtained in the above steps to derive a series of fundamen-tal solutions υ(ξ) of equation (2.4) depending on (w/g). Then we can obtain theexact solutions of equation (2.1) by integrating each of the obtained fundamentalsolutions υ(ξ) with respect to ξ and r times as follows:

(2.6) u(ξ) =

ξ∫ ξr∫...

ξ2∫υ(ξ1)dξ1...dξr−1dξr +

r∑j=1

djξr−j,

where dj(j = 1, ..., r) are arbitrary constants.

Remark 1. Let us now examine equation (1.2) carefully as follows:

(1) If we choose w = g′, a = −µ, b = −λ, c = −1 and r = 0, then u(ξ) canbe expressed as:

(2.7) u(ξ) =m∑i=0

αi

(g′

g

)i

,

where g(ξ) satisfies the linear ODE:

(2.8) g′′ + λg′ + µg = 0.


This is just the well-known (G′/G)-expansion method proposed byWang et al. [12].

(2) If w = tanh ξ, g = 1, a = 1, b = 0, c = −1 and r = 0, then

(2.9) u(ξ) =m∑i=0

αitanhi(ξ),

which is the well-known tanh-function expansion method, see [1]–[3], [6].

(3) If w = g′/g, b = 0 and r = 0, then we have the new expansion

(2.10) u(ξ) =m∑i=0

αi

(g′

g2

)i

,

where g(ξ) satisfies the nonlinear ODE:

(2.11) g′′g2 − 2g(g′)2 = ag4 + c(g′)2,

which is called (g′/g2)-expansion method and proposed in [26].

(4) If w = gg′ and r = 0, then we have the new expansion

(2.12) u(ξ) =m∑i=0

αi(g′)i,

where g(ξ) satisfies the nonlinear ODE:

(2.13) g′′ = a+ bg′ + c(g′)2,

which is called g′-expansion method and proposed in [26]. Li et al. [26] haveapplied the two expansions (2.10) and (2.12) for finding exact solutions to theVakhnenko equation

utx + u2x + uuxx + u = 0.

Remark 2. where r ≥ 1, the ansatz (2.5) is new and is not reported in [26]. So,in the next section, we apply the two expansions (g′/g2)-expansion method andthe (g′)-expansion method using (2.5) with r ≥ 1 to find new exact solutions of amodified generalized Vakhnenko equation.

3. New solutions of a modified generalized Vakhnenko equation

Consider a modified generalized Vakhnenko equation (mGVE) [27]–[29]:

(3.1)∂

∂x

(℘2u+

1

2pu2 + βu

)+ q℘u = 0, ℘ =

∂

∂t+ u

∂

∂x,


where p, q, β are arbitrary non-zero constants. equation (3.1) can be traced tothe well-known Vakhnenko equation (VE) which was initially presented to modelhigh-frequent wave motion in a relaxing medium [29]. Recently, equation (3.1) hasbeen discussed using the (G′/G)-expansion method [27] and using the auxiliaryequation method [28]. To calculate the exact solutions for equation (3.1), a sen-sible step is to transform variables. We introduce two new independent variablesX and T defined by:

(3.2) x = T +

X∫−∞

U(X ′, T )dX ′ + x0, t = X,

where u(x, t) = U(X,T ) and x0 is a constant. We introduce a new function Wdefined by

(3.3) W (X,T ) =

X∫−∞

U(X ′, T )dX ′.

Then

(3.4) WX(X,T ) = U(X,T ), ,WT (X,T ) =

X∫−∞

UT (X′, T )dX ′.

It is easy to see that

(3.5)∂

∂T=

∂

∂x

∂x

∂T+∂

∂t

∂t

∂T,

∂

∂X=

∂

∂x

∂x

∂X+∂

∂t

∂t

∂X.

From (3.2) and (3.5) we have

(3.6)∂

∂T= (1 +WT )

∂

∂x,

∂

∂X=

∂

∂t+ u

∂

∂x,

and hence ℘u = Ux, ℘2u = UXX . Now, equation (3.1) reduces to

(3.7) WXXXT + pWXWXT + q(1 +WT )WXX + βWXT = 0.

Assume that W (X,T ) = W (ξ), where ξ is given by (2.2), then equation (3.7)reduces to the equation

(3.8) k2VW ′′′ +1

2(p+ q)kV (W ′)2 + (β V − q)W ′ = 0,

with zero constants of integration. Setting r = 1 and W ′ = υ, we have W (ξ) =∫υ(ξ)dξ + d1, where υ(ξ) satisfies the following ODE:

(3.9) k2V υ′′ +1

2(p+ q)kV υ2 + (β V − q)υ = 0.


Recently, equation (3.9) has been solved in [27] using the (G′/G)-expansion methodvia the expansion (2.7). Let us now solve equation (3.9) using the expansion (2.10)and (2.12).

3.1. The new modified (g′/g2) expansion method

Balancing υ′′ with υ2 in equation (3.9) we getm = 2. Consequently, the expansion(2.10) reduces to

(3.10) υ(ξ) = α2

(g′

g2

)2

+ α1

(g′

g2

)+ α0, α2 = 0.

where g(ξ) satisfies equation (2.11). It is easy to see that

(3.11) υ′(ξ) = 2α2c

(g′

g2

)3

+ α1c

(g′

g2

)2

+ 2α2a

(g′

g2

)+ α1a,

(3.12) υ′′(ξ) = 6α2c2

(g′

g2

)4

+2α1c2

(g′

g2

)3

+8α2ac

(g′

g2

)2

+2α1ac

(g′

g2

)+2α2a

2.

Substituting (3.10)–(3.12) into (3.9) and collecting all terms with the same powersof (g′/g2)i, i = 0, 1, 2, 3, 4 together, the left hand side of equation (3.9) is convertedinto a polynomial in (g′/g2)i. Setting each coefficient of this polynomial to zero,we get the following algebraic equations:

(3.13) 0 : 2α2a2k2V +

1

2(p+ q)kV α2

0 + (V β − q)α0 = 0,

(3.14) 1 : 2α1ack2V + α0α1(p+ q)kV + (V β − q)α1 = 0,

(3.15) 2 : 8α2ack2V +

1

2(p+ q)kV α2

1 + α0α2(p+ q)kV + (V β − q)α2 = 0,

(3.16) 3 : 2α1c2k2V + α1α2(p+ q)kV = 0,

(3.17) 4 : 6α2c2k2V +

1

2(p+ q)kV α2

2 = 0.

Solving the above algebraic equations, we have the following results:

Case 1.

(3.18) α2 = −12c2k

p+ q, α1 = 0, α0 = −12ack

p+ q, V =

q

β − 4ack2, β = 4ack2

Case 2.

(3.19) α2 = −12c2k

p+ q, α1 = 0, α0 = − 4ack

p+ q, V =

q

β + 4ack2, β = −4ack2

where p+ q = 0. It is well known [26], the solution of equation (2.11) is given, andhence g′/g2 has the form:


(1) If ac > 0, then

(3.20)g′

g2=

√a

c

[c1 cos(

√ac ξ) + c2 sin(

√ac ξ)

c1 sin(√ac ξ)− c2 cos(

√ac ξ)

],

(2) If ac < 0, then

(3.21)g′

g2=

1

2c

2√|ac| −4c1

√|ac| exp

(2ξ√

|ac|)

c1 exp(2ξ√|ac|

)− c2

,(3) If a = 0, c = 0, then

(3.22)g′

g2=

− c1c (c1ξ + c2)

,

where c1 and c2 are arbitrary constants. Now, we have the following exact solutionsof equation (3.9)

(i) If ac > 0, then for case1, we have

(3.23) υ(ξ) =−12kac

p+ q

[c1 cos (

√ac ξ) + c2 sin (

√ac ξ)

c1 sin (√ac ξ)− c2 cos (

√ac ξ)

]2− 12kac

p+ q,

and consequently, we get

(3.24) W (ξ) =−12kac

p+ q

∫ [c1 cos (

√ac ξ) + c2 sin (

√ac ξ)

c1 sin (√ac ξ)− c2 cos (

√ac ξ)

]2dξ − 12kac

p+ qξ + d1,

where d1 is a constant of integration. Simplifying (3.24), we get two values ofW (ξ) as follows:

The first value is

(3.25) W1(ξ) =−12k

√ac

p+ qtan

(ξ1 +

√ac ξ

)+ d1,

where ξ1 = tan−1(

c1c2

), c21 + c22 = 0, and ξ is given by

(3.26) ξ = k(X − V T ) = k

(t− qT

β − 4ack2

), T = x−W1(ξ)− x0.

In this case, equation (3.1) has the general periodic solution

(3.27) u1(x, t) = W1x(ξ) =−12k2ac

p+ qsec2

(ξ1 +

√ac ξ

).

The second value is

(3.28) W2(ξ) =−12k

√ac

p+ qcot

(ξ2 −

√ac ξ

)+ d1,


where ξ2 = cot−1(

c1c2

), c21+ c

22 = 0, and ξ is given by (3.26), while T has the form

(3.29) T = x−W2(ξ)− x0.

In this case, equation (3.1) has the general periodic solution

(3.30) u2(x, t) = W2x(ξ) =−12k2ac

p+ qcsc2

(ξ2 −

√ac ξ

),

(ii) If a c < 0, then for case 1, we have

(3.31) υ(ξ) =−12k

p+ q

√|ac| −2c1

√|ac| exp

(2ξ√|ac|

)c1 exp

(2ξ√|ac|

)− c2

2

− 12kac

p+ q,

and, consequently, we get

(3.32)

W (ξ) =−12k |ac|p+ q

∫ c1 exp(ξ√

|ac|)+ c2 exp

(−ξ

√|ac|

)c1 exp

(ξ√|ac|

)− c2 exp

(−ξ

√|ac|

)2

dξ − 12kac

p+ qξ + d1.

Setting c1 =12(A+B), c2 =

−12(A−B), ϕ = ξ

√|ac|, where A and B are constants,

then (3.32) reduces to

(3.33) W (ξ) =−12k |ac|p+ q

∫ [A sinhϕ+B coshϕ

A coshϕ+B sinhϕ

]2dξ +

12k |ac|p+ q

ξ + d1.

Using the formulas (8), (10), (12) and (14) obtained in Peng [30], we have respec-tively the following general forms of soliton solutions:

(1) If|A| > |B| , then

(3.34) W1(ξ) =12k

p+ q

√|a c| tanh[ϕ+ sgn(AB)ψ1] + d1,

and, consequently, the solution of equation (3.1) has the form:

(3.35) u1(x, t) = W1x(ξ) =12k2 |a c|p+ q

sech2[ϕ+ sgn(AB)ψ1],

where

(3.36) ξ = k(X − V T ) = k

(t− qT

β − 4ack2

), T = x−W1(ξ)− x0,

andψ1 = tanh−1(

|B||A|

), sgn(AB) is the sign function. Note that the solution

(3.35) is in agreement with the obtained in [27].


(2) If |B| > |A| = 0, then

(3.37) W2(ξ) =12k

√|a c|

p+ qcoth[ϕ+ sgn(AB)ψ2] + d1,

and consequently, the solution of equation (3.1) has the form

(3.38) u2(x, t) = W2x(ξ) =−12k2 |a c|p+ q

csch2[ϕ+ sgn(AB)ψ2],

where

(3.39) ξ = k(X − V T ) = k

(t− qT

β − 4ack2

), T = x−W2(ξ)− x0,

and ψ2 = coth−1(

|B||A|

).

(3) If |B| > |A| = 0, then

(3.40) W3(ξ) =12k

√|a c|

p+ qcothϕ+ d1,

and, consequently, the solution of equation (3.1) has the form

(3.41) u3(x, t) = W3x(ξ) =−12k2 |ac|p+ q

csch2ϕ,

where

(3.42) ξ = k(X − V T ) = k

(t− qT

β − 4ack2

), T = x−W3(ξ)− x0,

(4) If |A| = |B| , then

(3.43) W4(ξ) = d1,

and, consequently, the solution of equation (3.1) is the zero solution

(3.44) u4(x, t) = W4x(ξ) = 0,

where

(3.45) ξ = k(X − V T ) = k

(t− qT

β − 4ack2

), T = x−W4(ξ)− x0.

(iii) If a = 0, c = 0, then for case 1, we have

(3.46) υ(ξ) =−12k

p+ q

(c1

c1ξ + c2

)2

,


and, consequently, we get

(3.47) W5(ξ) =12k

p+ q

(c1

c1ξ + c2

)+ d1.

The solution of equation (3.1) in this case has the form:

(3.48) u5(x, t) = W5x(ξ) =−12k2

(p+ q)

(c21

(c1ξ + c2)2

),

where

(3.49) ξ = k(X − V T ) = k

(t− qT

β

), T = x−W5(ξ)− x0,

Similarly, we can find the exact solutions for case 2, which are omitted here forsimplicity.

3.2. The new modified (g′)-expansion method

Setting m = 2, in (2.12) we have the formal solution of equation (3.9) in the form

(3.50) υ(ξ) = α2(g′)2+ α1 (g

′) + α0, α2 = 0,

where g satisfies equation (2.13). It is easy to see that

(3.51) υ′(ξ) = 2α2c(g′)3+ (2α2b+ α1c)(g

′)2+ (2α2a+ α1b) (g

′) + α1a,

(3.52)

υ′′(ξ) = 6α2c(g′)4 + (10α2bc+ 2α1c

2)(g′)3

+(4α2b2 + 3α1bc+ 8α2ac)(g

′)2

+(6α2ab+ 2α1ac+ α1b2) (g′) + (2α2a

2 + α1ab).

Substituting (3.50)-(3.52) into (3.9) and collecting all terms with the same powersof (g′)i, (i = 0, 1, 2, 3, 4) together, the left-hand side of equation (3.9) is convertedinto a polynomial in (g′)i. Setting each coefficient of this polynomial to zero, weget the following algebraic equations:

(3.53) 0 : k2V(2α2a

2 + α1ab)+

1

2(p+ q)kV α2

0 + (βV − q)α0 = 0,

(3.54) 1 : k2V(6α2ab+ 2α1ac+ α1b

2)+ α1α0(p+ q)kV + (βV − q)α1 = 0,

(3.55)2 : k2V (4α2b

2 + 3α1bc+ 8α2ac) +12(p+ q)kV α2

1+α2α0(p+ q)kV + (βV − q)α2 = 0,

(3.56) 3 : k2V (10α2bc+ 2α1c2) + α2α1(p+ q)kV = 0,

(3.57) 4 : 6k2V α2c2 +

1

2(p+ q)kV α2

2 = 0.

Solve these algebraic equations using the Maple or Mathematica we get the fol-lowing cases:


Case 1.

(3.58) α2 =−12kc2

p+q, α1 =

−12kbc

p+q, α0 =

−2k

p+q

(b2+2ac

), V =

q

4ack2+β−k2b2.

Case 2.

(3.59) α2 =−12kc2

p+ q, α1 =

−12kbc

p+ q, α0 =

−3kb2

p+ q, b = −2

√ac, V =

q

β.

Case 3.

(3.60) α2 =−12kc2

p+ q, α1 = 0, α0 =

−12ack

p+ q, b = 0, V =

q

β − 4ack2.

Case 4.

(3.61) α2 =−12kc2

p+ q, α1 = 0, α0 =

−4ack

p+ q, b = 0, V =

q

β + 4ack2.

It is well-known [26], the solution of equation (2.13) is given, and hence g′(ξ) hasthe forms:

If ∆ = 4ac− b2 < 0, then

(3.62) g′(ξ) =1

2c

[√−∆tanh

(−ξ2

√−∆

)− b

].

If ∆ = 4ac− b2 > 0, then

(3.63) g′(ξ) =1

2c

[√∆tan

(ξ

2

√∆

)− b

].

If ∆ = 4ac− b2 = 0, then

(3.64) g′(ξ) =−1

c

(1

ξ+b

2

).

Now, we deduce the following exact solutions of equation (3.9) as follows:

(i) If ∆ < 0, then for case 1, we have

(3.65) υ(ξ) =−3k∆

p+ qsech2

(−ξ2

√−∆

)+

2k∆

p+ q,


(3.66) W1(ξ) =

∫υ(ξ)dξ + d1,

which can be written as

(3.67) W1(ξ) =−6k

√−∆

p+ qtanh

(−ξ2

√−∆

)+

2k∆

p+ qξ + d1.


Now, the solution of equation (3.1) in this case is the soliton solution:

(3.68) u1(x, t) = W1x(ξ) =k2∆

p+ q

[2− 3 sech2

(−ξ2

√−∆

)],

where

(3.69) ξ = k(X − V T ) = k

(t− qT

4ack2 + β − k2b2

), T = x−W1(ξ)− x0.

(ii) If ∆ > 0, then for case 1, we have

(3.70) υ(ξ) =−3k∆

p+ qsec2

(ξ

2

√∆

)+

2k∆

p+ q,


(3.71) W2(ξ) =

∫υ(ξ)dξ + d1,


(3.72) W2(ξ) =−6k

√∆

p+ qtan

(ξ

2

√∆

)+

2k∆

p+ qξ + d1.


(3.73) u2(x, t) = W2x(ξ) =k2∆

p+ q

[2− 3sec2

(ξ

2

√∆

)],

where

(3.74) ξ = k(X − V T ) = k

(t− qT

4ack2 + β − k2b2

), T = x−W2(ξ)− x0.

(iii) If ∆ = 0, then for case 1, we have

(3.75) υ(ξ) =

(−12k

p+ q

)1

ξ2,


(3.76) W3(ξ) =

∫υ(ξ)dξ + d1,


(3.77) W2(ξ) =

(12k

p+ q

)1

ξ+ d1.


(3.78) u3(x, t) = W3x(ξ) =

(−12k2

p+ q

)1

ξ2,

where

(3.79) ξ = k(X − V T ) = k

(t− qT

β

), T = x−W3(ξ)− x0.

Similarly, we can find exact solutions for the other cases, which are omitted herefor simplicity.


4. Physical explanations of some obtained solutions

In this section, we will present some graphs for three types of solutions of equation(3.1) namely hyperbolic, trigonometric and rational function solutions by selectingsome special values of the parameters in the exact solutions using the mathemati-cal software Maple, which can be shown below in Figures 1–6. From these explicitsolutions, we see that the results (3.27),(3.30) and (3.73) are periodic solutions,the results (3.35) and (3.68) are bell-shaped soliton solutions, the results (3.38)and (3.41) are singular bell-shaped soliton solutions while the results (3.48) and(3.78) are rational function solutions.

The plot of some solutions


5. Some conclusions

In this paper, we have introduced and presented the new modified (w/g)-expansionmethod when w and g are taken special values and then we proposed two expan-sion methods via the modified (g′/g2)-expansion method and the modified (g′)-expansion method. We have applied these two methods to find the exact solutionsfor the modified generalized Vakhnenko equation (3.1) such as the periodic, solitonand rational function solutions. Finally, with the aid of Maple or Mathematicawe have assured the correctness of the obtained solutions by putting them backinto the original equation (3.1).

References

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[2] Wazwaz, A.M., Exact solutions for the ZK-MEW equations using the tanhand sine cosine methods, Int. J. Computer Math., 82 (2005), 699-708.

[3] Zayed, E.M.E., Abdelaziz, M.A.M., The tanh-function method using ageneralized wave transformation for nonlinear equations, Int. J. NonlinearSci. Numer. Simula., 11 (2010), 595-601.

[4] Lu, D., Jacobi elliptic function solutions for two variant Boussinesq equa-tions, Chaos, Solitons and Fractals, 24 (2005), 1373-1385.

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[6] Zayed, E.M.E., Abdelaziz, M.A.M., Exact solutions for the nonlinearShrodinger equation with variable coefficients using the generalized tanh-function, the sine-cosine and the exp-function methods, Appl. Math. Com-put., 218 (2011), 2259-2268.

[7] He, J.H., Wu, X.H., Exp-function method for nonlinear wave equations,Chaos, Solitons and Fractals, 30 (2006), 700-708.

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[10] Hirota, R., Exact solutions of KdV equation for multiple collisions of soli-tons, Phys. Rev. Lett., 27 (1971), 1192-1194.


[11] Miura, M.R., Backlund transformation, Berlin, Springer, 1978.

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[17] Zayed, E.M.E., Arnous, A.H., Exact solutions of the nonlinear ZK-MEWand the Potential YTSF equations using the modified simple equation method,AIP Conf. Proc., 1479 (2012a), 2044-2048.

[18] Zayed, E.M.E., Hoda Ibrahim, S.A., Exact solutions of nonlinear evo-lution equations in mathematical physics using the modified simple equationmethod, Chinese Phys, Lett., 29 (2012b), 060201-060204.

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ROUGH NEUTROSOPHIC SETS

Said Broumi

Faculty of Arts and HumanitiesHay El Baraka Ben M’sik Casablanca B.P. 7951Hassan II University Mohammedia-CasablancaMorocco


Department of MathematicsUniversity of New Mexico705 Gurley Avenue, Gallup, NM 87301USA

Mamoni Dhar

Department of MathematicsScience CollegeKokrajhar-783370, AssamIndia

Abstract. Both neutrosophic sets theory and rough sets theory are emerging as power-

ful tool for managing uncertainty, indeterminate, incomplete and imprecise information.

In this paper we develop an hybrid structure called rough neutrosophic sets and studied

their properties.

Keywords: Rough set, rough neutrosophic set.

1. Introduction

In 1982, Pawlak [1] introduced the concept of rough set (RS), as a formal tool formodeling and processing incomplete information in information systems. Thereare two basic elements in rough set theory, crisp set and equivalence relation,which constitute the mathematical basis of RSs. The basic idea of rough setis based upon the approximation of sets by a pair of sets known as the lowerapproximation and the upper approximation of a set. Here, the lower and upperapproximation operators are based on equivalence relation. After Pawlak, therehas been many models built upon different aspect, i.e., universe, relations, object,operators by many scholars [2], [3], [4], [5], [6], [7]. Various notions that combinerough sets and fuzzy sets, vague set and intuitionistic fuzzy sets are introduced,such as rough fuzzy sets, fuzzy rough sets, generalize fuzzy rough, intuitionisticfuzzy rough sets, rough intuitionistic fuzzy sets, rough vagues sets. The theory of

494 s. broumi, f. smarandache, m. dhar

rough sets is based upon the classification mechanism, from which the classificationcan be viewed as an equivalence relation and knowledge blocks induced by it bea partition on universe.

One of the interesting generalizations of the theory of fuzzy sets and intuitio-nistic fuzzy sets is the theory of neutrosophic sets introduced by F. Smarandache[8], [9]. Neutrosophic sets described by three functions: a membership functionindeterminacy function and a non-membership function that are independentlyrelated. The theory of neutrosophic set have achieved great success in variousareas such as medical diagnosis [10], database [11], [12], topology [13], image pro-cessing [14], [15], [16], and decision making problem [17]. While the neutrosophicset is a powerful tool to deal with indeterminate and inconsistent data, the theoryof rough sets is a powerful mathematical tool to deal with incompleteness.

Neutrosophic sets and rough sets are two different topics, none conflicts theother. Recently many researchers applied the notion of neutrosophic sets to re-lations, group theory, ring theory, soft set theory [23], [24], [25], [26], [27], [28],[29], [30], [31], [32] and so on. The main objective of this study was to introducea new hybrid intelligent structure called rough neutrosophic sets. The significanceof introducing hybrid set structures is that the computational techniques basedon any one of theses structures alone will not always yield the best results but afusion of two or more of them can often give better results.

The rest of this paper is organized as follows. Some preliminary conceptsrequired in our work are briefly recalled in Section 2. In Section 3, the concept ofrough neutrosophic sets is investigated. Section 4 concludes the paper.

2. Preliminaries

In this section we present some preliminaries which will be useful to our work inthe next section. For more details the reader may refer to [1], [8], [9].

Definition 2.1. [8] Let X be an universe of discourse, with a generic element inX denoted by x, the neutrosophic (NS) set is an object having the form

A = ⟨x : µA(x), νA(x), ωA(x)⟩ , x ∈ X,

where the functions µ, ν, ω : X →]−0, 1+[ define respectively the degree of member-ship (or Truth), the degree of indeterminacy, and the degree of non-membership(or Falsehood) of the element x ∈ X to the set A with the condition

(1) −0 ≤ µA(x) + νA(x) + ωA(x) ≤ 3+.

From a philosophical point of view, the neutrosophic set takes the value from realstandard or non-standard subsets of ]−0, 1+[. So, instead of ]−0, 1+[ we need totake the interval [0, 1] for technical applications, because ]−0, 1+[ will be difficultto apply in the real applications such as in scientific and engineering problems.

rough neutrosophic sets 495

For two NS,

A = ⟨x, µA(x), νA(x), ωA(x)⟩ | x ∈ X and

B = ⟨x, µB(x), νB(x), ωB(x)⟩ | x ∈ X,

the relations are defined as follows:

(i) A ⊆ B if and only if µA(x) ≤ µB(x), νA(x) ≥ µB(x), ωA(x) ≥ ωB(x),

(ii) A = B if and only if µA(x) = µB(x), νA(x) = µB(x), ωA(x) = ωB(x),

(iii) A ∩B = ⟨x,min(µA(x), µB(x)),max(νA(x), νB(x)),max(ωA(x), ωB(x))⟩ |x ∈ X,

(iv) A ∪B = ⟨x,max(µA(x), µB(x)),min(νA(x), νB(x)),min(ωA(x), ωB(x))⟩ |x ∈ X,

(v) AC = ⟨x, ωA(x), 1− νA(x), µA(x)⟩ | x ∈ X

(vi) 0n = (0, 1, 1) and 1n = (1, 0, 0).

As an illustration, let us consider the following example.

Example 2.2. Assume that the universe of discourse U = x1, x2, x3, where x1

characterizes the capability, x2 characterizes the trustworthiness and x3 indicatesthe prices of the objects. It may be further assumed that the values of x1, x2 andx3 are in [0, 1] and they are obtained from some questionnaires of some experts.The experts may impose their opinion in three components viz. the degree ofgoodness, the degree of indeterminacy and that of poorness to explain the cha-racteristics of the objects. Suppose A is a neutrosophic set (NS) of U , such that,

A = ⟨x1, (0.3, 0.5, 0.6)⟩ , ⟨x2, (0.3, 0.2, 0.3)⟩ , ⟨x3, (0.3, 0.5, 0.6)⟩,

where the degree of goodness of capability is 0.3, degree of indeterminacy ofcapability is 0.5 and degree of falsity of capability is 0.6 etc.

Definition 2.3. [1] Let U be any non-empty set. Suppose R is an equivalencerelation over U. For any non-null subset X of U , the sets

A1(x) = x : [x]R ⊆ X and A2(x) = x : [x]R ∩X = ∅

are called the lower approximation and upper approximation, respectively of X,where the pair S = (U,R) is called an approximation space. This equivalentrelation R is called indiscernibility relation.

The pair A(X) = (A1(x), A2(x)) is called the rough set of X in S. Here [x]Rdenotes the equivalence class of R containing x.


Definition 2.4. [1] Let A = (A1, A2) and B = (B1, B2) be two rough sets in theapproximation space S = (U,R). Then,

A ∪B = (A1 ∪B1, A2 ∪B2),

A ∩B = (A1 ∩B1, A2 ∩B2),

A ⊆ B if A ∩B = A,

∼ A = U − A2, U − A1.

3. Rough neutrosophic sets

In this section we introduce the notion of rough neutrosophic sets by combi-ning both rough sets and nuetrosophic sets. and some operations viz. union,intersection, inclusion and equalities over them. Rough neutrosophic set are thegeneralization of rough fuzzy sets [2] and rough intuitionistic fuzzy sets [22].

Definition 3.1. Let U be a non-null set and R be an equivalence relation on U .Let F be neutrosophic set in U with the membership function µF , indetermi-nacy function νF and non-membership function ωF . The lower and the upperapproximations of F in the approximation (U,R) denoted by N(F ) and N(F ) arerespectively defined as follows:

N(F ) = < x, µN(F )(x), νN(F )(x), ωN(F )(x) >| y ∈ [x]R, x ∈ U,N(F )) = < x, µN(F )(x), νN(F )(x), ωN(F )(x) >| y ∈ [x]R, x ∈ U,

where:

µN(F )(x) =∧

y∈[x]R

µF (y), νN(F )(x) =∨

y∈[x]R

νF (y), ωN(F )(x) =∨

y∈[x]R

ωF (y),

µN(F )(x) =∨

y∈[x]R

µF (y), νN(F )(x) =∧

y∈[x]R

νF (y), ωN(F )(x) =∧

y∈[x]R

ωF (y).

So

0 ≤ µN(F )(x) + νN(F )(x) + ωN(F )(x) ≤ 3

and

µN(F )(x) + νN(F )(x) + ωN(F )(x) ≤ 3,

where ”∨” and ”∧” mean ”max” and ”min” operators respectively, µF (x), νF (y)and ωF (y) are the membership, indeterminacy and non-membership of y withrespect to F . It is easy to see that N(F ) and N(F ) are two neutrosophic sets inU , thus the NS mappings N,N : N(U → N(U) are, respectively, referred to as theupper and lower rough NS approximation operators, and the pair (N(F ), N(F ))is called the rough neutrosophic set in (U,R).


From the above definition, we can see that N(F ) and N(F ) have constantmembership on the equivalence classes of U , if N(F ) = N(F ); i.e.,

µN(F ) = µN(F ),

νN(F ) = νN(F ),

ωN(F ) = ωN(F ).

For any x ∈ U, we call F a definable neutrosophic set in the approximation (U,R).It is easily to be proved that Zero ON neutrosophic set and unit neutrosophicsets 1N are definable neutrosophic sets. Let us consider a simple example in thefollowing.

Example 3.2. Let U = p1, p2, p3, p4, p5, p6, p7, p8 be the universe of discourse.Let R be an equivalence relation its partition of U is given by

U/R = p1, p4, p2, p3, p6, p5, p7, p8.

LetN(F )=(p1, (0.2, 0.3, 0.4), (p4, (0.3, 0.5, 0.4)), (p5, (0.4, 0.6, 0.2)),

(p7, (0.1, 0.3, 0.5))be a neutrosophic set of U . By Definition 3.1, we obtain:

N(F ) = (p1, (0.2, 0.5, 0.4)), (p4, (0.2, 0.5, 0.4)), (p5, (0.4, 0.6, 0.2));N(F ) = (p1, (0.2, 0.3, 0.4)), (p4, (0.2, 0.3, 0.4)), (p5, (0.4, 0.6, 0.2)),

(p7, (0.1, 0.3, 0.5)), (p8, (0.1, 0.3, 0.5)).

For another neutrosophic sets

N(G) = (p1, (0.2, 0.3, 0.4)), (p4, (0.2, 0.3, 0.4)), (p5, (0.4, 0.6, 0.2)).

The lower approximation and upper approximation of N(G) are calculated as

N(G) = (p1, (0.2, 0.3, 0.4)), (p4, (0.2, 0.3, 0.4)), (p5, (0.4, 0.6, 0.2));N(G) = (p1, (0.2, 0.3, 0.4)), (p4, (0.2, 0.3, 0.4)), (p5, (0.4, 0.6, 0.2)).

Obviously N(G) = N(G) is a definable neutrosophic set in the approximationspace (U,R).

Definition 3.3. If N(F ) = (N(F ), N(F )) is a rough neutrosophic set in (U,R),the rough complement of N(F ) is the rough neutrosophic set denoted ∼ N(F ) =(N(F )c, N(F )c), where N(F )c, N(F )c are the complements of neutrosophic setsN(F ) and N(F ), respectively,

N(F )c = < x, ωN(F ), 1− νN(F )(x), µN(F )(x) >| x ∈ U,

andN(F )c = < x, ωN(F ), 1− νN(F )(x), µN(F )(x) >| x ∈ U.

Definition 3.4. If N(F1) and N(F2) are two rough neutrosophic set of theneutrosophic sets F1 and F2 respectively in U , then we define the following:


(i) N(F1) = N(F2) iff N(F1) = N(F2) and N(F1) = N(F2).

(ii) N(F1) ⊆ N(F2) iff N(F1) ⊆ N(F2) and N(F1) ⊆ N(F2).

(iii) N(F1) ∪N(F2) =⟨N(F1) ∪N(F2), N(F1) ∪N(F2)

⟩.

(iv) N(F1) ∩N(F2) =⟨N(F1) ∩N(F2), N(F1) ∩N(F2)

⟩.

(v) N(F1) +N(F2) =⟨N(F1) +N(F2), N(F1) +N(F2)

⟩.

(vi) N(F1) ·N(F2) =⟨N(F1) ·N(F2), N(F1) ·N(F2)

⟩.

If N,M,L are rough neutrosophic set in (U,R), then the results in the followingproposition are straightforward from definitions.

Proposition 3.5.

(i) ∼ N(∼ N) = N

(ii) N ∪M = M ∪N, N ∩M = M ∩N

(iii) (N ∪M) ∪ L = N ∪ (M ∪ L) and (N ∩M) ∩ L = N ∩ (M ∩ L)

(iv) (N ∪M)∩L = (N ∪M)∩ (N ∪L) and (N ∩M)∪L = (N ∩M)∪ (N ∩L).

De Morgan ’s Laws are satisfied for neutrosophic sets:

Proposition 3.6.

(i) ∼ (N(F1) ∪N(F2)) = (∼ N(F1)) ∩ (∼ N(F2))

(ii) ∼ (N(F1) ∩N(F2)) = (∼ N(F1)) ∪ (∼ N(F2)).

Proof. (i) (N(F1) ∪ N(F2)) =∼ (N(F1) ∪ N(F2), N(F1) ∪ N(F2)) =

(∼N(F1)∪N(F2),∼N(F1)∪N(F2))=(N(F1)∪N(F2)c, N(F1)∪N(F2)c)= (∼N(F1) ∩N(F2),∼N(F1) ∩N(F2)) = (∼N(F1)) ∩ (∼ N(F2)).

(ii) Similar to the proof of (i).

Proposition 3.7. If F1 and F2 are two neutrosophic sets in U such that F1 ⊆ F2,then N(F1) ⊆ N(F2)

(i) N(F1 ∪ F2) ⊇ N(F1) ∪N(F2),

(ii) N(F1 ∩ F2) ⊆ N(F1) ∩N(F2).


Proof.

µN(F1∪F2)(x) = infµF1∪F2)(x) | x ∈ Xi= inf(maxµF1(x), µF2(x) | x ∈ Xi)≥ maxinfµF1(x) | x ∈ Xi, infµF2(x) | x ∈ Xi= maxµN(F1)(xi), µN(F2)(xi)= µN(F1) ∪ µN(F2)(xi).

Similarly,νN(F1∪F2)(xi) ≤ (νN(F1) ∪ νN(F2))(xi)

ωN(F1∪F2)(xi) ≤ (ωN(F1) ∪ ωN(F2))(xi)

Thus,N(F1 ∪ F2) ⊇ N(F1) ∪N(F2).

We can also see thatN(F1 ∪ F2) = N(F1) ∪N(F2).

Hence,N(F1 ∪ F2) ⊇ N(F1) ∪N(F2).

(ii) The proof of (ii) is similar to the proof of (i).

Proposition 3.8.

(i) N(F ) = ∼ N(∼ F )

(ii) N(F ) = ∼ N(∼ F )

(iii) N(F ) ⊇ N(F ).

Proof. According to Definition 3.1, we can obtain

(i) F = ⟨x, µF (x), νF (x), ωF (x)⟩ | x ∈ X∼ F = ⟨x, ωF (x), 1− νF (x), µF (x)⟩ | |x ∈ X

N(∼ F ) =⟨

x, ωN(∼F )(x), 1− νN(∼F )(x), µN(∼F )(x)⟩| y ∈ [x]R, x ∈ U

∼ N(∼ F ) =

⟨x, µN(∼F )(x), 1−(1−νN(∼F )(x)), ωN(∼F )(x)

⟩|y ∈ [x]R, x ∈ U

=

⟨x, µN(∼F )(x), νN(∼F )(x), ωN(∼F )(x)

⟩| y ∈ [x]R, x ∈ U

where

µN(∼F )(x) =∧

y∈[x]R

µF (y), νN(∼F )(x) =∨

y∈[x]R

νF (y), ωN(∼F )(x) =∨

y∈[x]R

ωF (y).

Hence N(F ) =∼ N(∼ F ).

(ii) The proof is similar to the proof of (i).


(iii) For any y ∈ N(F ), we can have

µN(F )(x) =∧

y∈[x]R

µF (y) ≤∨

y∈[x]R

µF (y), νN(F )(x) =∨

y∈[x]R

νF (y) ≥∧

y∈[x]R

νF (y)

and ωN(F )(x) =∨

y∈[x]R

ωF (y) ≥∧

y∈[x]R

ωF (y).

Hence N(F ) ⊆ N(F ).

4. Conclusion

In this paper we have defined the notion of rough neutrosophic sets. We havealso studied some properties on them and proved some propositions. The conceptcombines two different theories which are rough sets theory and neutrosophictheory. While neutrosophic set theory is mainly concerned with, indeterminateand inconsistent information, rough set theory is with incompleteness; but boththe theories deal with imprecision. Consequently, by the way they are defined,it is clear that rough neutrosophic sets can be utilized for dealing with both ofindeterminacy and incompleteness.

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[32] Deli, I., Interval-valued neutrosophic soft sets and its decision making,http://arxiv.org/abs/1402.3130



NEUTROSOPHIC PARAMETRIZED SOFT SET THEORYAND ITS DECISION MAKING

Said Broumi

Faculty of Arts and HumanitiesHay El Baraka Ben M’sik Casablanca B.P. 7951Hassan II University Mohammedia-CasablancaMorocco

Irfan Deli

Muallim Rıfat Faculty of EducationKilis 7 Aralık University79000 KilisTurkey


Department of MathematicsUniversity of New Mexico705 Gurley Avenue, Gallup, NM 87301USA

Abstract. In this work, we present definition of neutrosophic parameterized (NP) soft

set and its operations. Then we define NP-aggregation operator to form NP-soft de-

cision making method which allows constructing more efficient decision processes. We

also give an example which shows that they can be successfully applied to problem that

contain indeterminacy.

Keywords: soft set, neutrosophic set, neutrosophic soft set, neutrosophic parameter-

ized soft set, aggregation operator.

1. Introduction

In 1999, Smarandache firstly proposed the theory of neutrosophic set (NS) [28],which is the generalization of the classical sets, conventional fuzzy set [30] andintuitionistic fuzzy set [5]. After Smarandache, neutrosophic sets has been success-fully applied to many fields such as;control theory [1], databases [2], [3], medicaldiagnosis problem [4], decision making problem [21], topology [22], and so on.

In 1999, a Russian researcher [27] firstly gave the soft set theory as a generalmathematical tool for dealing with uncertainty and vagueness and how soft settheory is free from the parameterization inadequacy syndrome of fuzzy set theory,rough set theory, probability theory. Then, many interesting results of soft set

504 s. broumi, i. deli, f. smarandache

theory have been studied on fuzzy soft sets [8], [12], [23], on intuitionistic fuzzysoft set theory [14], [25], on possibility fuzzy soft set [7], on generalized fuzzysoft sets [26], [29], on generalized intuitionistic fuzzy soft [6], on interval-valuedintuitionistic fuzzy soft sets [20], on intuitionistic neutrosophic soft set [9], ongeneralized neutrosophic soft set [10], on fuzzy parameterized soft set theory [17],[18], on fuzzy parameterized fuzzy soft set theory [13], on intuitionistic fuzzy pa-rameterized soft set theory [15], on IFP-fuzzy soft set theory [16], on neutrosophicsoft set [24], interval-valued neutrosophic soft set [11], [19]. In this paper, ourmain objective is to introduce the notion of neutrosophic parameterized soft setwhich is a generalization of fuzzy parameterized soft set and intuitionistic fuzzyparameterized soft set. The paper is structured as follows. In Section 2, we firstrecall the necessary background on neutrosophic and soft set. In Section 3, wegive neutrosophic parameterized soft set theory and their respective properties.In Section 4, we present a neutrosophic parameterized aggregation operator. InSection 5, a neutrosophic parameterized decision methods is presented withexample. Finally, we conclude the paper.

2. Preliminaries

Throughout this paper, let U be a universal set and E be the set of all pos-sible parameters under consideration with respect to U , usually, parameters areattributes, characteristics, or properties of objects in U .

We now recall some basic notions of neutrosophic set and soft set. For moredetails, the reader could refer to [23], [27].

Definition 1. [27] Let U be a universe of discourse then the neutrosophic set Ais an object having the form

A = ⟨x : µA(x), νA(x), ωA(x)

⟩, x ∈ U,

where the functions µ, ν, ω : U →]−0, 1+[ define respectively the degree of mem-bership, the degree of indeterminacy, and the degree of non-membership of theelement x ∈ X to the set A with the condition.

(1) −0 ≤ µA(x) + νA(x) + ωA(x) ≤ 3+.

From the philosophical point of view, the neutrosophic set takes the value fromreal standard or non-standard subsets of ]−0, 1+[. So, instead of ]−0, 1+[, we needto take the interval [0, 1] for technical applications, because ]−0, 1+[ will be difficultto apply in the real applications such as in scientific and engineering problems.

For two NS,

ANS = ⟨x, µA(x), νA(x), ωA(x)⟩ | x ∈ X

andBNS = ⟨x, µB(x), νB(x), ωB(x)⟩ | x ∈ X.

Then,

neutrosophic parametrized soft set theory... 505

1. ANS ⊆ BNS if and only if

µA(x) ≤ νB(x), νA(x) ≥ νB(x), ωA(x) ≥ ωB(x).

2. ANS = BNS if and only if

µA(x) = µB(x), νA(x) = νB(x), ωA(x) = ωB(x) for any x ∈ X.

3. The complement of ANS is denoted by ANS and is defined by

ANS = ⟨x, ωA(x), 1− νA(x), µA(x)⟩ | x ∈ X.

4. A ∩ B=⟨x,minµA(x), µB(x),maxνA(x), νB(x),maxωA(x), ωB(x)⟩: x ∈ X.

5. A ∪ B=⟨x,maxµA(x), µB(x),minνA(x), νB(x), minωA(x), ωB(x)⟩ :x ∈ X.


Example 1. Assume that the universe of discourse U = xx, x2, x3, x4. It maybe further assumed that the values of x1, x2, x3 and x4 are in [0, 1]. Then, A is aneutrosophic set (NS) of U , such that,

A = ⟨x1, 0.4, 0.6, 0.5⟩ , ⟨x2, 0.3, 0.4, 0.7⟩ , ⟨x3, 0.4, 0.4, 0.6⟩ , ⟨x4, 0.5, 0.4, 0.8⟩.

Definition 2. [23] Let U be an initial universe set and E be a set of parameters.Let P (U) denotes the power set of U . Consider a nonempty set A ,A ⊂ E. A pair(K,A) is called a soft set over U , where K is a mapping given by K : A → P (U).


Example 2. Suppose that U is the set of houses under consideration, say

U = h1, h2, h3, h4, h5.

Let E be the set of some attributes of such houses, say

E = e1, e2, e3, e4, e5,

where e1, e2, e3, e4, e5 stand for the attributes ”beautiful”, ”costly”, ”in the greensurroundings”, ”moderate” and technically, respectively. In this case, to define asoft set means to point out expensive houses, beautiful houses, and so on. Forexample, the soft set (K,A) that describes the ”attractiveness of the houses” inthe opinion of a buyer, says Thomas, and may be defined like this:

A = e1, e2, e3, e4, e5,K(e1) = h2, h3, h5,K(e2) = h2, h4,K(e3) = h1,K(e4) = U,

K(e5) = h3, h5.


3. Neutrosophic parameterized soft set theory

In this section, we define neutrosophic parameterized soft set and their operations.

Definition 3.1. Let U be an initial universe, P (U) be the power set of U,E be aset of all parameters and K be a neutrosophic set over E. Then, a neutrosophicparameterized soft sets

ΨK = (⟨x, µK(x), νK(x), ωK(x), fK(x)⟩ : x ∈ E,

where µK : E → [0, 1], νK : E → [0, 1], ωK : E → [0, 1] and fK : E → P (U)such that fK(x) = Φ if µK(x) = 0, νK(x) = 1 and ωK(x) = 1. Here, the func-tions µK , νK and ωK are called membership function, indeterminacy function andnon-membership function of neutrosophic parameterized soft set (NP-soft set),respectively.

Example 3.2. Assume that U = u1, u2, u3 is a universal set and E = x1, x2is a set of parameters. If K = ⟨x1, 0.2, 0.3, 0.4⟩ , ⟨x2, 0.3, 0.5, 0.4⟩ and fK(x1) =u2, u3, fK(x2) = U, then, a neutrosophic parameterized soft set ΨK is written by

ΨK = (⟨x1, 0.2, 0.3, 0.4⟩ , u2, u3), (⟨x2, 0.3, 0.5, 0.4⟩ , U).

Definition 3.3. Let ΨK ∈ NP -soft set. If fK(x) = U, µK(x) = 0, νK(x) = 1and ωK(x) = 1, for all x ∈ E, then ΨK is called a K-empty NP-soft set, denotedby ΨΦK

. If K = Φ, then the K-empty NP-soft set is called an empty NP-soft set,denoted by ΨΦ.

Definition 3.4. Let ΨK ∈ NP -soft set. If fK(x) = U, µK(x) = 1, νK(x) = 0 andωK(x) = 0, for all x ∈ E, then ΨK is called a K- universal NP-soft set, denotedby ΨK . If K = E, then the K-universal NP-soft set is called a universal NP-softset, denoted by ΨE.

Definition 3.5. ΨK and ΩL are two NP-soft sets. Then, ΨK is an NP-subsetof ΩL, denoted by ΨK ⊑ ΩL if and only if µK(x) ≤ µL(x), νK(x) ≥ νL(x) andωK(x) ≥ ωL(x), and fK(x) ⊑ gL(x) for all x ∈ E.

Definition 3.6. ΨK and ΩL are two NP-soft sets. Then, ΨK = ΩL, if and onlyif ΨK ⊑ ΩL and ΩL ⊑ ΨK , for all x ∈ E.

Definition 3.7. Let ΨK ∈ NP -soft set. Then, the complement of ΨK , denotedby Ψc

K , is defined by

ΨcK = (⟨x, ωK(x), νK(x), µK(x)⟩ , fKc(x)) : x ∈ E,

where fKc(x) = U \ fK(x).


Definition 3.8. Let ΨK and ΩL be two NP-soft sets. Then, the union of ΨK andΩL, denoted by ΨK ⊔ ΩL, is defined by

ΨK ⊔ ΩL = (⟨x,maxµK(x), µL(x),minνK(x), νL(x),minωK(x), ωL(x)⟩ , fK∪L(x)) : x ∈ E,

where fK∪L(x) = fK(x) ∪ fL(x).

Definition 3.9. Let ΨK and ΩL be two NP-soft sets. Then, the intersection ofΨK and ΩL, denoted by ΨK ⊓ ΩL, is defined by

ΨK ⊓ ΩL = (⟨x,minµK(x), µL(x),maxνK(x), νL(x),maxωK(x), ωL(x)⟩ , fK∩L(x)) : x ∈ E,

where fK∩L(x) = fK(x) ∩ fL(x).

Example 3.10. Let U = u1, u2, u3, u4, E = x1, x2, x3. Then,

ΨK = (⟨x1, 0.2, 0.3, 0.4⟩ , u1, u2), (⟨x2, 0.3, 0.5, 0.4⟩ , u2, u3)ΩL = (⟨x2, 0.1, 0.2, 0.4⟩ , u3, u4), (⟨x3, 0.5, 0.2, 0.3⟩ , u3).

Then

ΨK ⊔ ΩL = (⟨x1, 0.2, 0.3, 0.4⟩ , u1, u2), (⟨x2, 0.3, 0.2, 0.4⟩ , u2, u3, u4),(⟨x3, 0.5, 0.20.3⟩ , u3)

ΨK ⊓ ΩL = (⟨x2, 0.1, 0.5, 0.4⟩ , u3, u4)Ψc

K = (⟨x1, 0.4, 0.3, 0.2⟩ , u3, u4), (⟨x2, 0.4, 0.5, 0.3⟩ , u1, u4).

Remark 3.11. ΨK ⊑ ΩL does not imply that every element of ΨK is an elementof ΩL as in the definition of classical subset. For example, assume that U =u1, u2, u3, u4 is a universal set of objects and E = x1, x2, x3 is a set of allparameters, if NP-soft sets ΨK and ΩL are defined as

ΨK = (⟨x1, 0.2, 0.3, 0.4⟩ , u1, u2), (⟨x2, 0.3, 0.5, 0.4⟩ , u2)ΩL = (⟨x1, 0.3, 0.2, 0.4⟩ , U), (⟨x2, 0.5, 0.20.3⟩ , u1, u4).

It can be seen that ΨK ⊑ ΩL, but every element of ΨK is not an element of ΩL.

Proposition 3.12. Let ΨK ,ΩL ∈ NP -soft sets. Then

(i) ΨK ⊑ ΨE.

(ii) ΨΦ ⊑ ΨK.

(iii) ΨK ⊑ ΨK .


Proof. It is clear from Definitions 3.3-3.5.

Proposition 3.13. Let ΨK ,ΩL and γM ∈ NP -soft sets. Then

(i) ΨK = ΩL and ΩL = γM ⇐⇒ ΨK = γM ,

(ii) ΨK ⊑ ΩL and ΩL ⊑ ΨK ⇐⇒ ΨK = ΩL,

(iii) ΨK ⊑ ΩL and ΩL ⊑ γM =⇒ ΨK ⊑ γM .

Proof. It can be proved by Definitions 3.3-3.5.

Proposition 3.14. Let ΨK ∈ NP -soft set. Then

(i) (ΨcK)

c = ΨK,

(ii) ΨcΦ = ΨE,

(iii) ΨcE = ΨΦ.

Proof. It is trial.


(i) ΨK ⊔ΨK = ΨK,

(ii) ΨK ⊔ΨΦ = ΨK,

(iii) ΨK ⊔ΨE = ΨE,

(iv) ΨK ⊔ ΩL = ΩL ⊔ΨK,

(v) (ΨK ⊔ ΩL) ⊔ γM = ΨK ⊔ (ΩL ⊔ γM).

Proof. It is clear.


(i) ΨK ⊓ΨK = ΨK,

(ii) ΨK ⊓ΨΦ = ΨΦ,

(iii) ΨK ⊓ΨE = ΨK,

(iv) ΨK ⊓ ΩL = ΩL ⊓ΨK,

(v) (ΨK ⊓ ΩL) ⊓ γM = ΨK ⊓ (ΩL ⊓ γM).

Proof. It is clear.


(i) ΨK ⊔ (ΩL ⊓ γM) = (ΨK ⊔ ΩL) ⊓ (ΨK ⊔ γM),


(ii) ΨK ⊓ (ΩL ⊔ γM) = (ΨK ⊔ ΩL) ⊔ (ΨK ⊔ γM).

Proof. It can be proved by Definitions 3.8 and 3.9.

Proposition 3.18. Let ΨK ,ΩL ∈ NP -soft set. Then

(i) (ΨK ⊔ ΩL)c = Ψc

K ⊓ ΩcL,

(ii) (ΨK ⊓ ΩL)c = Ψc

K ⊔ ΩcL.

Proof. It is clear.

Definition 3.19. Let ΨK ,ΩL ∈ NP -soft set. Then

(i) An OR-product of ΨK and ΩL denoted by ΨK∨ΩL is defined as follows

ΨK∨ΩL = (⟨(x, y), (maxµK(x), µL(y),minνK(x), νL(x),minωK(x), ωL(y)⟩ ,ΨK ∪ ΩL(x, y))) : x, y ∈ E,

where ΨK ∪ ΩL(x, y) = ΨK(x) ∪ ΩL(y).

(ii) An AND-product of ΨK and ΩL, denoted by ΨK∧ΩL, is defined as follows

ΨK∧ΩL = (⟨(x, y), (minµK(x), µL(y),maxνK(x), νL(y),maxωK(x), ωL(y)⟩ ,ΨK ∩ ΩL(x, y))) : x, y ∈ E,

where ΨK ∩ ΩL(x, y) = ΨK(x) ∩ ΩL(y).


(i) ΨK∧ΨΦ = ΨΦ

(ii) (ΨK∧ΩL)∧γM = ΨK∧(ΩL∧γM)

(iii) (ΨK∨ΩL)∨γM = ΨK∨(ΩL∨γM).

Proof. It can be proved by Definition 3.15.

4. NP-aggregation operator

In this section, we define an NP-aggregation operator of an NP-soft set to con-struct a decision method by which approximate functions of a soft set are combinedto produce a single neutrosophic set that can be used to evaluate each alternative.

Definition 4.1. Let ΨK ∈ NP -soft set. Then an NP-aggregation operator ofΨK , denoted by Ψagg

K , is defined by

ΨaggK = (⟨u, µagg

K (u), νaggK (u), ωagg

K (u)⟩) : u ∈ U,


which is a neutrosophic set over U ,

µaggK : U → [0, 1] µagg

K (u) =1

|U |∑x∈Eu∈U

µK(x)γfK(x)(u),

νaggK : U → [0, 1] νagg

K (u) =1

|U |∑x∈Eu∈U

νK(x)γfK(x)(u),

and

ωaggK : U → [0, 1] µagg

K (u) =1

|U |∑x∈Eu∈U

ωK(x)γfK(x)(u),

and where

γfK(x)(u) =

1, x ∈ fK(x),

0, otherwise.

|U | is the cardinality of U .

Definition 4.2. Let ΨK ∈ NP -soft set and ΨaggK an aggregation neutrosophic

parameterized soft set, then a reduced fuzzy set of ΨaggK is a fuzzy set over U

denoted by

ΨaggK =

µaggΨKf

(u)

u: u ∈ U

,

where µaggΨKf

(u) : U → [0, 1] and µaggΨKf

(u) + νaggK (u)− ωagg

K (u).

5. NP-Decision methods

Inspired by the decision making methods regard in [13]-[15]. In this section, wealso present an NP-decision method to a neutrosophic parameterized soft set.Based on Definitions 4.1 and 4.2, we construct an NP-decision making method bythe following algorithm. Now, we construct an NP-soft decision making methodby the following algorithm to produce a decision fuzzy set from a crisp set of thealternatives.

According to the problem, the decision maker

(i) constructs a feasible Neutrosophic subsets K over the parameters set E,

(ii) constructs an NP-soft set ΨK over the alternatives set U ,

(iii) computes the aggregation neutrosophic parameterized soft set ΨaggK of ΨK ,

(iv) computes the reduced fuzzy set µΨaggKf (u)

of ΨaggK ,

(v) chooses the element of µΨaggKf (u)

that has maximum membership degree.

Now, we can give an example for the NP-soft decision making method.


Example. Assume that a company wants to fill a position. There are four can-didates who fill in a form in order to apply formally for the position. There is adecision maker (DM) that is from the department of human resources. He wantsto interview the candidates, but it is very difficult to make it all of them. There-fore, by using the NP-soft decision making method, the number of candidates arereduced to a suitable one. Assume that the set of candidates U = u1, u2, u3, u4which may be characterized by a set of parameters E = x1, x2, x3. For i = 1, 2, 3,the parameters i stand for experience, computer knowledge and young age, respec-tively. Now, we can apply the method as follows:

Step i. Assume that DM constructs a feasible neutrosophic subsets K over theparameters set E as follows:

K = ⟨x1, 0.2, 0.3, 0.4⟩ , ⟨x2, 0.3, 0.2, 0.4⟩ , ⟨x3, 0.5, 0.20.3⟩.

Step ii. DM constructs an NP-soft set ΨK over the alternatives set U as follows:

ΨK = (⟨x1, 0.2, 0.3, 0.4⟩ , u1, u2), (⟨x2, 0.3, 0.2, 0.4⟩ , u2, u3, u4),(⟨x3, 0.5, 0.2, 0.3⟩ , u3).

Step iii. DM computes the aggregation neutrosophic parameterized soft set ΨaggK

of ΨK as follows¿

ΨaggK = ⟨u1, 0.05, 0.075, 0.1⟩ , ⟨u2, 0.1, 0.125, 0.2⟩ , ⟨u3, 0.2, 0.1, 0.175⟩ ,

⟨u4, 0.125, 0.05, 0.075⟩.

Step iv. Computes the reduced fuzzy set µΨaggKf (u)

of ΨaggK as follows:

µΨaggKf (u1) = 0.025




Step v. Finally, DM chooses u3 for the position from µΨaggKf (u)

since it has the

maximum degree 0.125 among the others.

6. Conclusion

In this work, we have introduced the concept of neutrosophic parameterized softset and studied some of its properties. The complement, union and intersectionoperations have been defined on the neutrosophic parameterized soft set. The defi-nition of NP-aggregation operator is introduced with application of this operationin decision making problems.


References

[1] Aggarwal, S., Biswas, R., Ansari, A.Q., Neutrosophic Modeling andControl, Computer and Communication Technology, (2010), 718-723.

[2] Arora, M., Biswas, R., Pandy, U.S., Neutrosophic Relational DatabaseDecomposition, International Journal of Advanced Computer Science andApplications, 2/ 8 (2011), 121-125.

[3] Arora, M., Biswas, R., Deployment of Neutrosophic Technology to Re-trieve Answers for Queries Posed in Natural Language, in 3rd Interna-tional Conference on Computer Science and Information Technology, (2010),435-439.

[4] Ansari, Biswas, Aggarwal, Proposal for Applicability of NeutrosophicSet Theory in Medical AI, International Journal of Computer Applications,27/5 (2011), 5-11.

[5] Atanassov, K., Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, 20(1986), 87-96.

[6] Babitha, K.V., Sunil, J.J., Generalized Intuitionistic Fuzzy Soft Setsand Its Applications, Gen. Math. Notes, 7/ 2 (2011), 1-14.

[7] Bashir, M., Salleh, A.R., Alkhazaleh, S., Possibility IntuitionisticFuzzy Soft Set, Advances in Decision Sciences, doi:10.1155/2012/404325.

[8] Borah, M., Neog, T.J., Sut, D.K., A study on some operations of fuzzysoft sets, International Journal of Modern Engineering Research, 2/2 (2012),157-168.

[9] Broumi, S., Smarandache, F., Intuitionistic Neutrosophic Soft Set, Jour-nal of Information and Computing Science, 8/ 2 (2013), 130-140.

[10] Broumi, S., Generalized Neutrosophic Soft Set, International Journal ofComputer Science, Engineering and Information Technology, 3/2 (2013),17-30.

[11] Broumi, S., Deli, I., Smarandache, F., Relations on Interval ValuedNeutrosophic Soft Sets, Journal of New Results in Science, 5 (2014), 1-20.

[12] Cagman, N., Enginoglu, S., Cıtak F., Fuzzy Soft Set Theory and ItsApplications, Iran J. Fuzzy Syst., 8 (3) (2011), 137-147.

[13] Cagman, N., Cıtak , F., Enginoglu, S., Fuzzy parameterized fuzzy softset theory and its applications, Turkish Journal of Fuzzy System, 1/1 (2010),21-35.


[14] Cagman, N., Karatas, S., Intuitionistic fuzzy soft set theory and its de-cision making, Journal of Intelligent and Fuzzy Systems, DOI:10.3233/IFS-2012-0601.

[15] Cagman, N., Deli, I., Intuitionistic fuzzy parametrized soft set theory andits decision making, Submitted.

[16] Cagman, N., Karaaslan, F., IFP fuzzy soft set theory and its applica-tions, Submitted.

[17] Cagman, N., Deli, I., Product of FP-Soft Sets and its Applications,Hacettepe Journal of Mathematics and Statistics, 41/3 (2012), 365-374.

[18] Cagman, N., Deli, I., Means of FP-Soft Sets and its Applications,Hacettepe Journal of Mathematics and Statistics, 41/5 ( 2012), 615-625.

[19] Deli, I., Interval-valued neutrosophic soft sets and its decision making,submitted.

[20] Jiang, Y., Tang, Y., Chen, Q., Liu, , H., Tang, J., Interval-valued in-tuitionistic fuzzy soft sets and their properties, Computers and Mathematicswith Applications, 60 (2010), 906-918.

[21] Kharal, A., A Neutrosophic Multicriteria Decision Making Method, NewMathematics and Natural Computation, Creighton University, USA,2013.

[22] Lupianez, F.G., On neutrosophic topology, Kybernetes, 37/6 (2008),797-800.

[23] Maji, P.K., Roy, A.R., Biswas, R., Fuzzy soft sets, Journal of FuzzyMathematics, 9/3 (2001), 589-602.

[24] Maji, P.K., Neutrosophic Soft Set, Annals of Fuzzy Mathematics and Infor-matics, 5/ 1 (2013), 287-623.

[25] Maji, P.K., Biswas, R., Roy, A.R., Intuitionistic fuzzy soft sets,The Journal of Fuzzy Mathematics, 9/3 (2001), 677-692.

[26] Majumdar, P., Samanta, S.K., Generalized Fuzzy Soft Sets, Computersand Mathematics with Applications, 59 (2010), 1425-1432.

[27] Molodtsov, D.A., Soft Set Theory First Result, Computers and Mathe-matics with Applications, 37 (1999), 19-31.

[28] Smarandache, F., A Unifying Field in Logics. Neutrosophy: NeutrosophicProbability, Set and Logic, Rehoboth: American Research Press, 1999.


[29] Yang, H.L., Notes On Generalized Fuzzy Soft Sets, Journal of Mathema-tical Research and Exposition, 31/3 (2011), 567-570.

[30] Zadeh, L.A., Fuzzy sets, Information and Control, 8 (1965), 338-353.



YOUNG TYPE INEQUALITIES FOR MATRICES

Yang Peng

School of Mathematics and StatisticsChongqing Three Gorges UniversityChongqing, 404100P.R. Chinae-mail: peng [email protected]

Abstract. In this note, we present a refinement of an inequality due to Hirzallah and

Kittaneh [Linear Algebra Appl., 308 (2000), 77-84]. Meanwhile, we also obtain an im-

provement of a result shown by Kittaneh and Manasrah [Linear Multilinear Algebra, 59

(2011), 1031-1037].

Keywords: Young type inequalities; Hilbert-Schmidt norm; positive semidefinite ma-

trices; Kantorovich constant.

MSC (2010) Subject Classification: 15A60.

1. Introduction

Let Mn be the space of n×n complex matrices. For A = (aij) ∈ Mn, the Hilbert-Schmidt norm of A is defined by

∥A∥2 =

√√√√ n∑i,j=1

|aij|2.

It is known that the Hilbert-Schmidt norm is unitarily invariant.The classical Young inequality for scalar says that if a, b ≥ 0 and 0 ≤ v ≤ 1,

thenavb1−v ≤ va+ (1− v) b

with equality if and only if a = b. By using Young’s inequality, we can obtain someresults of Heinz mean. For more information on Heinz inequality for matrices thereader is referred to [1]–[3].

The Kontorovich constant is defined as

K (t, 2) =(t+ 1)2

4t

for t > 0. Zuo, Shi, Fujii [4] proved that if a, b ≥ 0 and 0 ≤ v ≤ 1, then

K (h, 2)r avb1−v ≤ va+ (1− v) b,(1.1)

516 y. peng

where h =a

b, r = min v, 1− v. This is a refinement of the classical Young

inequality.Let A,X,B ∈ Mn such that A and B are positive semidefinite. Kosaki [5]

and Bhatia-Parthasarathy [6] proved that if 0 ≤ v ≤ 1, then∥∥AvXB1−v∥∥2

2≤ ∥vAX + (1− v)XB∥22 .(1.2)

This is a matrix version of Young inequality. Hirzallah and Kittaneh [7] provedthat if 0 ≤ v ≤ 1, then∥∥AvXB1−v

∥∥2

2+ v20 ∥AX −XB∥22 ≤ ∥vAX + (1− v)XB∥22 ,(1.3)

where v0 = min v, 1− v. Inequality (1.3) is an improvement of inequality (1.2).Kittaneh-Manasrah [8] and He-Zou [9] proved that if 0 ≤ v ≤ 1, then

∥vAX + (1− v)XB∥22 ≤∥∥AvXB1−v

∥∥2

2+ s20 ∥AX −XB∥22 ,(1.4)

where s0 = max v, 1− v. Inequality (1.4) is a reverse inequality of (1.3).In this note, we present refinements of inequalities (1.3) and (1.4).

2. Main results

In this section, we first give a refinement of inequality (1.3). To achieve it, weneed the following lemma.

Lemma 2.1. If a, b ≥ 0 and 0 ≤ v ≤ 1, then

K (h, 2)r(avb1−v

)2+ v20 (a− b)2 ≤ (va+ (1− v) b)2 ,(2.1)

where h =a

b, v0 = min v, 1− v, r = min 2v0, 1− 2v0.

Proof. If v =1

2, inequality (2.1) becomes equality. If v <

1

2, then by (1.1), we

have

(va+ (1− v) b)2 − v20 (a− b)2 = (va+ (1− v) b)2 − v2 (a− b)2

= 2vab+ (1− 2v) b2

≥ K (h, 2)r (avb1−v)2.

If v >1

2, then by (1.1), we have

(va+ (1− v) b)2 − v20 (a− b)2 = (va+ (1− v) b)2 − (1− v)2 (a− b)2

= (2v − 1) a2 + 2 (1− v) ab

≥ K (h, 2)r (avb1−v)2.


young type inequalities for matrices 517

Theorem 2.2. Let A,X,B ∈ Mn such that A and B are positive semidefinite.Suppose that the spectral decomposition of A,B are A = UΛ1U

∗, B = V Λ2V∗

respectively, where Λ1 = diag (λ1, ..., λn) ,Λ2 = diag (µ1, ..., µn) , λi, µi ≥ 0,i = 1, ..., n. Let

K = min

K

(λi

µj

, 2

), i, j = 1, ..., n

.

Then

Kr∥∥AvXB1−v

∥∥2

2+ v20 ∥AX −XB∥22 ≤ ∥vAX + (1− v)XB∥22 ,(2.2)

where v0 = min v, 1− v, r = min 2v0, 1− 2v0.Proof. Let Y = U∗XV = [yij]. We have as in [10],

∥vAX + (1− v)XB∥22 =n∑

i,j=1

(vλi + (1− v)µj)2 |yij|2,

∥AvXB1−v∥22 =n∑

i,j=1

(λviµ

1−vj

)2 |yij|2,∥AX −XB∥22 =

n∑i,j=1

(λi − µj)2 |yij|2.

Inequality (2.2) deduces from inequality (2.1) and above equalities. This completesthe proof.

Next, we show an improvement of inequality (1.4). To do this, we need thefollowing lemma.

Lemma 2.3. If a, b ≥ 0 and 0 ≤ v ≤ 1, then

(va+ (1− v) b)2 ≤ K (h, 2)−r (avb1−v)2

+ s20 (a− b)2 ,(2.3)

where h =a

b, s0 = max v, 1− v, r = min 2s0 − 1, 2− 2s0.

Proof. If v =1

2, inequality (2.3) becomes equality. If v <

1

2, then by (1.1), we

have

s20 (a− b)2 − (va+ (1− v) b)2 = (1− v)2 (a− b)2 − (va+ (1− v) b)2

= (1− 2v) a2 − 2 (1− v) ab= (1− 2v) a2 + 2vab− 2ab

≥ K (h, 2)r (a1−vbv)2 − 2ab

≥ −K (h, 2)−r (avb1−v)2.

If v >1

2, then by (1.1), we have

s20 (a− b)2 − (va+ (1− v) b)2 = v2 (a− b)2 − (va+ (1− v) b)2

= (2v − 1) b2 − 2vab= (2v − 1) b2 + 2 (1− v) ab− 2ab

≥ K (h, 2)r (a1−vbv)2 − 2ab

≥ −K (h, 2)−r (avb1−v)2.

518 y. peng


Theorem 2.4. Let A,X,B ∈ Mn such that A and B are positive semidefinite.Suppose that the spectral decomposition of A,B are A = UΛ1U

∗, B = V Λ2V∗

respectively, where Λ1 = diag (λ1, ..., λn) ,Λ2 = diag (µ1, ..., µn) , λi, µi ≥ 0,i = 1, ..., n. Let

K = min

K

(λi

µj

, 2

), i, j = 1, ..., n

.

Then

∥vAX + (1− v)XB∥22 ≤ K−r∥∥AvXB1−v

∥∥2

2+ s20 ∥AX −XB∥22 ,(2.4)

where s0 = max v, 1− v, r = min 2v0, 1− 2v0.Proof. The result follows from inequality (2.3) and by using an argument similarto that used for the proof of Theorem 2.2. This completes the proof.

Since K (t, 2) =(t+ 1)2

4t≥ 1 for t > 0, it follows that inequalities (2.2) and

(2.4) are refinements of inequalities (1.3) and (1.4) respectively.

References

[1] Hu, X., Some inequalities for unitarily invariant norms, J. Math. Inequal.,6 (2012), 615-623.

[2] Fu, X., He, C., On some inequalities for unitarily invariant norms, J. Math.Inequal., 7 (2013), 727-737.

[3] Zou, L., Inequalities related to Heinz and Heron means, J. Math. Inequal.,7 (2013), 389-397.

[4] Zuo, H., Shi, G., Fujii, M., Refined Young inequality with Kantorovichconstant, J. Math. Inequal., 5 (2011), 551-556.

[5] Kosaki, H., Arithmetic-geometric mean and related inequalities for opera-tors, J. Funct. Anal., 156 (1998), 429-451.

[6] Bhatia, R., Parthasarathy, K.R., Positive definite functions and ope-rator inequalities, Bull. London Math. Soc., 32 (2000), 214-228.

[7] Hirzallah, O., Kittaneh, F., Matrix Young inequalities for the Hilbert-Schmidt norm, Linear Algebra Appl., 308 (2000), 77-84.

[8] Kittaneh, F., Manasrah, Y., Reverse Young and Heinz inequalities formatrices, Linear Multilinear Algebra, 59 (2011), 1031-1037.

[9] He, C., Zou, L., Some inequalities involving unitarily invariant norms,Math. Inequal. Appl., 15 (2012), 767-776.

[10] Kittaneh, F., Manasrah, Y., Improved Young and Heinz inequalities formatrices, J. Math. Anal. Appl., 361 (2010), 262-269.



BOUNDS FOR THE EIGENVALUES OF MATRICES

Limin Zou1

Youyi Jiang

School of Mathematics and StatisticsChongqing Three Gorges UniversityChongqing, 404100P.R. China

Abstract. In this paper, we prove that all the eigenvalues of arbitrarily complex matrix

lies in a closed disk with the radius involving the sum of the squares of the absolute

values of the eigenvalues. So, any known upper bound on this sum of squares yields a

”new” eigenvalues inclusion region. As applications, some existing results are obtained

or improved.

Keywords: eigenvalues; Frobenius norm; trace; disk.

MSC (2010) Subject Classification: 15A18; 15A60.

1. Introduction

Let M be an n× n complex matrix and λi (i = 1, ..., n) be the eigenvalues of M .We denote by ||M ||F , M∗ and trM the Frobenius norm, conjugate transpose and

trace of M , respectively. Let i ∈ N and Ri (M) =n∑

j=1,j =i

|mij| be the sum of the

absolute values of the non-diagonal entries in the ith row of M = [mij]n×n. Forz = a+ bi ∈ C, the conjugate of z is denoted by z = a− bi.

Estimation of eigenvalues has been, and still is, a hot topic of matrix analysis.There are many research papers published in a variety of journals each year, anddifferent approaches have been taken for different purposes. Here, we brieflyreview three types of estimation methods for eigenvalues of matrices.

The well-known upper bound forn∑

i=1

|λi|2 was presented by Schur in 1909 as

follows:

1Corresponding author. E-mail address: [email protected] (L. Zou)

520 l. zou, y. jiang

(1.1)n∑

i=1

|λi|2 ≤ ∥M∥2F ,

which is the first bound forn∑

i=1

|λi|2.

The famous Gersgorin disk theorem says that every eigenvalue of M lies inat least one of the Gersgorin discs centered at mii with radius Ri (M); i.e.,

λi ∈n∪

i=1

z ∈ C : |z −mii| ≤ Ri (M), i = 1, ..., n.

The Gersgorin disks are a particular class of easily computed regions in the planethat are guaranteed to include the eigenvalues of a given matrix. Many authors,perhaps attracted by the geometrical elegance of the Gersgorin theory, have ge-neralized the ideas and methods of this theory to obtain other types of inclusion[1, Chapter 6].

In 1994, Gu [2, Theorem 1] proved that every eigenvalue of M lies in the

following disk centered attrM

n:

(1.2)

z ∈ C :

∣∣∣∣z − trM

n

∣∣∣∣ ≤√√√√n− 1

n

(∥M∥2F − |trM |2

n

) .

Bhatia [3, p.24] pointed that results such as (1.2) are interesting because theygive some information about the location of the eigenvalues of a matrix in termsof more easily computable function like the Frobenius norm and the trace.

In this paper, we prove that all the eigenvalues of M lies in the following disk

centered attrM

n:

(1.3)

z ∈ C :

∣∣∣∣z − trM

n

∣∣∣∣ ≤√√√√n− 1

n

(n∑

i=1

|λi|2 −|trM |n

2) .

The radius involvesn∑

i=1

|λi|2, thus (1.3) gives the relationship between the first and

third types of estimation methods. Any known upper bound forn∑

i=1

|λi|2 yields

a ”new” estimation of eigenvalues such as (1.2). As applications, some existingestimations are directly obtained by (1.3).

bounds for the eigenvalues of matrices 521

2. Main result

Theorem 2.1. Let M be an n × n complex matrix. Then all the eigenvalues ofM are located in the following disk:z ∈ C :

∣∣∣∣z − trM

n

∣∣∣∣ ≤√√√√n− 1

n

(n∑

i=1

|λi|2 −|trM |n

2) .

Proof. We show that λ1 is located in this disk, and similarly for others. By theCauchy–Schwarz inequality, we have∣∣∣∣∣

n∑i=2

λi

∣∣∣∣∣2

≤n∑

i=2

12 ·n∑

i=2

|λi|2 ≤ (n− 1)n∑

i=2

|λi|2.

Then

(2.1)(n− 1) |λ1|2 +

∣∣∣∣ n∑i=2

λi

∣∣∣∣2 ≤ (n− 1) |λ1|2 + (n− 1)n∑

i=2

|λi|2

= (n− 1)n∑

i=1

|λi|2.

Since ∣∣∣∣∣n∑

i=2

λi

∣∣∣∣∣2

=

∣∣∣∣∣n∑

i=1

λi

∣∣∣∣∣2

− λ1

n∑i=1

λi − λ1

n∑i=1

λi + |λ1|2 ,

by (2.1), we have

n |λ1|2 − λ1

n∑i=1

λi − λ1

n∑i=1

λi +

∣∣∣∣∣n∑

i=1

λi

∣∣∣∣∣2

≤ (n− 1)n∑

i=1

|λi|2,

which implies

n |λ1|2 − λ1

n∑i=1

λi − λ1

n∑i=1

λi +1

n

∣∣∣∣∣n∑

i=1

λi

∣∣∣∣∣2

≤ (n− 1)n∑

i=1

|λi|2 −n− 1

n

∣∣∣∣∣n∑

i=1

λi

∣∣∣∣∣2

,

and then

|λ1|2 −λ1

n

n∑i=1

λi −λ1

n

n∑i=1

λi +1

n2

∣∣∣∣∣n∑

i=1

λi

∣∣∣∣∣2

≤ n− 1

n

n∑i=1

|λi|2 −n− 1

n2

∣∣∣∣∣n∑

i=1

λi

∣∣∣∣∣2

.

Consequently,


(λ1 −

1

n

n∑i=1

λi

)(λ1 −

1

n

n∑i=1

λi

)≤ n− 1

n

n∑i=1

|λi|2 −1

n

∣∣∣∣∣n∑

i=1

λi

∣∣∣∣∣2 ,

which is equivalent to

∣∣∣∣λ1 −trM

n

∣∣∣∣ ≤√√√√n− 1

n

(n∑

i=1

|λi|2 −|trM |n

2).

The proof is completed.

3. Applications

In section, we give some applications of our result.

Corollary 3.1. [2, Theorem 1]. Let M be an n×n complex matrix. Then all theeigenvalues of M are located in the following disk:z ∈ C :

∣∣∣∣z − trM

n

∣∣∣∣ ≤√√√√n− 1

n

(∥M∥2F − |trM |2

n

) .

This result is a corollary of Theorem 2.1 and the Schur’s inequality (1.1).

Let M be an n× n complex matrix partitioned as

M =

Ak×k Bk×(n−k)

C(n−k)×k D(n−k)×(n−k)

, 1 ≤ k ≤ n− 1,

where Ak×k is the principal submatrix of M . Tu [4, Theorem 1] obtained thefollowing result:

(3.1)n∑

i=1

|λi|2 ≤ ∥M∥2F − max1≤k≤n−1

(∥∥Bk×(n−k)

∥∥− ∥∥C(n−k)×k

∥∥)2 .The following result obtained by Zou and Jiang is a corollary of Theorem 2.1 and(3.1).

bounds for the eigenvalues of matrices 523

Corollary 3.2. [5, Theorem 2.1] Let M be an n × n complex matrix. Then allthe eigenvalues of M are located in the following disk:

z ∈ C :

∣∣∣∣z − trM

n

∣∣∣∣ ≤ R

,

where

R =

√√√√n− 1

n

(∥M∥2F − max

1≤k≤n−1

(∥∥Bk×(n−k)

∥∥F−∥∥C(n−k)×k

∥∥F

)2 − |trM |2

n

).

Kress et al. [6, Theorem 1] obtained an upper bound forn∑

i=1

|λi|2 as follows:

(3.2)n∑

i=1

|λi|2 ≤√∥M∥4F − 1

2∥MM∗ −M∗M∥2F .

If M is non-normal, then (3.2) is obviously sharper than (1.1). The followingresult is a corollary of Theorem 2.1 and (3.2).

Corollary 3.3. Let M be an n× n complex matrix. Then all the eigenvalues ofM are located in the following disk:

(3.3)

z ∈ C :

∣∣∣∣z − trM

n

∣∣∣∣ ≤ R

,

where

R =

√√√√n− 1

n

(√∥M∥4F − 1

2∥MM∗ −M∗M∥2F − |trM |2

n

).

Note that the inequality (3.3) is a refinement of (1.2) for non-normal matrices.

Acknowledgments. The author wishes to express his heartfelt thanks to the ref-eree and Professor Piergiulio Corsini for their detailed and helpful suggestions forrevising the manuscript. This research was supported by Scientific and Techno-logical Research Program of Chongqing Municipal Education Commission (GrantNo. KJ131122) and Scientific Research Project of Chongqing Three Gorges Uni-versity (No. 12ZD-17).


References

[1] Horn, R.A., Johnson, C.R.,Matrix analysis, Cambridge University Press,Cambridge, 1985.

[2] Gu, Y., The distribution of eigenvalues of a matrix, Acta MathematicaeApplicatae Sinica, 17 (1994), 501-511.

[3] Bhatia, R., Matrix Analysis Springer-Verlag, New York, 1997.

[4] Tu, B., The lower bound of the rank of a matrix and the sufficient conditionsof non-singularity of a matrix. (I), Journal of Fu Dan University, 21 (1982),416- 422.

[5] Zou, L., Jiang, Y., Estimation of the eigenvalues and the smallest singularvalue of matrices, Linear Algebra Appl., 433 (2010), 1203-1211.

[6] Kress, R., Ludwig, H., Wegmann, R., On nonnormal matrices, LinearAlgebra Appl., 8 (1974), 109-120.



ON THOMPSON’S CONJECTUREFOR ALTERNATING GROUP A26

Shitian Liu

Yanhua Huang

School of ScienceSichuan University of Science and EngineeringZigong Sichuan, 643000Chinae-mails: [email protected]

[email protected]

Abstract. LetG be a group. Denote byN(G) the set of nonidentity orders of conjugacy

classes of elements in G. For groups A10, A16 and A22 which are uniquely determined

by N(G), theses degrees are p + 3 and p + 4 is a prime with p = 7, 13, 19. If p + 4 is

composite, then whether can the groups Ap+3 be characterized by N(G). In this paper,

we give an example for Ap+3 with p + 4 composite, namely, we proved that if G is a

group with trivial center and N(G) = N(A26), then G ∼= A26.

Keywords and phrases: element order, alternating group, thompson’s conjecture,

conjugacy classes, simple group.

AMS Subject Classification: 20D05, 20D06, 20D20.

1. Introduction

Let G be a finite group. Denote byN(G) the set of nonidentity orders of conjugacyclasses of elements in G. Related to N(G), J.G. Thompson gave the followingconjecture.

Thompson’s Conjecture. (see [11, Question 12.38]). If L is a finite simplenon-Abelian group, G is a finite group with trivial center, and N(G) = N(L),then G ∼= L.

Let π(G) denote the set of all prime divisors of |G|. Let GK(G) be a graphwith vertex set π(G) such that two primes p and q in π(G) are joined by an edgeif G has an element of order p · q. We set s(G) denote the number of connectedcomponents of the prime graph GK(G). A classification of all finite simple groupswith disconnected prime graph was obtained in [10], [14]. Based on these results,

526 s. liu, y. huang

Thompson’s conjecture was proved valid for all finite simple groups with s(G) ≥ 2(see [2], [3]). So whether there is a group with connected prime graph for whichThompson’s conjecture would be true? Recently, the groups A10, A16 and A22

were proved valid for this conjecture (see [13], [6] and [15], respectively). As thedevelopment of this topics, we will prove that Thompson’s conjecture is true forthe alternating group A26 of degree 26.

We will introduce some notations used to the proof of the main theorem. LetAn and Sn denote the alternating and symmetric groups of degree n, respectively.Let G be a group. Set Aut(G) denotes the automorphism group of G. Let ω(G)denote the set of element order of G. The notations are standard (see [5], forinstance).

2. Some lemmas

In this section we will give some preliminary results.

Lemma 2.1 [13, Lemma 1.2] and [1, Lemma 2.3] Let x, y ∈ G, (|x|, |y|) = 1,and xy = yx. Then

(1) CG(xy) = CG(x) ∩ CG(y);

(2) |xG| divides |(xy)G|;

(3) If |xG| = |(xy)G|, then CG(x) ≤ CG(y).

Lemma 2.2 [13, Lemma 3] If P and H are finite groups with trivial centers,and N(P ) = N(G), then π(P ) = π(H).

Lemma 2.3 [13, Lemma 4] Suppose that G is a finite group with trivial centerand p is a prime from π(G) such that p2 does not divide |xG| for all x in G. Thena Sylow p-subgroup of G is elementary abelian.

Lemma 2.4 [13, Lemma 5] Let K be a normal subgroup of G, and G = G/K.

(1) If x is the image of an element x of G in G. Then |xG| divides |xG|.

(2) If (|x|, |K|) = 1, then CG(x) = CG(x)K/K.

(3) If y ∈ K, then |yK | divides |yG|.

3. Main theorem and its proof

In this section, we give the main theorem and its proof.

Theorem 3.1 Let G be a finite group with trivial center and N(G) = N(A26).Then G is isomorphic to A26.

on thompson’s conjecture for alternating group A26 527

Proof. We divide the proof into the following lemmas.

Lemma 3.2 Let L = A26. Then the following hold.

(1) |L| = 222 · 310 · 56 · 73 · 112 · 132 · 17 · 19 · 23.

(2) numbers 26!23·3 and 26·25·24

3are maximal and minimal with respect to divisibility

in the set N(L), respectively.

(3) for any n ∈ N(L), numbers 172, 192 and 232 do not divide n.

(4) 23′-numbers in N(L) \ 1 are

24 · 52 · 13,222 · 39 · 56 · 73 · 112 · 132 · 17 · 19.19′-numbers in N(L) \ 1 are

24 · 52 · 13,27 · 32 · 53 · 11 · 13 · 23,24 · 3 · 53 · 7 · 11 · 13 · 23,22 · 32 · 52 · 7 · 11 · 13 · 23,24 · 52 · 7 · 11 · 13 · 23,25 · 3 · 5 · 11 · 13 · 23,2 · 3 · 52 · 13 · 23,220 · 310 · 56 · 73 · 112 · 132 · 17 · 23,222 · 38 · 56 · 73 · 112 · 132 · 17 · 23,219 · 39 · 56 · 73 · 112 · 132 · 17 · 19 · 23,220 · 39 · 56 · 73 · 112 · 132 · 17 · 23,222 · 310 · 56 · 72 · 112 · 132 · 17 · 23.17′-numbers in N(L) \ 1 are

24 · 52 · 13,27 · 32 · 53 · 11 · 13 · 23,24 · 3 · 53 · 7 · 11 · 13 · 23,22 · 32 · 52 · 7 · 11 · 13 · 23,24 · 52 · 7 · 11 · 13 · 23,25 · 3 · 5 · 11 · 13 · 23,2 · 3 · 52 · 13 · 23,28 · 32 · 53 · 7 · 11 · 13 · 19 · 23,27 · 53 · 7 · 11 · 13 · 19 · 23,25 · 33 · 53 · 7 · 11 · 13 · 19 · 23,


25 · 34 · 52 · 7 · 11 · 13 · 19 · 23,25 · 3 · 53 · 7 · 11 · 13 · 19 · 23,27 · 3 · 52 · 7 · 11 · 13 · 19 · 23,222 · 38 · 56 · 73 · 112 · 132 · 19 · 23,220 · 39 · 56 · 73 · 112 · 132 · 19 · 23,220 · 310 · 55 · 73 · 112 · 132 · 19 · 23,222 · 36 · 56 · 73 · 112 · 132 · 19 · 23,218 · 310 · 56 · 73 · 112 · 132 · 19 · 23,222 · 310 · 56 · 72 · 112 · 132 · 19 · 23,219 · 39 · 56 · 73 · 112 · 132 · 19 · 23,221 · 37 · 56 · 73 · 112 · 132 · 19 · 23,222 · 39 · 55 · 73 · 112 · 132 · 19 · 23,221 · 39 · 56 · 73 · 112 · 132 · 19 · 23,220 · 39 · 55 · 73 · 112 · 132 · 19 · 23,217 · 39 · 55 · 73 · 112 · 132 · 19 · 23,219 · 38 · 55 · 73 · 112 · 132 · 19 · 23,215 · 36 · 55 · 72 · 112 · 132 · 19 · 23.

(5) the numbers

222 · 310 · 56 · 73 · 112 · 17 · 19 · 23,222 · 38 · 56 · 73 · 112 · 132 · 19 · 23,222 · 310 · 56 · 72 · 112 · 132 · 17 · 23,222 · 39 · 56 · 73 · 112 · 132 · 17 · 19are maximal elements of N(L) by divisibility.

(6) for any n ∈ N(L)\1, n is divisible by 11 or 13.

Proof. The information is obtained via GAP [7] or [8].

Lemma 3.3 Let G be a finite group with trivial center and N(G) = N(L). Then|L| | |G| and π(G) = π(L) = 2, 3, 5, 7, 11, 13, 17, 19, 23.

Proof. Since |xG||CG(x)| = |G|, then every member form N(G) divides the orderof G and |L| | |G|. So by Lemmas 2.2 and 3.2, we have that π(G) = π(L).

Lemma 3.4 Suppose that G is a finite group with trivial center and N(G) = N(L).Let p ∈ 17, 19, 23. Then the Sylow p-subgroup S of G is of order p. There doesnot exist an element of order 17 · 19, 17 · 23 and 19 · 23.


Proof. By Lemma 2.3, S is elementary abelian. If |x| = p, then we have that|xG| is a p′-number.

Suppose that p = 23 and |S| ≥ 232. We can assume that there exists anelement y of G with that |yG| = 222 · 310 · 56 · 73 · 112 · 17 · 19 · 23 by Lemma 3.2.

Assume that 23 | |y|. Let |y| = 23t. Since S is elementary abelian, thenumbers 23 and t are coprime. Put u = y23, v = yt. Then y = uv, CG(uv) =CG(u)∩CG(v) by Lemma 2.1. Therefore |vG| | |yG| = 222 ·310 ·56 ·73 ·112 ·17·19·23.On the other hand, the element v is of order 23, and therefore, |vG| = 24 · 52 · 13,222 · 39 · 56 · 73 · 112 · 132 · 17 · 19. It follows that 13 | |vG|, a contradiction

Assume that 23 - |y|. Let x be an element of CG(y) of order 23. ThenCG(xy) = CG(x) ∩ CG(y), and therefore |xG| | |(xy)G| and |yG| | |(xy)G|. SinceS is abelian, S ≤ CG(x). It follows that 23 - |xG| and so, |xG| = 24 · 52 · 13,222 · 39 · 56 · 73 · 112 · 132 · 17 · 19, a contradiction.

For cases “p = 19 and p = 17”, we can get the same result as “p = 23”.There is no element of order 17 · 19, 17 · 23, or 19 · 23 in G.

Lemma 3.5 Suppose that G is a finite group with trivial center and N(G) = N(L).Assume that p ∈ 2, 5, 7, 11, 13, P is a Sylow p-subgroup of G, and Z(P ) is itscenter. Then for every element x ∈ Z(P )\1, the order of the centralizer CG(x)is coprime to 23 if p = 2, coprime to 23 if p = 5, coprime to 23 if p = 7, coprimeto 17 · 19 · 23 if p = 11, and coprime to 17 · 19 · 23 if p = 13.

Proof. By Lemma 3.4, |G|p = |L|p for p ∈ 17, 19, 23. For any 1 = x ∈ Z(P ),we get the desired results since N(G) = N(L).

Lemma 3.6 Let G be a finite group and p ∈ π(G). If p2 - |G|, then G has anormal series 1 ≤ K ≤ L ≤ G, such that L/K is a simple group and p ∈ π(L/K).

Proof. Since G is a finite group, G has a chief series. So let G0 ≤ G1 ≤ G2 · · · ≤Gr = G be a chief series of G. There exists some t, such that 1 ≤ t ≤ r andp ∈ π(Gt) \ π(Gt−1). Let L = Gt and K = Gt−1, then 1 ≤ K ≤ L ≤ G is anormal series of G and L/K is a chief factor of G. Therefore L/K is a minimalnormal subgroup of G/K. We know that the chief factors are characteristicallysimple. Also every characteristically simple group is a simple group or a productof isomorphic simple groups. So L/K is a simple group or a product of isomorphicsimple groups. Since p ∈ π(L/K) and p2 - |L/K|, it follows that L/K is a simplegroup.

Lemma 3.7 There is a normal series 1 ≤ K ≤ L ≤ G such that L/K ∼= A26.

Proof. By Lemma 3.6, we have that there is a normal series 1 ≤ K ≤ L ≤ Gsuch that M := L/K is isomorphic to a direct product of nonabelian simplegroups S1, S2, · · · , Sn which are listed in [16]. Since G has not a Hall 17, 19, 23-subgroup, then the numbers 17, 19 and 23 divide the order of exactly one of thesegroups. Without generality, we assume that the numbers 17, 19, 23 divide S1.Obviously, S1 G. Let G∗ = G/S1 and M∗ = M/S1. We prove that n = 1.


Assume that n ≥ 2. Then a Sylow 2-subgroup P2 of G∗ is nontrivial and itscenter Z(P2) has a nontrivial intersection with M∗. Let x be a nontrivial elementof S2 × · · · × Sn with that its image in G lies in Z(P2). Since y ∈ Z(S1), thenthere is an element of order 2 · 23. It follows that there exists an element y with|yG| = 219 ·310 ·56 ·73 ·112 ·132 ·17 ·19, which contradicts Lemma 3.2(4). So n = 1.

ThereforeL/K ≤ G ≤ Aut(L/K)

with L/K a nonabelian simple group.For p ∈ 17, 19, 23, we have that |G|p = p by Lemma 3.4 and by Lemma 3.6,

p ∈ π(L/K). It follows from [16], that L/K ∼= An where n = 23, 24, 25, 26, 27, 28.If L/K ∼= A23, then there exist an element x of G with that

xG = 218 · 39 · 54 · 73 · 112 · 13 · 17 · 19.

Let x be the preimage of x of G in G. Then since |xG| is maximal in N(G), this

forces |xG| = |xG|. It follows that 13 | |CG(x)|. Then there is an element of order13 · 23, and by the proof of Lemma 3.4, we get a contradiction.

Similarly, we can rule out these cases “L/K ∼= A24, and L/K ∼= A25”.If L/K ∼= A27, then there exist an element x of G with that

xG = 220 · 312 · 56 · 73 · 112 · 132 · 17 · 19

which contradicts Lemma 3.2(4).If L/K ∼= A28, then there exist an element x of G with that

xG = 222 · 313 · 55 · 75 · 112 · 132 · 17 · 19

which contradicts Lemma 3.2(4).Therefore, L/K ∼= A26.

Lemma 3.8 G ∼= A26.

Proof. By Lemma 3.7,

A26 ≤ G ≤ Aut(A26) ∼= S26.

If G ∼= S26, then there exist an element x of G with that

xG = 219 · 38 · 56 · 73 · 112 · 132 · 17 · 19

which contradicts Lemma 3.2(4).So G ∼= A26. Then we define the normal series 1 ≤ K ≤ G into the chief

ones. We prove that K = 1. If K = 1, then order consideration, we have thatG/K ∼= A26 and |K| = 2. It follows that Z(G) = K and there is an element oforder 2 · 23, a contradiction.

Therefore, K = 1 and G ∼= A26.This completes the proof of the Lemma and also of the main theorem.


4. Some applications and problem

Chen et al in [4] proved that the group A10 can be characterized by its order andtwo special conjugacy classes sizes. Then obviously, we also have the followingresult.

Corollary 4.1 Let G be a finite group with trivial center. Assume that N(G) =N(A26) and |G| = |A26|. Then G ∼= A26.

We know that the alternating group An with n = 10, 16, 22, 26, are characterizedby N(G). Then, by [2], [3], we have the following.

Corollary 4.2 Let G be a finite group with trivial center. Assume that N(G) =N(An) with n ≤ 26. Then G ∼= An.

Related to Thompson’s conjecture, we give the following problem.

Problem. Are the alternating groups Ap+3 of degree p + 3 where p is a prime,characterized by N(G)?

Shi gave the following conjecture.

Conjecture [12] Let G be a group and H a finite simple group. Then G ∼= H ifand only if (a) ω(G) = ω(H) and (b) |G| = |H|.

Then, we have the following corollary.

Corollary 4.3 Let G is a group and n an integer with n ≤ 26. Then G ∼= An ifand only if ω(G) = ω(An) and |G| = |An|.

Acknowledgments. The object was supported by the Department of Educationof Sichuan Province (Grant No: 12ZB291), by the Opening Project of SichuanProvince University Key Laborstory of Bridge Non-destruction Detecting andEngineering Computing (Grant No: 2013QYJ02) and by the Scientific ResearchProject of Sichuan University of Science and Engineering (Grant No: 2014RC02).The authors are very grateful for the helpful suggestions of the referee especiallyin Lemma 10 which give the proof more simple.

References

[1] Ahanjideh, N., Ahanjideh, M., On the Validity of Thompson’s Conjec-ture for Finite Simple Groups, Comm. Algebra, 41 (11) (2013), 4116–4145.

[2] Chen, G., On Thompson’s conjecture, J. Algebra, 185 (1) (1996), 184–193.

[3] Chen, G., Further reflections on Thompson’s conjecture, J. Algebra, 218 (1)(1999), 276–285.


[4] Chen, Y., Chen, G., Recognition of A10 and L4(4) by two special conjugacyclass sizes, Ital. J. Pure Appl. Math., 29 (2012), 387–394.

[5] Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wil-son, R.A., Atlas of finite groups, Oxford University Press, Eynsham, 1985.Maximal subgroups and ordinary characters for simple groups, With compu-tational assistance from J.G. Thackray.

[6] Gorshkov, I.B., Thompson’s conjecture for simple groups with a connectedprime graph, Alg. Log., 51 (2) (2012), 111–127.

[7] Group, T.G., GAP-Groups, Algorithms, and Programming, Vers. 4.4.12(2008), http://www.gap–system.org.

[8] James, G., Kerber, A., The representation theory of the symmetric group,vol. 16 of Encyclopedia of Mathematics and its Applications. Addison-WesleyPublishing Co., Reading, Mass., 1981. With a foreword by P.M. Cohn. Withan introduction by Gilbert de B. Robinson.

[9] Kleidman, P., Liebeck, M., The subgroup structure of the finite classi-cal groups, vol. 129 of London Mathematical Society Lecture Note Series.Cambridge University Press, Cambridge, 1990.

[10] Kondrat’ev, A.S., On prime graph components of finite simple groups,Mat. Sb., 180 (6) (1989), 787–797, 864.

[11] Mazurov, V.D., Khukhro, E.I. (Eds.), The Kourovka notebook, 7th ed.Russian Academy of Sciences Siberian Division, Institute of Mathematics,Novosibirsk, 2010. Unsolved problems in group theory, Including archive ofsolved problems.

[12] Shi, W.J., A new characterization of the sporadic simple groups, GroupTheory (Singapore, 1987), de Gruyter, Berlin, 1989, 531–540.

[13] Vasil’ev, A.V., On Thompson’s conjecture, Sib. Elektron. Mat. Izv., 6(2009), 457–464.

[14] Williams, J.S., Prime graph components of finite groups, J. Algebra, 69 (2)(1981), 487–513.

[15] Xu, M., Thompson’s conjecture for alternating group of degree 22, Front.Math. China, 8 (5) (2013), 1227–1236.

[16] Zavarnitsine, A.V., Finite simple groups with narrow prime spectrum, Sib.Elektron. Mat. Izv., 6 (2009), 1–12.



APPLICATION OF BIPOLAR FUZZY SOFT SETS IN K-ALGEBRAS

Muhammad Akram

Department of MathematicsUniversity of the PunjabNew Campus, LahorePakistane-mail: [email protected], [email protected]

Noura O. Alsherei

Department of MathematicsFaculty of Sciences (Girls)King Abdulaziz University, JeddahSaudi Arabiae-mail: [email protected]

K.P. Shum

Institute of MathematicsYunnan UniversityKunming, 650091Chinae-mail: [email protected]

Adeel Farooq

Department of MathematicsCOMSATS Institute of information technologyLahorePakistane-mail: [email protected]

Abstract. On the basis of the concept of bipolar fuzzy soft sets, a new kind of K-

algebra is introduced in this paper. The concepts of bipolar fuzzy soft K-algebras are

described and some related properties are investigated. The notion of a generalized

bipolar fuzzy soft K-algebra is also introduced and discussed.

Keywords: Soft K-algebras; bipolar fuzzy soft K-subalgebras; (∈,∈ ∨q)-bipolar fuzzysoft K-subalgebras.

2000 Mathematics Subject Classification: 20N15, 94D05.

1. Introduction

The notion of a K-algebra (G, ·,⊙, e) was first introduced by Dar and Akram[12] in 2003 and published in 2005. A K-algebra is an algebra built on a group(G, ·, e) by adjoining an induced binary operation ⊙ on G which is attached toan abstract K-algebra (G, ·,⊙, e). This system is, in general non-commutativeand non-associative with a right identity e, if a group is non-commutative. For

534 m. akram, n.o. alsherei, k.p. shum, a. farooq

a given group, the K-algebra is proper if group is not an elementary abelian 2-group. Thus, a K-algebra is abelian and non-abelian purely depends on the basegroup. In 2004, Dar and Akram further renamed a K-algebra on a group G asa K(G)-algebra [13] due to its structural basis G. The K(G)-algebras have beencharacterized by their left and right mappings in [13, 14] when group is abelianand non-abelian.

In 1994, Zhang [29] initiated the concept of bipolar fuzzy sets as a genera-lization of fuzzy sets [28]. Bipolar fuzzy sets are an extension of fuzzy sets [28]whose membership degree range is [−1, 1]. In a bipolar fuzzy set, the membershipdegree 0 of an element means that the element is irrelevant to the correspondingproperty, the membership degree (0, 1] of an element indicates that the elementsomewhat satisfies the property, and the membership degree [−1, 0) of an elementindicates that the element somewhat satisfies the implicit counter-property. Al-though bipolar fuzzy sets and intuitionistic fuzzy sets look similar to each other,they are essentially different sets. In many domains, it is important to be ableto deal with bipolar information. It is noted that positive information representswhat is granted to be possible, while negative information represents what is con-sidered to be impossible. This domain has recently motivated new research inseveral directions. In particular, fuzzy and possibilistic formalisms for bipolarinformation have been proposed [16], because when we deal with spatial informa-tion in image processing or in spatial reasoning applications, this bipolarity alsooccurs. For instance, when we assess the position of an object in a space, wemay have positive information expressed as a set of possible places and negativeinformation expressed as a set of impossible places.

In 1999, Molodtsov [23] initiated the novel concept of soft set theory to dealwith uncertainties which can not be handled by traditional mathematical tools.Applications of soft set theory in real life problems are now catching momentumdue to the general nature parametrization expressed by a soft set. Maji et al.[22] gave first practical application of soft sets in decision making problems. Theyalso presented the definition of fuzzy soft set [21]. Following the concept of softsets, several authors have applied this concept to algebraic structures [17-20, 30].Al-Shehri et al.[3] has applied this concept to K-algebras. Akram et al. intro-duced the notions of fuzzy soft K-subalgebras in [6]. Moreover, Akram et al. [7]has introduced the notions of intuitionist fuzzy soft K-algebras and studied someof their properties. Bipolar fuzzy sets and soft sets are two different methods forrepresenting uncertainty and vagueness. In this article, we apply these methods incombination to study uncertainty and vagueness inK-algebras. We first introducethe concept of bipolar fuzzy soft K-algebras and investigate some of their proper-ties. Then we introduce the notion of a generalized bipolar fuzzy soft K-algebraand studied some of its related properties. The definitions and terminologies thatwe used in this paper are standard.

2. Preliminaries

In this section, we review some elementary concepts that are necessary for thispaper.

application of bipolar fuzzy soft sets in K-algebras 535

Let (G, ·, e) be a group in which each non-identity element is not of order2. Then a K- algebra [12] is a structure K = (G, ·,⊙, e) on a group G in whichinduced binary operation ⊙ : G × G → G is defined by ⊙(a, b) = a ⊙ b = a.b−1

and satisfies the following axioms:

(K1) (a⊙ b)⊙ (a⊙ c) = (a⊙ ((e⊙ c)⊙ (e⊙ b)))⊙ a,

(K2) a⊙ (a⊙ b) = (a⊙ (e⊙ b))⊙ a,

(K3) (a⊙ a) = e,

(K4) (a⊙ e) = a,

(K5) (e⊙ a) = a−1,

for all a, b, c ∈ G. A K-algebra K is called abelian if and only if x ⊙ (e ⊙ b) =b⊙ (e⊙ a) for all a, b ∈ G. If a K-algebra K is abelian, then the axioms (1) and(2) can be written as:

(K1) (a⊙ b)⊙ (a⊙ c) = c⊙ b .

(K2) a⊙ (a⊙ b) = b.

In what follows, we denote a K-algebra by K unless otherwise specified.A nonempty subset H of a K-algebra K is called a subalgebra [12] of the

K-algebra K if a ⊙ b ∈ H, for all a, b ∈ H. Note that every subalgebra of aK-algebra K contains the identity e of the group (G, ·, e). Naturally, the mappingf : K1 → K2 of K-algebras is called a homomorphism [15] if f(a⊙b) = f(a)⊙f(b),for all a, b ∈ K1. We refer the reader to the book [10] for further informationregarding K-algebras.

Definition 2.1. [29] Let X be a nonempty set. A bipolar fuzzy set B in X is anobject having the form

B = (x, µ+(x), µ−(x)) |x ∈ X

where µ+ : X → [0, 1] and µ− : X → [−1, 0] are mappings.

We use the positive membership degree µ+(x) to denote the satisfaction de-gree of an element x to the property corresponding to a bipolar fuzzy set B, andthe negative membership degree µ−(x) to denote the satisfaction degree of an ele-ment x to some implicit counter-property corresponding to a bipolar fuzzy set B.If µ+(x) = 0 and µ−(x) = 0, it is the situation that x is regarded as having onlypositive satisfaction for B. If µ+(x) = 0 and µ−(x) = 0, it is the situation that xdoes not satisfy the property of B but somewhat satisfies the counter property ofB . It is possible for an element x to be such that µ+(x) = 0 and µ−(x) = 0 whenthe membership function of the property overlaps that of its counter property oversome portion of X.

For the sake of simplicity, we shall use the symbol B = (µ+, µ−) for thebipolar fuzzy set B = (x, µ+(x), µ−(x)) | x ∈ X.


Definition 2.2. [29] For every two bipolar fuzzy sets A = (µ+A, µ

−A) and B =

(µ+B, µ

−B) in X, we define

• (A∩B)(x) = (min(µ+

A(x), µ+B(x)),max(µ−

A(x), µ−B(x))),

• (A∪B)(x) = (max(µ+

A(x), µ+B(x)),min(µ−

A(x), µ−B(x))).

Definition 2.3. For x ∈ X, define Ax = (y, z) ∈ X × X|x = yz. For twobipolar fuzzy subsets A = (µ+

A, µ−A) and B = (µ+

B, µ−B) of X, the product of two

bipolar fuzzy subsets is denoted by A B and is defined as:

µ+A µ+

B(x) =

∨

(s,t)∈Ax

µ+A(s) ∧ µ

+B(t) if Ax = ∅,

0 if Ax = ∅

and

µ−A µ−

B(x) =

∧

(s,t)∈Ax

µ−A(s) ∨ µ

−B(t) if Ax = ∅,

0 if Ax = ∅.

Molodtsov [23] defined the notion of soft set in the following way: Let U bean initial universe and E be a set of parameters. Let P (U) denotes the power setof U and let A be non-empty subset of E.

Definition 2.4. [23] A pair (Φ, A) is called a soft set over U , where Φ is a mappinggiven by Φ : A→ P (U).

In other words, a soft set over U is a parameterized family of subsets of theuniverse U . For ε ∈ A, Φ(ε) may be considered as the set of ε− approximateelements of the soft set (Φ, A). Clearly, a soft set is not just a subset of U .The concept of bipolar fuzzy soft set, which was originally proposed by [27]. LetBF (U) denote the family of all bipolar fuzzy sets in U .

Definition 2.5. [27] Let U be an initial universe and A ⊆ E be a set of pa-rameters. A pair (ϕ,A) is called an bipolar fuzzy soft set over U , where ϕ is amapping given by ϕ : A → BF (U). A bipolar fuzzy soft set is a parameterizedfamily of bipolar fuzzy subsets of U . For any ε ∈ A, ϕε is referred to as the set ofa− approximate elements of the bipolar fuzzy soft set (ϕ,A), which is actually abipolar fuzzy set on U and can be

ϕε = (µ+ϕε(x), µ−

ϕε(x)) | x ∈ U

where µ+ϕε(x) denotes the degree of x keeping the parameter ε, µ−

ϕε(x) denotes the

degree of x keeping the non-parameter ε.

Definition 2.6. [27] Let (ϕ,A) and (ψ,B) be two bipolar fuzzy soft sets over U .We say that (ϕ,A) is a bipolar fuzzy soft subset of (ψ,B) and write(ϕ,A) b (ψ,B) if


(i) A ⊆ B,

(ii) For any ε ∈ A, ϕ(ε) ⊆ ψ(ε).

(ϕ,A) and (ψ,B) are said to be bipolar fuzzy soft equal and write (ϕ,A) = (ψ,B)if (ϕ,A) b (ψ,B) and (ψ,B) b (ϕ,A).

Definition 2.7. [27] Let (ϕ,A) and (ψ,B) be two bipolar fuzzy soft sets overU . Then their extended intersection is a bipolar fuzzy soft set denoted by (φ,C),where C = A ∪B and

φ(ε) =

ϕ(ε) if ε ∈ A−B,

ψ(ε) if ε ∈ B − A,

ϕ(ε) ∩ ψ(ε) if ε ∈ A ∩B,

for all ε ∈ C. This is denoted by (φ,C) = (ϕ,A)∩(ψ,B).

Definition 2.8. [27] If (ϕ,A) and (ψ,B) are two bipolar fuzzy soft sets over thesame universe U then “(ϕ,A)AND(ψ,B)” is a bipolar fuzzy soft set denoted by(ϕ,A) ∧ (ψ,B), and is defined by (ϕ,A) ∧ (ψ,B) = (φ,A × B) where, φ(a, b) =φ(a) ∩ ψ(b) for all (a,b) ∈ A × B. Here ∩ is the operation of a bipolar fuzzyintersection.

Definition 2.9. [27] Let (ϕ,A) and (ψ,B) be two bipolar fuzzy soft sets over U .Then their extended union denoted by (φ,C), where C = A ∪B and

φ(ε) =


ψ(ε) if ε ∈ B − A,

ϕ(ε) ∪ ψ(ε) if ε ∈ A ∩B,

for all ε ∈ C. This is denoted by (φ,C) = (ϕ,A)∪(ψ,B).

Definition 2.10. [24] Let (ϕ,A) and (ψ,B) be two fuzzy soft sets over a commonuniverse U with A∩B = ∅, then their restricted intersection is a bipolar fuzzy softset (φ,A∩B) denoted by (ϕ,A)e (ψ,B) = (φ,A∩B) where, φ(ε) = ϕ(ε)∩ ψ(ε)for all ε ∈ A ∩B.

Definition 2.11. [24] Let (ϕ,A) and (ψ,B) be two bipolar fuzzy soft sets overa common universe U with A ∩ B = ∅, then their restricted union is denoted by(ϕ,A) d (ψ,B) and is defined as (ϕ,A) d (ψ,B) = (φ,C) where C = A ∩ B andfor all ε ∈ C, φ(ε) = ϕ(ε) ∪ ψ(ε).

Definition 2.12. [24] The extended product of two bipolar fuzzy soft sets (ϕ,A)and (ψ,B) over U is a fuzzy soft set, denoted by (ϕψ,C), where C = A∪B and

(ϕ ψ)(ε) =


ψ(ε) if ε ∈ B − A,

ϕ(ε) ψ(ε) if ε ∈ A ∩B,

for all ε ∈ C. This is denoted by (ϕ ψ,C) = (ϕ,A)(ψ,B).


Definition 2.13. [24] If A ∩ B = ∅, then the restricted product (φ,C) of twobipolar fuzzy soft sets (ϕ,A) and (ψ,B) over U is defined as the bipolar fuzzysoft set, (φ,A ∩B) denoted by (ϕ,A) oR (ψ,B) where φ (ε) = ϕ (ε) ψ (ε), for allε ∈ A ∩B. Here ϕ (ε) ψ (ε) is the product of two bipolar fuzzy subsets of U .

3. Bipolar fuzzy soft K-algebras

Definition 3.1. Let (ϕ,A) be a bipolar fuzzy soft set over K. Then (ϕ,A) issaid to be a bipolar fuzzy soft K-subalgebra over K if ϕ(x) is a bipolar fuzzy K-subalgebra of K for all x ∈ A, that is, a bipolar fuzzy soft set (ϕ,A) over K is calleda bipolar fuzzy soft K-subalgebra of K if the following conditions are satisfied:

(1) µ+ϕε(x⊙ y) > minµ+

ϕε(x), µ+

ϕε(y),

(2) µ−ϕε(x⊙ y) 6 maxµ−

ϕε(x), µ−

ϕε(y)

for all x, y ∈ G.

Definition 3.2. Let (ϕ,A) and (ψ,B) be bipolar fuzzy soft K-subalgebras overK. Then (ϕ,A) is a bipolar fuzzy subalgebra of (ψ,B) if (i) A ⊂ B and (ii) ϕ(x)is a bipolar fuzzy subalgebra of ψ(x) for all x ∈ A.

Example 3.3. Consider the K-algebra K = (G, ·,⊙, e), where G = e, a, a2, a3is the cyclic group of order 4 and ⊙ is given by the following Cayley’s table:

⊙ e a a2 a3

e e a3 a2 aa a e a3 a2

a2 a2 a e a3

a3 a3 a2 a e

Let A = e1, e2, e3 and ϕ : A→ P(G) be a set-valued function defined by

ϕ(e1) =

e

(0.7,−0.2),

a

(0.3,−0.4),

a2

(0.6,−0.2),

a3

(0.3,−0.6)

,

ϕ(e2) =

e

(0.5,−0.4),

a

(0.3,−0.3),

a2

(0.4,−0.3),

a3

(0.4,−0.5)

,

ϕ(e3) =

e

(0.5,−0.4),

a

(0.5,−0.2),

a2

(0.5,−0.4),

a3

(0.2,−0.5)

.

Let B = e2, e3 and ψ : B → P(G) be a set-valued function defined by

ψ(e2) =

e

(0.2,−0.2),

a

(0.2,−0.3),

a2

(0.1,−0.6),

a3

(0.3,−0.0)

,

ψ(e3) =

e

(0.2,−0.2),

a

(0.3,−0.3),

a2

(0.3,−0.3),

a3

(0.4,−0.4)

.


Clearly, (ϕ,A) and (ψ,B) are bipolar fuzzy soft sets over K. By routine calcu-lations, it is easy to see that ϕ(x) and ψ(x) are bipolar fuzzy K-subalgebras forx ∈ A and x ∈ B, respectively. Hence (ϕ,A) and (ψ,B) are bipolar fuzzy softK-subalgebras.

We state the following propositions without their proofs.

Proposition 3.4. Let (ϕ,A) and (ψ,B) be bipolar fuzzy soft K-subalgebras overK, then (ϕ,A)∩(ψ,B) and (ϕ,A) ∧ (ψ,B) are bipolar fuzzy soft K-subalgebrasover K.

Proposition 3.5. Let (ϕ,A) and (ψ,B) be bipolar fuzzy soft K-subalgebras overK. If A ∩B = ∅ then (ϕ,A)∪(ψ,B) is a bipolar fuzzy soft K-subalgebra of K.

Proposition 3.6. Let (ϕ,A) and (ψ,B) be bipolar fuzzy soft K-subalgebras overK. If ϕ(x) ⊆ ψ(x) for all x ∈ A, (ϕ,A) is a bipolar fuzzy soft subalgebra of (ψ,B).

Theorem 3.7. Let (ϕ,A) be a bipolar fuzzy soft K-subalgebra over K and let(φi, Bi) | i ∈ I be a nonempty family of bipolar fuzzy soft K-subalgebras of(ϕ,A). Then

(a)∩i∈I

(φi, Bi) is a bipolar fuzzy soft K-subalgebra of (ϕ,A),

(b)∧i∈I

(φi, Bi) is a bipolar fuzzy soft K-subalgebra of∧i∈I

(ϕ,A),

(c) If Bi ∩ Bj = ∅ for all i, j ∈ I, then∨i∈I

(φi, Bi) is a bipolar fuzzy soft

K-subalgebra of∨i∈I

ϕ,A).

Definition 3.8. Let (ϕ,A) be a bipolar fuzzy soft set over U . For each s ∈ [0, 1],the set (ϕ,A)s = (ϕs, A) is called an s-level soft set of (ϕ,A), where ϕsε = x ∈ U |µ+ϕε(x) ≥ s, µ−

ϕε(x) ≤ −s for all ε ∈ A. Clearly, (ϕ,A)s is a soft set over U .

Example 3.9. Consider the K-algebra K = (G, ·,⊙, e), where G = e, a, a2, a3is the cyclic group of order 4 and ⊙ is given by Cayley’s table in Example 3.3.0.2-level soft set of (ϕ,A) = e1, e2, e3.

Theorem 3.10. Let (ϕ,A) be a bipolar fuzzy soft set over K. (ϕ,A) is a bipolarfuzzy soft K-subalgebra if and only if (ϕ,A)s is a soft K-subalgebra over K foreach s ∈ [0, 1].

Proof. Suppose that (ϕ,A) is a bipolar fuzzy soft K-subalgebra. For eachs ∈ [0, 1], ε ∈ A and x1, x2 ∈ ϕsε, then µ

+ϕε(x1) ≥ s, µ+

ϕε(x2)≥ s and µ−

ϕε(x1) ≤ −s,

µ−ϕε(x2)

≤ −s. From definition 3.1, it follows that ϕsε is a bipolar fuzzy K-

subalgebra over K. Thus µ+ϕε(x1 ⊙ x2) ≥ min(µ+

ϕε(x1), µ

+ϕε(x2)), µ

+ϕε(x1 ⊙ x2) ≥ s,

µ−ϕε(x1 ⊙ x2) ≤ max(µ−

ϕε(x1), µ

−ϕε(x2)), µ

−ϕε(x1 ⊙ x2) ≤ −s. This implies that


x1 ⊙ x2 ∈ ϕsε, i.e., ϕsε is a K-subalgebra over K. According to Definition 3.8,

(ϕ,A)s is a soft K-subalgebra over K for each s ∈ [0, 1].

Conversely, assume that (ϕ,A)s is a soft K-subalgebra over K for eachs ∈ [0, 1]. For each ε ∈ A and x1, x2 ∈ G, let s = minµ+

ϕε(x1), µ

+ϕε(x2) and

let −s = maxµ−ϕε(x1), µ

−ϕε(x2), then x1, x2 ∈ ϕsε. Since ϕsε is a K-subalgebra

over K, then x1 ⊙ x2 ∈ ϕsε. This means that µ+ϕε(x1 ⊙ x2) ≥ min(µ+

ϕε(x1), µ

+ϕε(x2)),

µ−ϕε(x1⊙x2) ≤ max(µ−

ϕε(x1), µ

−ϕε(x2)), i.e., ϕ

sε is a bipolar fuzzy K-subalgebra over

K. According to Definition 3.1, (ϕ,A) is a bipolar fuzzy soft K-subalgebra overK. This completes the proof.

Proposition 3.11. Let K be a K-algebra built on Abelian group. Let (ϕ,A),(ψ,B) and (φ,C) be bipolar fuzzy soft K-subalgebras over K where A, B and Care subsets of E. Then

(1) (ϕ,A)o((ψ,B)∪(φ,C)) = ((ϕ,A)o(ψ,B))∪((ϕ,A)o(φ,C)),

(2) (ϕ,A)o((ψ,B) d (φ,C)) = ((ϕ,A)o(ψ,B)) d ((ϕ,A)o(φ,C)),

(3) ((ϕ,A)∪(ψ,B))o(φ,C) = ((ϕ,A)o(φ,C))∪((ψ,B)o(φ,C)),

(4) ((ϕ,A) d (ψ,B))o(φ,C) = ((ϕ,A)o(φ,C)) d ((ψ,B)o(φ,C)).

Proof. The proofs follow from the definitions of operations.

Proposition 3.12. Let K be a K-algebra built on Abelian group. Let (ϕ,A),(ψ,B) and (φ,C) be bipolar fuzzy soft K-subalgebras over K where A, B and Care subsets of E. Then

(5) (ϕ,A)oR((ψ,B) d (φ,C))⊂((ϕ,A)oR(ψ,B)) d ((ϕ,A)oR(φ,C)),

(6) (ϕ,A)oR((ψ,B)∪(φ,C))⊂((ϕ,A)oR(ψ,B))∪((ϕ,A)oR(φ,C)),

(7) ((ϕ,A)∪(ψ,B))oR(φ,C)⊂((ϕ,A)oR(φ,C))∪((ψ,B)oR(φ,C)),

(8) ((ϕ,A) d (ψ,B))oR(φ,C)⊂((ϕ,A)oR(φ,C)) d ((ψ,B)or(φ,C)).

Proof. The proofs follow from the definitions of operations.

Definition 3.13. Let f : X → Y and g : A → B be two functions, A and B areparametric sets from the crisp sets X and Y , respectively. Then the pair (f, g) iscalled a bipolar fuzzy soft function from X to Y .

Definition 3.14. Let (ϕ,A) and (ψ,B) be two bipolar fuzzy soft sets over G1

and G2, respectively and let (f, g) be a bipolar fuzzy soft function from G1 to G2.


(1) The image of (ϕ,A) under the bipolar fuzzy soft function (f, g), denotedby (f, g)(ϕ,A), is the bipolar fuzzy soft set on K2 defined by (f, g)(ϕ,A) =(f(ϕ), g(A)), where for all k ∈ g(A), y ∈ G2

µ+f(ϕ)k

(y) =

∨

f(x)=y

∨g(a)=k

ϕa(x) if x ∈ g−1(y),

1 otherwise,

µ−f(ϕ)k

(y) =

∧

f(x)=y

∧g(a)=k

ϕa(x) if x ∈ g−1(y),

−1 otherwise.

(2) The preimage of (ψ,B) under the bipolar fuzzy soft function (f, g), de-noted by (f, g)−1(ψ,B), is the bipolar fuzzy soft set over K1 defined by(f, g)−1(ψ,B) = (f−1(ψ), g−1(B)), where for all a ∈ g−1(A), for all x ∈ G1,

µ+f−1(ψ)a

(x) = µ+ψg(a)

(f(x)),

µ−f−1(ψ)a

(x) = µ−ψg(a)

(f(x)).

Definition 3.15. Let (f, g) be a bipolar fuzzy soft function from K1 to K2. Iff is a homomorphism from K1 to K2 then (f, g) is said to be bipolar fuzzy softhomomorphism, If f is a isomorphism from K1 to K2 and g is one-to-one mappingfrom A onto B then (f, g) is said to be bipolar fuzzy soft isomorphism.

Theorem 3.16. Let (ψ,B) be a bipolar fuzzy soft K-subalgebra over K2 and let(f, g) be a bipolar fuzzy soft homomorphism from K1 to K2. Then (f, g)−1(ψ,B)is a bipolar fuzzy soft K-subalgebra over K1.

Proof. Let x1, x2 ∈ G1, then

f−1(µ+ψε)(x1 ⊙ x2) = µ+

ψg(ε)(f(x1 ⊙ x2)) = µ+

ψg(ε)(f(x1)⊙ f(x2))

>minµ+ψg(ε)

(f(x1)), µ+ψg(ε)

(f(x2))

= minf−1(µ+gε)(x1), f

−1(µ+gε)(x2),

f−1(µ−ψε)(x1 ⊙ x2) = µ−

ψg(ε)(f(x1 ⊙ x2)) = µ−

ψg(ε)(f(x1)⊙ f(x2))

6maxµ−ψg(ε)

(f(x1)), µ−ψg(ε)

(f(x2))

= maxf−1(µ−gε)(x1), f

−1(µ−gε)(x2).

Hence (f, g)−1(ψ,B) is a bipolar fuzzy soft K-subalgebra over K1.

Remark. Let (ϕ,A) be a bipolar fuzzy soft K-subalgebra over K1 and let (f, g)be a bipolar fuzzy soft homomorphism from K1 to K2. Then (f, g)(ϕ,A) may notbe a bipolar fuzzy soft K-subalgebra over K2.


4. Generalized bipolar fuzzy soft K− algebras

Definition 4.1. Let c be a point in a non-empty setG. If γ ∈ (0, 1] and δ ∈ [−1, 0)are two real numbers, then the bipolar fuzzy c(γ, δ) =< x, cγ, cδ > is called abipolar fuzzy point in G, where γ(resp, δ) is the positive degree of membership(resp, negative degree of membership) of c(γ, δ) and c ∈ G is the support ofc(γ, δ). Let c(γ, δ) be a bipolar fuzzy in G and let A =< x, µ+

A, µ−A > be a

bipolar fuzzy in G. Then c(γ, δ) is said to belong to A, written c(γ, δ) ∈ A ifµ+A(c) ≥ γ and µ−

A(c) ≤ δ. We say that c(γ, δ) is quasicoincident with A, writtenc(γ, δ)qA, if µ+

A(c) + γ > 1 and µ−A(c) + δ < −1. To say that c(γ, δ) ∈ ∨qA

(resp, c(γ, δ) ∈ ∧qA) means that c(γ, δ) ∈ A or c(γ, δ)qA (resp, c(γ, δ) ∈ A andc(γ, δ)qA) and c(γ, δ)∈ ∨qA means that c(γ, δ) ∈ ∨qA does not hold.

Definition 4.2. A bipolar fuzzy set A = (µ+A, µ

−A) in K is called an (∈,∈ ∨q)-

bipolar fuzzy K− algebra of K if it satisfies the following conditions:

(a) x(s, t) ∈ A⇒ e(s, t) ∈ ∨qA,

(b) x(s1, t1) ∈ A, y(s2, t2) ∈ A⇒ (x⊙ y)(min(s1, s2),max(t1, t2)) ∈ ∨qA,

for all x, y ∈ G, s, s1, s2 ∈ (0, 1], t, t1, t2 ∈ [−1, 0).

Example 4.3. Consider the K-algebra K = (G, ·,⊙, e), where G = e, a, a2, a3is the cyclic group of order 4 and ⊙ is given by Cayley’s table in Example 3.3.We define a bipolar fuzzy set A : G→ [0, 1]× [−1, 0] by

µ+A(x) :=

1 if x = e,

0.5 otherwise,

µ−A(x) :=

0 if x = e,

−0.3 otherwise.

Take s = 0.4 ∈ (0, 1] and t = −0.5 ∈ [−1, 0). By routine computations, it is easyto see that A is not an (∈,∈ ∨q- bipolar fuzzy K-subalgebra of K.

Theorem 4.4. Let A be a bipolar fuzzy set in a K-algebra K. Then A is an(∈,∈ ∨q)− bipolar fuzzy K-subalgebra of K if and only if µ+

A(x ⊙ y) >min(µ+

A(x), µ+A(y), 0.5), µ

−A(x⊙y) 6 max(µ−

A(x), µ−A(y),−0.5) hold for all x, y ∈ G.

Proof. The proof is straightforward and thus omitted.

Theorem 4.5. Let A be a bipolar fuzzy set of K-algebra of K. Then A is an(∈,∈ ∨q)- bipolar fuzzy K-subalgebra of K if and only if each nonempty A(s,t),s ∈ (0.5, 1], t ∈ [−1,−0.5) is a K-subalgebra of K.

Proof. Assume that A is an (∈,∈ ∨q)− bipolar fuzzy K-subalgebra of K andlet s ∈ (0.5, 1], t ∈ [−1, 0.5). If x, y ∈ A(s,t), then µ+

A(x) ≥ s and µ+A(y) ≥ s,

µ−A(x) ≤ t and µ−

A(y) ≤ t. Thus,

µ+A(x⊙ y) > min(µ+

A(x), µ+A(y), 0.5) > min(s, 0.5) = s,

µ−A(x⊙ y) 6 max(µ−

A(x), µ−A(y),−0.5) 6 max(t,−0.5) = t,


and so x ⊙ y ∈ A(s,t). This shows that A(s,t) are K- subalgebras of K. The proofof converse part is obvious. This ends the proof.

Theorem 4.6. Let A be a bipolar fuzzy set in a K-algebra K. Then A(s,t) isa K-subalgebra of K if and only if max(µ+

A(x ⊙ y), 0.5) > min(µ+A(x), µ

+A(y)),

min(µ−A(x⊙ y),−0.5) 6 max(µ−

A(x), µ−A(y)) for all x, y ∈ G.


Definition 4.7. Let (ϕ,A) be a bipolar fuzzy soft set over a K-algebra K. Then(ϕ,A) is called an (∈,∈∨q)-bipolar fuzzy softK-subalgebra if ϕ(α) is an (∈,∈∨q)-bipolar fuzzy K-subalgebra of K for all α ∈ A.

Theorem 4.8. Let (ϕ,A) and (ψ,B) be two (∈,∈ ∨q)-bipolar fuzzy soft K-subalgebras over a K-algebra K. Then (ϕ,A) ∧ (ψ,B) is an (∈,∈ ∨q)-bipolarfuzzy soft K-subalgebra over K.

Proof. Since (ϕ,A)∧(ψ,B) = (φ,C), where C = A×B and φ(α, β) = ϕ(α)∩ψ(β)for all (α, β) ∈ C. Now for any (α, β) ∈ C, since (ϕ,A) and (ψ,B) are (∈,∈∨q)-bipolar fuzzy soft K-subalgebras over K, we have both ϕ(α) and ψ(β) are (∈,∈∨q)-bipolar fuzzyK-subalgebras of K. Thus φ(α, β) = ϕ(α)∩ψ(β) is an (∈,∈∨q)-bipolar fuzzy K-subalgebra of K. Therefore, (ϕ,A)∧(ψ,B) is an (∈,∈∨q)-bipolarfuzzy soft K-subalgebra over K.

Theorem 4.9. Let (ϕ,A) and (ψ,B) be two (∈,∈ ∨q)-bipolar fuzzy soft K-subalgebras over a K-algebra K. If A∩B = ∅, then (ϕ,A)⊓(ψ,B) is an (∈,∈∨q)-bipolar fuzzy soft K-subalgebra over K.

Proof. Since (ϕ,A)∩(ψ,B) = (φ,C), where C = A ∩B and φ(α) = ϕ(α) ∩ ψ(α)for all α ∈ C. Now, for any α ∈ C, since (ϕ,A) and (ψ,B) are (∈,∈∨q)-bipolarfuzzy softK-subalgebras overK, we have both ϕ(α) and ψ(α) are (∈,∈∨q)-bipolarfuzzy K-subalgebras of K. Thus φ(α) = ϕ(α)∩ψ(α) is an (∈,∈∨q)-bipolar fuzzyK-subalgebra of K. Therefore, (ϕ,A)∩(ψ,B) is an (∈,∈ ∨q)-bipolar fuzzy softK-subalgebra over K.

As a generalization of Theorem 3.7, we have the following result.

Theorem 4.10. Let (ϕ,A) be an (∈,∈ ∨q)-bipolar fuzzy soft K-subalgebra overK and let (φi, Bi) | i ∈ I be a nonempty family of (∈,∈∨q)-bipolar fuzzy softK-subalgebras of (ϕ,A). Then

(a)∩i∈I

(φi, Bi) is an (∈,∈∨q)-bipolar fuzzy soft K-subalgebra of (ϕ,A),

(b)∧i∈I

(φi, Bi) is an (∈,∈∨q)-bipolar fuzzy soft K-subalgebra of∧i∈I

(ϕ,A),

(c) If Bi ∩ Bj = ∅ for all i, j ∈ I, then∨i∈I

(φi, Bi) is an (∈,∈∨q)-bipolar fuzzy

soft K-subalgebra of∨i∈I

ϕ,A).



Theorem 4.11. Let (ϕ,A) and (ψ,B) be two (∈,∈ ∨q)-bipolar fuzzy soft K-subalgebras over a K-algebra K. Then (ϕ,A)∩(ψ,B) is an (∈,∈∨q)-bipolar fuzzysoft K-subalgebra over K.

Proof. Since (ϕ,A)∩(ψ,B) = (φ,C), where C = A ∪B and

φ(α) =

ϕ(α) if α ∈ A−B,

ψ(α) if α ∈ B − C,

ϕ(α) ∩ ψ(α) if α ∈ A ∩B.

for all α ∈ C.Now, for any α ∈ C, we consider the following cases.

Case 1. α ∈ A−B. Then φ(α) = ϕ(α) is an (∈,∈∨q)-bipolar fuzzyK-subalgebraof K since (ϕ,A) is an (∈,∈∨q)-bipolar fuzzy soft K-subalgebra over K.

Case 2. α ∈ B−A. Then φ(α) = ψ(α) is an (∈,∈∨q)-bipolar fuzzyK-subalgebraof K since (ψ,B) is an (∈,∈∨q)-bipolar fuzzy soft K-subalgebra over K.

Case 3. α ∈ A ∩ B. Then φ(α) = ϕ(α) ∩ ψ(α) is an (∈,∈ ∨q)-bipolar fuzzyK-subalgebra of K by the assumption. Thus, in any case, φ(α) is an (∈,∈∨q)-bipolar fuzzy K-subalgebra of K. Therefore, (ϕ,A)∩(ψ,B) is an (∈,∈∨q)-bipolarfuzzy soft K-subalgebra over K.

Theorem 4.12. Let (ϕ,A) and (ψ,B) be two (∈,∈ ∨q)-bipolar fuzzy softK-subalgebras over a K-algebra K. If A and B are disjoint, then (ϕ,A)∪(ψ,B)is an (∈,∈∨q)-bipolar fuzzy soft K-subalgebra over K.


5. Conclusions

Molodtsov introduced the concept of soft set theory as a general mathematical toolfor dealing with uncertainty and vagueness, and many researchers have createdsome models to solve problems in decision making and medical diagnosis. Insoft computing and uncertain modeling, soft sets can be combined with othermathematical tools. Bipolar fuzzy sets and soft sets are two different mathematicaltools for representing vagueness and uncertainty. In this paper we have appliedthese mathematical tools in combination to study vagueness and uncertainty inK- algebras. We have introduced the notion of an (∈,∈ ∨q)− bipolar fuzzy softK-subalgebra. The natural extension of this research work is connected with thestudy of: (i) fuzzy soft rough K-algebras, and (ii) bipolar fuzzy rough K-algebras.


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SEMIGROUP DISTANCES OF FINITE GROUPOIDS

Barbora Batıkova

Sarka Dvorakova

Milan Trch

Department of MathematicsTechnical FacultyCzech University of Life SciencesKamycka 129, 165 21 Praha 6 – SuchdolCzech Republice-mails: [email protected]

[email protected]@tf.czu.cz

Abstract. The simplest cases of non-associative groupoids are presented by groupoids

(so called SH-groupoids) having just one non-associative (ordered) triple of elements.

In this paper, only SH-groupoids having the simplest possible non-associative triple

(a, a, a) are investigated. For each positive integer n finite SH-groupoids En(·) generatedby at most two elements are constructed and their semigroup distances are described.

It is proved that there are finite non-associative groupoids having their semigroup dis-

tance equal just to the arbitrary given positive integer n.

Keywords: groupoid, non-associative triple, distance of groupoids, semigroup distance.

1991 Mathematics Subject Classification: 20N05.

Introduction

Groupoids having only one non-associative (ordered) triple were studied at firstby G. Szasz in [10] and [11], and by P. Hajek in [2] and [3]. Therefore theyare called Szasz-Hajek groupoids and shortly denoted as SH-groupoids. Sets ofnon-associative triples and semigroup distances of some finite groupoids were alsoinvestigated in [1], [4] and [5]. The structure of SH-groupoids was studied later byT. Kepka and M. Trch, and the main properties of SH-groupoids were describedin [6], [7], [8] and [9] for each of possible non-associative triples. Further, it wasproved in [6] that the semigroup distance of a minimal SG-groupoid having theonly non-associative triple (a, a, a) is equal to the number 1 or to the number2. In [13], a minimal SH-groupoid with the semigroup distance equal to 3 isconstructed. This paper is an immediate continuation of [12], [14] and [15].

548 b. batıkova, s. dvorakova, m. trch

1. Preliminaries

A groupoid G(·) is called σ-stratified groupoid if there exists a mapping σ of theset G to the set of positive integers such that σ(x · y) = σ(x) + σ(y) for eachcouple x, y ∈ G. The mapping σ is called a mapping stratifying the underlyingset G.

For each positive integer n, the corresponding subset LnG = x ∈ G|σ(x) = nis called n-th layer of the underlying set G. Of course, we have

G = L1(G) ∪ L2(G) ∪ · · · ∪ Ln(G) ∪ Ln+1(G) . . . .

Furthermore, if G(·) is a σ-stratified groupoid then each subgroupoid H(·) of thegroupoid G(·) is also a σ-stratified groupoid.

If G(·) is a σ-stratified groupoid and κ is a congruence on G(·) then thecongruence κ is called σ-stratified congruence if σ(x) = σ(y) for each (x, y) ∈ κ.In this case the corresponding groupoid G/κ(·) is a σ-stratified groupoid, too.

Let G(·) be a σ-stratified groupoid. Further, let k be an arbitrary givenpositive integer and, finally, let t ∈ G be an arbitrary chosen element havingσ(t) ≥ k. Then there exists a groupoid Gk() such that x y = x · y for everyx, y ∈ G having σ(x) + σ(y) < k and x y = t whenever σ(x) + σ(y) ≥ k. Thecorresponding groupoid Gk() is called restriction of the k-th order of the groupoidG(·). If the groupoid G(·) is finitely generated then the corresponding groupoidGk() is a finite groupoid which can be seen as a σ-stratified groupoid up to thek-layer.

If G(·) is a non-associative σ-stratified groupoid, then the set M containingpositive integers σ(x) + σ(y) + σ(z) for every x, y, z ∈ G such that x · yz = xy · zis non-empty. This set contains the least element m having the same propertyand the corresponding restriction of the (m + 1)-th order is a non-associativegroupoid. The corresponding groupoid will be called fundamental restriction ofthe non-associative σ-stratified groupoid and it will be denoted as G(·).

The simplest type of non-associative groupoids are groupoids containing justone non-associative ordered triple of elements. A groupoid G(·) is called an SH-groupoid of the type (a, a, a) if there exists an element a ∈ G such that the orderedtriple (a, a, a) is the only non-associative triple of the groupoid G(·). An SH-groupoid G(·) of the type (a, a, a) is called minimal SH-groupoid if the groupoidG(·) is generated by the one-element set a.

Let H(·) be a subgroupoid of an SH-groupoid G(·) having the non-associativetriple (a, a, a). Then either (a, a, a) ∈ H3 and H(·) is an SH-groupoid having thenon-associative triple (a, a, a), or H(·) is a semigroup in the opposite case.

Let κ be a congruence on SH-groupoid G(·). If (a, a, a) is the correspondingnon-associative triple then either (a · aa, aa · a) ∈ κ and then the correspondinggroupoid G(·)/κ is a semigroup, or (a · aa, aa · a) /∈ κ and then the correspondinggroupoid G(·)/κ is an SH-groupoid of the same type (a, a, a).

The following main theorem concerning SH-groupoids was proved by G. Szasz:

1.1. Theorem. Let G(·) be an SH-groupoid. If (a, b, c) is the only non-associativetriple of G(·) and let x, y ∈ G be such that x · y ∈ a, b, c. Then x · y ∈ x, y.

semigroup distances of finite groupoids 549

Let G(⋄) and G(∗) be a couple of groupoids having the same underlying setG. Then dist(G(⋄), G(∗)) denotes card(x, y) ∈ G2 |x ⋄ y = x ∗ y.

Let G(·) be a groupoid. Let sdist(G(·)) be the minimum of cardinal numbersdist(G(·), G(∗)), where G(∗) runs through the set of all semigroups having theunderlying set G. The number sdist(G(·)) is called semigroup distance of thegroupoid G(·).

Let G(·) be a groupoid and let G(⋆) be a semigroup having the same un-derlying set G. If sdist(G(·)) = dist(G(·), G(⋆)) then the semigroup G(⋆) will befurther denoted as nearest semigroup of the groupoid G(·).

2. Stratified minimal SH-groupoids of the type (a, a, a)

In this part, let F (·) denote the absolutely free groupoid generated by a one-element set a. Let λ(u) denote the length of the term u for each element u ∈ F .It is obvious that λ(x · y) = λ(x) + λ(y) for every x, y ∈ F . Thus, the groupoidF (·) is a λ-stratified groupoid.

If G(·) is an arbitrary minimal SH-groupoid of the type (a, a, a) then thereexists a congruence κ on F (·) such that G(·) is an isomorphic image of thegroupoid F/κ(·). Of course, the corresponding congruence κ satisfies the followingtwo conditions: (a · aa, aa · a) /∈ κ and (x · yz, xy · z) ∈ κ for every x, y, z ∈ G,(x, y, z) = (a, a, a).

Suppose at first that there exists at least one stratified SH-groupoud G(·) ofthe type (a, a, a). If σ is the corresponding stratifying mapping of the underlyingset G then the inequality σ(a · a) = 2 · σ(a) > σ(a) has to be valid.

2.1. Construction. Suppose that the elements a, b, c, d, f, g and, further,a4, a5, ..., an, an+1, ... are pairwise different.

(i) Put V = a, b, c, d, a4, a5, . . . , ak, ak+1, . . . . Define a mapping λ of the set Vto the set of positive integers such that λ(a) = 1, λ(b) = 2, λ(c) = 3 = λ(d)and let λ(ak) = k for each positive integer k ≥ 4.

Define, further, a binary operation x · y on the set V such that a · a = b,a · b = c, b · a = d and put x · y = ak whenever λ(x) + λ(y) = k ≥ 4. ThenV (·) becomes a groupoid and it is easy to check that V (·) is a λ-stratifiedgroupoid.

(ii) Put W = a, b, c, d, f, g, a5, a6, . . . , am, am+1, . . . and define a mapping λof the set W to the set of positive integers such that λ(a) = 1, λ(b) = 2,λ(c) = 3 = λ(d), λ(f) = 4 = λ(g) and let λ(am) = m for each positiveinteger m ≥ 5.

Define, further, a binary operation x · y on the set W such that a · a = b,a · b = c, b · a = d, f = a · c = b · b = d · a, a · d = g = c · a and, further, putx · y = am whenever λ(x) + λ(y) = m ≥ 5. Then W (·) becomes a groupoidand it is easy to check that W (·) is again a λ-stratified groupoid.


2.2. Lemma. The groupoids V (·) and W (·) are the only two different λ-stratifiedminimal SH-groupoids of the type (a, a, a).

Proof. It is obvious that both groupoids V (·), W (·) are generated by the one-element set a and a · aa = aa · a. From the definition it follows immediatelythat λ(x · y) = 1. Thus, x · y = a for every x, y ∈ V . The same is also true forevery x, y ∈ W .

(i) At first consider the groupoid V (·). It is easy to check that x · yz = xy · zwhenever (x, y, z) = (a, a, a). From the construction it follows that λ(x · yz) =λ(xy · z) ≥ 4. But each layer Lk(V ) contains only one element for each k ≥ 4.Thus, V (·) is a λ-stratified minimal SH-groupoid of the type (a, a, a). It is easy tosee that sdist(V (·)) = 1. In fact, if we put a ⋆ b = d = c = a · b and x ⋆ y = x · yin the remaining cases then the groupoid V (⋆) is a semigroup. Therefore the SH-groupoid V (·) of the type (a, a, a) will be further called minimal SH-groupoid ofthe first kind.

(ii) Consider, further, the groupoidW (·), in which a(a·aa) = aa·aa = (aa·a)a,a(aa · a) = (a · aa)a and, thus, f = a · c = b · b = d · a, a · d = g = c · a. We haveλ(x · yz) = λ(xy · z) ≥ 5 in the remaining cases. But each layer Lk(W ) containsonly one element for each k ≥ 5 and, thus, x · yz = xy · z if (x, y, z) = (a, a, a).Therefore, W (·) is the other λ-stratified minimal SH-groupoid of the type (a, a, a).Now, it is easy to check that sdist(W (·)) = 2. In fact, if we put c ⋆ a = f = g =c · a, and b ⋆ a = c = d = b · a and x ⋆ y = x · y in the remaining cases thenthe groupoid W (⋆) is a semigroup. Therefore, the SH-groupoid W (·) of the type(a, a, a) will be further called minimal SH-groupoid of the second kind.

(iii) Finally, let κ be an arbitrary λ-stratified congruence on F (·). It followsfrom λ(x · y) > 1 that x · y = a for every x, y ∈ F/κ. Therefore, a · f = a · bb =a · ac = b · c = ba · b = (ba · a) · a = f · a. Similarly, we get f · a = bb · a =ac · a = a · ca = a · g = g · a = ad · a = a · da = a · f . Thus, the layer L5(F/κ)contains only one element a5 and the rest is clear.

2.3. Lemma. Let k be an arbitrary given positive integer such that k ≥ 5.Consider the k-th restriction V k(·) for the SH-groupoid V (·). Then the underlyingset of the restricted groupoid V k(·) is finite and the groupoid V k(·) is an SH-groupoid of the type (a, a, a) having sdist(Vk(·)) = 1.

Proof. It follows from the construction that the underlying set V k of the groupoidV k(·) is finite and it contains just k + 1 different elements. Further, define on V k

a new binary operation ⋆ such that b ⋆ a = c = d = b · a and x ⋆ y = x · y in theremaining cases. It is easy to see that V k(⋆) is a semigroup and the rest is clear.

2.4. Definition. Let V (·) be the minimal SH-groupoid of the first kind con-structed in 2.1 and let k = 4. The corresponding 4-th restriction of the SH-groupoid V (·) of the type (a, a, a) will be further called fundamental restrictionof the first kind of the minimal SH-groupoid V (·).


2.5. Example. The following five-element groupoid F (·) is just the fundamentalrestriction of the minimal SH-groupoid V (·) and sdist(F(·)) = 1.

F a b c d e

a b c e e e

b d e e e e

c e e e e e

d e e e e e

e e e e e e

It is easy to check that the groupoid F (·) is also the fundamental restriction ofthe SH-groupoid W (·).2.6. Lemma. Let k be an arbitrary given positive integer such that k ≥ 5. Con-sider the k-th restriction W k(·) for the SH-groupoid W (·). Then the underlying setof the restricted groupoid W k(·) is finite and the groupoid W k(·) is an SH-groupoidof the type (a, a, a) having sdist(Wk(·)) = 2.

Proof. It is obvious and similar to the proof of Lemma 2.3.

2.7. Definition. Let W (·) be the minimal SH-groupoid of the second kindconstructed in 2.1 and let k = 5. Then the corresponding 5-th restriction ofthe SH-groupoid W (·) of the type (a, a, a) will be further called fundamentalrestriction of the second kind of the minimal SH-groupoid W (·).

2.8. Example. The following seven-element groupoid H(·) is the fundamentalrestriction of the minimal SH-groupoidW (·) of the second kind and sdist(H(·)) = 2.

G a b c d f g h

a b c f g h h h

b d f h g h h h

c g h h h h h h

d f h h h h h h

f h h h h h h h

g h h h h h h h

h h h h h h h h

3. Stratified SH-groupoids generated by two-element set

In this part, let F (·) denote an absolutely free groupoid generated by a two-elementset a, p and let λ(u) denote the length of an arbitrary element u ∈ G.

3.1. Construction. Denote by κ the least congruence on F (·) such that a · aa =aa · a, aa · aa = a(aa · a) and x · yz = xy · z for every x, y, z ∈ G whenever(a, a, a) = (x, y, z). Consider the groupoid F/κ(·). It is obvious that λ(x · y) =λ(x) + λ(y) = 1 and, thus, x · y = a for every x, y ∈ F .


It follows from this that (tu·v)·w = tu·vw = t·(u·vw) for every t, u, v, w ∈ F .Therefore, x1 · x2 . . . xn = x1x2 · x3 . . . xn = · · · = x1x2 . . . xn−1 · xn, for each posi-tive integer n ≥ 5 and x1, x2, . . . xn ∈ F .

Denote, for the simplicity, the groupoid F/κ(·) as E(·). The groupoid E(·)contains a proper subgroupoid generated by the one-element set a. It is obviousthat this is just the λ-stratified SH-groupoid V (·) constructed in 2.1.

Similarly, E(·) contains a proper subgroupoid P (·) generated by the one-element set p. It follows immediately from the construction that it is an infinitesemigroup generated by the one-element set p. Further, it is easy to see thatfor each positive integer m ≥ 2 and each m-element ordered set (x1, x2, . . . , xm) ∈a, pm such that (a, a, . . . , a) = (x1, x2, . . . , xm) = (p, p, . . . , p) the underlying setE contains just 2m−2 different elements x1x2 . . . xm of the lengthm. Therefore, foreach positive integer n ≥ 4 the corresponding layer Ln(E) contains 2n elements.Further, it obvious that card(L3(E)) = 9, card(L2(E)) = 4 and card(L1(E)) = 2.

3.2. Lemma. Let E(·) be the groupoid constructed in 3.1 Then

(i) E(·) is the λ-stratified groupoid;

(ii) E(·) is a primitive extension of the SH-groupoid V (·), see [14];

(iii) E(·) is an SH-groupoid of the type (a, a, a);

(iv) sdist(E(·)) = 1.

Proof. It follows immediately from Construction 3.1.

3.3. Construction. Let E(·) be the λ-stratified SH-groupoid constructed in 3.1.For each positive integer n ≥ 2 and (a, a, . . . , a) = (x1, x2, . . . , xm) = (p, p, . . . , p),consider the element u = x1x2 . . . xn of the length λ(u) = n. Let α(u) denotethe number of all placements of the element a in the corresponding ordered set(x1, x2, . . . , xn). Similarly, let π(u) denote the number of all placements of theelement p in the corresponding ordered set (x1, x2, . . . , xn).

For arbitrary given positive integers k,m, define a mapping σ of the set E tothe set of positive integers such that σ(u) = k · α(u) +m · π(u). Especially, wehave σ(a) = k, σ(b) = 2k, σ(c) = 3k = σ(d) and σ(an) = n · k for each n ≥ 4.Finally, we have σ(pn) = n ·m for each positive integer n. It is easy to checkthat the mapping σ is also a mapping stratifying the underlying set E. Of course,some of the corresponding σ-layers Ln(E) could be empty subsets of the set E.

4. Reduced stratified SH-groupoids of type (a, a, a)

In this part, of the paper consider the groupoid E(·) constructed in 3.1. Let nbe an arbitrary given positive integer. Suppose, at first, that there is at leastone stratified groupoid En(·) satisfying the condition a2 = pn. If σ is a mappingstratifying the underlying set E then 2 · σ(a) = n · σ(p).


4.1. Construction. Let n = 2k where k is an arbitrary positive integer and letE(·) be the groupoid constructed in 3.1. Put σ(a) = k, σ(p) = 1 and, further, letσ(u) = k · α(u) + π(u) in the remaining cases. Now, consider the least congruenceκ on E containing the ordered pair (b, p2k). Then σ(b) = σ(p2k) and, thus, κ isa σ-stratified congruence on the groupoid E(·). Denote, for the simplicity, thecorresponding groupoid E/κ(·) as E2k(·).

4.2. Construction. Let n = 2k + 1 where k is an arbitrary positive integerand let E(·) be the groupoid constructed in 3.1. Put σ(a) = 2k + 1, σ(p) = 2and, further, let σ(u) = (2k + 1) · α(u) + 2 · π(u) in the remaining cases. Now,consider the least congruence κ on E(·) containing the ordered pair (b, p2k+1).Then σ(b) = σ(p2k+1) and, thus, κ is a σ-stratified congruence on the groupoidE(·). Denote, for the simplicity, the corresponding groupoid E/κ(·) as E2k+1(·).

4.3. Lemma. Let n ≥ 2 and let En(·) be the groupoid constructed in 4.1, 4.2.Then

(i) En(·) is a groupoid generated by the two-element set a, p;

(ii) En(·) is a σ-stratified SH-groupoid containing the non-associative triple(a, a, a);

(iii) the equation x · y = b is solvable in En(·) and it has just n different solutions;

(iv) the equation x · y = c is solvable in En(·) and it has just n different solutions;

(v) the equation x · y = d is solvable in En(·) and it has just n different solutions.

Proof. (i) It is obvious that c = a · b = a · aa = aa · a = b · a = d. Further, itfollows from the construction that x · yz = xy · z whenever x, y, z ∈ Hn are suchthat (a, a, a) = (x, y, z).

(ii) The mapping σ is a mapping stratifying the underlying set En. Thus,En(·) is a σ-stratified SH-groupoid of the type (a, a, a).

(iii) It is obvious that b = a · a = p · pn−1 = p2 · pn−2 = · · · = pn−1 · p. Thus,the equation x · y = b has just only n solutions (a, a),(p, pn−1), . . . ,(pn−1, p).

(iv) Similarly, c = a · b = ap · pn−1 = ap2 · pn−2 = · · · = apn−1 · p. Thus, theequation x · y = c has just only n solutions (a, b),(ap, pn−1), . . . ,(apn−1, p).

(v) It is obvious that d = b · a = p · pn−1a = p2 · pn−2a = · · · = pn−1 · pa. Theequation x · y = d has just only n solutions (b, a),(p, pn−1a), . . . ,(pn−1, pa).

4.4. Lemma. Let n ≥ 2 be a positive integer. Then sdist(En(·)) ≤ n.

Proof. Define on the set En a new binary operation such that b a = c = d =b · a and pk pn−ka = c = d = pk · pn−ka for each 1 ≤ k < n and, further, letx y = x · y in the remaining cases. Then a (a a) = (a a) a. It is possible tocheck that x (y z) = (x y) z whenever x, y, z ∈ En are such that (x, y, z) =(a, a, a). Thus, En() is a semigroup and the rest is clear.


4.5. Lemma. Let En(·) be the groupoid constructed in 4.1, 4.2. Further, letEn(⋆) be an arbitrary semigroup having sdist(En(·)) = dist(En(·)), (En(⋆)). Then

(i) either d = a ⋆ b = a · b = c, or c = b ⋆ a = b · a = d;

(ii) if b ⋆ a = b · a and x · y = b, then c = b ⋆ a = x ⋆ (y ⋆ a) = x · (y · a);

(iii) if a ⋆ b = a · b and x · y = b, then d = a ⋆ b = (a ⋆ x) ⋆ y = (a · x) · y.

Proof. (i) The subgroupoid V (·) generated by the one-element set a is aminimal SH-groupoid of the type (a, a, a) and the equation x · y = b is solvable inV (·) only if (x, y) = (a, a). Therefore, either d = a ⋆ b = a · b = c, or c = b ⋆ a =b · a = d.

(ii) Now, if c = b ⋆ a then also c = pk ⋆ (pn−k ⋆ a) = pk · (pn−k · a) = d foreach 1 ≤ k < n. Therefore, dist(En(⋆), En(·)) ≥ n and the rest is clear.

(iii) The other case d = a ⋆ b is similar to (ii).

4.6. Lemma. Let En(·) be the groupoid constructed in 4.1, 4.2. Then

sdist(En(·)) = n.

Proof. It follows from proof of the previous lemma.

5. Fundamental restrictions of reduced stratified SH-groupoids

In this part, consider the reduced stratified SH-groupoids En(·) constructed in 4.1and 4.2. For each positive integerm there are finite groupoids F(n,m)(·) constructedas a m-th restriction of the reduced stratified SH-groupoid En(·).

Put, further, m = 3 · σ(a) + 1 and consider the fundamental restriction of thereduced σ-stratified SH-groupoid En(·).

Especially, for each even positive integer n = 2k let σ be a mapping strati-fying the underlying set E2k such that σ(a) = k and σ(p) = 1. Then we havem = 3k + 1. Denote the corresponding fundamental restriction of the SH-groupoidE2k(·) as E2k(·).

Similarly, for an arbitrary given odd positive integer n = 2k + 1, n ≥ 3, letσ be a mapping stratifying the underlying set E2k+1 such that σ(a) = 2k + 1 andσ(p) = 2. Then m = 6k + 4. Denote the corresponding fundamental restrictionof the SH-groupoid E2k+1(·) as E2k+1(·).

It is obvious that the fundamental restrictions E2k(·) and E2k+1(·) are finitenon-associative SH-groupoids. In this section, it will be proved that

sdist(E2k(·)) = 2k and sdist(E2k+1(·)) = 2k + 1.

5.1 Lemma. For each positive integer n ≥ 2 we have sdist(En(·)) ≤ n.

Proof. The proof is similar to the proof of Lemma 4.4.


5.2 Proposition. For each positive integer k we have sdist(E2k+1(·)) = 2k + 1.

Proof. Consider the fundamental restriction E2k+1(·) and suppose that thereexists a semigroup E2k+1(∇) having dist(E2k+1(·),E2k+1(∇)) < 2k + 1. In thiscase we have a2 = p2k+1 and the corresponding stratifying mapping σ satisfies thecondition σ(a) = 2k + 1 and σ(p) = 2.

Denote by Dp the set describing ”different products” and containing justonly all ordered pairs (x, y) satisfying the condition x · y = x ⋆ y. An unknownsemigroup E2k+1(∇) will be investigated step by step. For each reconstructingstep denote by Kp the set of so far known pairs (x, y) having x · y = x ⋆ y and letUp denote the set of the remaining (still unknown) ordered pairs from the set Dp.

It is obvious that at least one of the following three inequalities has to bevalid: a∇a = a · a, a∇b = a · b, b∇a = b · a. Thus, the set Kp contains at leastone element at the starting step. Therefore, the set Up can contain at most 2k−1elements at the same time.

Finally, consider the following 2k-element set Iu of ”further investigated”ordered pairs (p, u), (p2, u), . . . , (pk, u), (u, pk), (u, pk−1), . . . , (u, p) for u ∈ Up. It isobvious that there is at least one positive integer 1 ≤ i ≤ k such that pi∇u = pi · uor u∇pi = u · pi.

(i) In the first part of the proof suppose that there exists an element u ∈ E2k+1

such that u = a∇a = b = a · a. Let, for example, u∇pi=u·pi. If a∇pi = a · pi = riand a∇ri = a · ri then σ(u∇pi) = σ(u) + σ(pi) = 2i+ σ(u). But we also haveσ(a∇a∇pi) = σ(a) + σ(a∇pi) = σ(a) + σ(a · pi) = 2σ(a) + σ(pi) = 2σ(a) + 2i. Itfollows from this that σ(u) = 2σ(a). But the corresponding layer contains onlyone element and this is just only the element b = a · a, a contradiction. Therefore,at least one of the following two conditions has to be valid: a∇pi = a · pi = rior a∇ri = a · ri. Thus, after this step, the set Kp contains at least two differentelements and the set Up can contain at most 2k − 2 elements, now.

It follows from this that pi∇u = pi · u or in the set Iu there has to be anotherpositive integer j that pj∇u = pj · u or u∇pj = u · pj. Suppose, for example, thatpj∇u = pj · u. Now, if pj∇a = pj · a = tj and tj∇a = tj · a then we obtain againσ(pj∇u) = σ(u) + σ(pj) = 2j + σ(u).

But we also have σ(pj∇a∇a) = σ(pj · a∇a) = σ(tj∇a) = σ(tj · a) = σ(pj · b)= 2j + 2σ(a). Therefore, we get σ(u) = σ(b). But the corresponding layer con-tains only one element and, thus, u = b, a contradiction. It follows from this againthat pj∇a = pj · a = tj or tj∇a = tj · a. That means, that after this step the setKp contains at least three different elements and the set Up can contain at most2k − 3 elements, now.

It is possible to continue in this way till the set Up is empty. But then theset Iu still contains another positive integer m such that it is valid u∇pm = u · pmor pm∇u = pm · u. In this case we get a∇pm = a · pm = rm, a∇rm = a · rm and,also, pm∇a = pm · a = tm, tm∇a = tm · a. It is easy to see that from this we geta contradiction. Thus, a∇a = a · a in any case.

(ii) In this part of the proof suppose, that there exists an element v ∈ E2k+1

such that v = a∇b = c = a · b and let a∇a = a · a. Of course, the set Dp contains


at most 2k elements and the set Up can contain at most 2k − 1 elements, now.Consider the following set Ip = (p, p2k), (p2, p2k−1), . . . , (p2k, p). It is obviousthat there exists a positive integer i such that pi∇p2k+1−i = p2k+1 = b = a · a.But then v = a∇b = a∇p2k+1 = a∇pi∇p2k+1−i.

Further, if a∇pi = a · pi = qi and qi∇p2k+1−i = qi · p2k+1−i, thenv = (a∇pi)∇p2k+1−i = qi · p2k+1−i = (api) · p2k+1−i = a · p2k+1 = a · b = c. This isa contradiction. Thus, at least one of the following two inequalities has to be valid:a∇pi = a · pi or qi∇p2k+1−i = qi · p2k+1−i. Therefore, the set Kp contains at leasttwo different elements and the set Up can contain at most 2k−2 elements, now. Itfollows from this that the set Iu contains another positive integer j = m such thatagain pj∇p2k+1−j = p2k+1 = b = a · a. Now, similarly as above, both equalitiesa∇pj = a · pj = qj and qj∇p2k+1−j = qj · p2k+1−j cannot be valid at the same time.The process of the reconstruction of the expected semigroup can continue step bystep in this way till the time when the set Up is empty. But, the set Iu containsagain another positive integer m such that pm∇p2k+1−m = p2k+1 = b = a · a. It isobvious that we get immediately another contradiction. Thus, a∇b = a · b.

(iii) Finally, it is clear that the case w = b∇a = d = b · a and a∇a = a · a issimilar to (ii). Thus, there is no semigroup E2k+1(∇)) havingdist(E2k+1(·),E2k+1(∇)) < 2k + 1 and the proof is finished.

5.3 Proposition. For each positive integer k we have sdist(E2k(·)) = 2k.

Proof. Consider the fundamental restriction E2k(·) and suppose that there existsa semigroup E2k(∇) having dist(E2k(·),E2k(∇)) < 2k. In this case we get a2 = p2k

and, thus, the corresponding stratifying mapping σ satisfies the condition σ(a) = kand σ(p) = 1.

Denote by Dp the set containing just only all ordered pairs (x, y) satisfyingthe condition x · y = x∇y. An unknown semigroup E2k(∇) will be investigatedstep by step during the proof. For each reconstructing step denote by Kp the setof all so far known pairs (x, y) having x · y = x∇y and denote by Up the set ofthe remaining (still unknown) ordered pairs from the set Dp.

It is obvious that at least one of the following three inequalities has to bevalid: a∇a = a · a, a∇b = a · b, b∇a = b · a. The set Kp contains at least oneelement in the beginning and, thus, the set Up can contain at most 2k−2 elementsat the same time.

(i) In the first part of the proof suppose that there exists an element u ∈ E2k

such that u = a∇a = b = a · a and σ(u) = 2σ(a). Now consider the 2k-element setIu containing the ordered pairs (p, u), (p2, u), . . . , (pk, u), (u, pk), (u, pk−1), . . . , (u, p).It is obvious that there has to be a positive integer 1 ≤ i ≤ k such that pi∇u =pi · u or u∇pi = u · pi. Let, for example, pi∇u = pi · u. Further, suppose thatpi∇a = pi · a = ti and ti∇a = ti · a. Then σ(pi∇u) = σ(pi · u) = i+ σ(u) =i+ 2σ(a). But we also have σ(pi∇u) = σ((pi∇a)∇a) = σ((pi · a)∇a) = σ(ti∇a) =σ(ti · a) = σ(pi · (aa)) = i+ 2σ(a), which is a contradiction. Thus, at least one ofthe following two conditions pi∇a = pi · a = ti, ti∇a = ti · a holds. Now the setKp contains at least two elements and, thus, the set Up can contain at most 2k−3elements. It is easy to see that u∇pi = u · pi or there is another positive integer


j = i such that u∇pj = u · pj or pj∇u = pj · u. Let, for example, u∇pj = u · pj.Similarly as above, it is easy to check that a∇pj = a · pj = rj or a∇rj = a · rj.

Of course, it is possible to continue in this way till the time when the setUp is empty. In this case there exists again another positive integer m suchthat pm∇u = pm · u or u∇pm = u · pm. But now we get pm∇a = pm · a = tm,tm∇a = tm · a and a∇pj = a · pj = rj, a∇rj = a · rj. It is clear that from this weget a contradiction again. Thus, we have either σ(u) = 2σ(a) or a∇a = a · a.

(ii) In this part of the proof suppose that there is an element u = a∇a =a · a having σ(u) = 2σ(a). Then there exists such a positive integer m thatu = pmapk−m. The set Dp contains at most 2k− 1 elements including the orderedpair (a, a). The set Kp contains now one element and, thus, the set Up can containat most 2k − 2 other elements.

The set of the ordered pairs (p, u), (p2, u), . . . , (pk, u), (u, pk), . . . , (u, p) con-tains at least one ordered pair (pi, u) or (u, pi) such that pi∇u = pi · u or u∇pi =u · pi. Let, for example, pi∇u = pi · u and suppose, furthermore, that pi∇a =pi · a = ti and ti∇a = ti · a. Then we get pi∇u = pi∇(a∇a) = (pi∇a)∇a =pia · a = pi · aa = pib = pi · p2k. But we also have pi∇u = pi · pmapk−m andp2k+i = pi+mapk−m in E2k. Thus, at least one of the following inequalities has tobe valid: pi∇a = pia = ti or ti∇a = ti · a. Then the set Kp contains at least twodifferent elements and, thus, the set Up can contain at most 2k − 3 elements.

Therefore u∇pi = u · pi or there has to be another positive integer j = isuch that u∇pj = u · pj or pj∇u = pj · u. Let, for example, u∇pj = upj. Now,suppose, furthermore, that a∇pj = apj = rj and a∇rj = a · rj = a · apj. It iseasy to check that, similarly as above, we get a∇pj = apj or a∇rj = a · rj. It ispossible to continue in this way till the set Up is empty and, then, we obtain againa contradiction. It is proved now that a∇a = b = a · a.

(iii) This part of the proof is similar to (ii) of the proof of the previousproposition. Let b = a∇a = a · a = p2k and put v = a∇b = a · b = c. Thena∇b = a∇a∇a = b∇a. The set Kp contains one element and, thus, the setUp can contain at most 2k − 2 elements. Now, consider 2k − 1 of the followingordered pairs (p, p2k−1), (p2, p2k−2), . . . , (p2k−1, p). It is obvious that there is apositive integer i such that pi∇p2k−i = pi · p2k−i = p2k = b. But then we getv = a∇p2k = a∇(pi∇p2k−i).

Now, suppose, that a∇pi = api = ri and ri∇p2k−i = ri · p2k−i = api · p2k−i =a · p2k = a · b = c. In this case we get v = a∇p2k = a∇(pi∇p2k−i) = (api)∇p2k−i =ri · p2k−i = (api) · p2k−i = a · (pip2k−i) = a · p2k = a · b = c, a contradiction. Thusa∇pi = api or ri∇p2k−i = ri · p2k−i. But then the set Kp contains at least twodifferent elements and, thus, the set Up can contain at most 2k−3 elements. There-fore, there exists another positive integer j = i such that pj∇p2k−j = pj · p2k−j =p2k = b. It is obvious that this process can continue again in the same way tillthe the set Up is empty. But then there has to be another positive integer m suchthat pm∇p2k−m = pm · p2k−m = p2k = b and we again obtain a contradiction.

(iv) In the remaining part let b = a∇a = a · a = p2k and put w = b∇a =b · a = d. It is clear that this case is similar to (iii). We can again, step by step,


obtain a system of contradictions and, finally, it is clear that there is no semigrouphaving the corresponding property.

It follows immediately from this that sdist(E2k(·)) ≥ 2k and the rest is clear.

5.4 Theorem. For an arbitrary given positive integer n there exists a finite non-associative groupoid G(·) generated by the two-element set a, p satisfying thecondition a = x · y = p and having sdist(G(·)) = n.

Proof. It follows immediately from the construction and from 5.2 and 5.3.

5.5 Example. It is easy to check that the following groupoid H(·) is just theleast minimal SH-groupoid of the type (a, a, a) and of the second kind havingsdist(H(·)) = 2, see Lemma 2.2.

H a b c d f g

a b c f g f f

b d f f f f f

c g f f f f f

d f f f f f f

f f f f f f f

g f f f f f f

5.6 Example. The following table is just the table of the 11-element SH-groupoidE2(·) having sdist(E2(·)) = 2.

2 a p b r t c d u v w e

a b r c u v e e e e e e

p t b u w d e e e e e e

b d u e e e e e e e e e

r v c e e e e e e e e e

t u w e e e e e e e e e

c e e e e e e e e e e e

d e e e e e e e e e e e

u e e e e e e e e e e e

v e e e e e e e e e e e

w e e e e e e e e e e e

e e e e e e e e e e e e

5.7 Remark. The minimal SH-groupoid G(·) constructed in 2.8 and having thesame semigroup distance has a seven-element underlying set. It follows from 5.5that there are finite non-associative groupoids with the semigroup distance equalto 2 and having smaller underlying set. Does there exist a finite non-associativegroupoid having at most five-element underlying set and having its semigroupdistance equal to 2?

5.8 Example. The following table is just the table of the 16-element SH-groupoidE3(·) having sdist(E3(·)) = 3.


3 p a f g h b q r s t u c d v w e

p f h b r q t v w d e e e e e e e

a g b q t u c e e e e e e e e e e

f b s t w d e e e e e e e e e e e

g q u c e e e e e e e e e e e e e

h r t t e e e e e e e e e e e e e

b t d e e e e e e e e e e e e e e

q c e e e e e e e e e e e e e e e

r v e e e e e e e e e e e e e e e

s w e e e e e e e e e e e e e e e

t e e e e e e e e e e e e e e e e

u e e e e e e e e e e e e e e e e

c e e e e e e e e e e e e e e e e

d e e e e e e e e e e e e e e e e

v e e e e e e e e e e e e e e e e

w e e e e e e e e e e e e e e e e

e e e e e e e e e e e e e e e e e

6. Comments and open problems

6.1. It was proved in 2.5, 2.6 and 2.8 that there are finite non-associativegroupoids having their semigroup distance equal to 1 resp. to 2. It follows from 5.5that the construction using a fundamental restriction of stratified SH-groupoidsEn(·) is not minimal with respect to the number of elements contained in theunderlying set En.

6.2. Examples 5.7 and 5.9 show that the number of elements of the underlyingset En quickly grows up with the increasing positive integer n.

6.3. The construction of finite non-associative groupoids En(·) is based on theuse of primitive extensions of the minimal SH-groupoids V (·) (see 2.1) having thesemigroup distance equal to 1. It is possible to use also primitive extensions ofthe minimal SH-groupoid W (·) (see 2.1) and having semigroup distance equal to2. It seems to be clear that a lesser number of elements in the underlying setcan bring a bigger semigroup distance in this case. But, in that case, the proofsshould be more complicated and the result should be the same.

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ON HOW TO CONSTRUCT LEFT SEMIMODULESFROM THE RIGHT ONES

Barbora Batıkova

Department of MathematicsCULSKamycka 129, 165 21 Praha 6 – SuchdolCzech Republice-mail: [email protected]

Tomas Kepka

Department of AlgebraMFF UKSokolovska 83, 186 75 Praha 8Czech Republice-mail: [email protected]

Petr NemecDepartment of MathematicsCULSKamycka 129, 165 21 Praha 6 – SuchdolCzech Republice-mail: [email protected]

Abstract. In the paper, various constructions of left semimodules from the right ones

are investigated.

Keywords: semiring, semimodule, ideal, congruence-simple.

2000 Mathematics Subject Classification: 16Y60.

1. Introduction

(Congruence-)simple semirings are studied, e.g., in [1], [2], [4], [5], [6], [7] andsome of them are constructed as endomorphism semirings of commutative semi-groups. That is, these semirings are characterized via semimodules. If S is asemiring and SM a left semimodule then the mapping a 7→ λa, λa(x) = ax, a ∈ S,x ∈ M , is a (semiring) homomorphism of the semiring S into the full endomor-phism semiring of the additive semigroupM(+). This “canonical” homomorphismis injective if and only if the semimodule SM is faithful. Of course, if the semiringS is simple then the homomorphism is either constant or injective. Now, in orderto get various more or less regular representations of (simple) semirings via endo-morphisms, we have to find “nice” left semimodules (once we compose mappingsfrom the right to the left). Unfortunately, it may happen that left semimodulesof such kind are not easily available but, contrarywise, useful right semimodulesare at hand. Therefore, we have to find a passage from the right side to the leftside.

562 b. batıkova, t. kepka, p. nemec

1. Preliminaries

A semiring is an algebraic structure possessing two associative binary operations(most frequently denoted as addition and multiplication) where the addition iscommutative and the multiplication distributes over the addition from either side.Basic specimens are endomorphism semirings of commutative semigroups andbasic pieces of information on semirings are available from [3].

A subset I of a semiring S is a left (right) ideal if SI ∪ (I + I) ⊆ I(IS ∪ (I + I) ⊆ I). We put R(S) = a ∈ S |Sa = a . If R(S) = ∅ thenR(S) is additively idempotent and it is the smallest (right) ideal of S. If, more-over, the right S-semimodule R(S) is faithful then the semiring S is additivelyidempotent.

A semiring S is said to be (congruence-))simple if it has just two congruencerelations. If S is simple and |R(S)| ≥ 2 then R(S)S is faithful (see 7.1) and S isadditively idempotent.

In the sequel, all semirings and all semimodules are assumed to be additivelyidempotent. If M(+) is a semilattice then the basic order is defined on M byx ≤ y if and only if x+ y = y. An element w ∈ M is the greatest element in theordered set M(≤) iff x + w = w for every x ∈ M . That is, w = oM is the (only)additively absorbing element. The existence of such an element will be denoted byoM ∈ M . On the other hand, oM /∈ M means that M has no absorbing element.Symmetrically, w is the smallest element iff w = 0M is additively neutral. Again,0M ∈ M menas that such an element is present and 0M /∈ M means that not.

2. From the right to the left (a)

Let S be a non-trivial semiring and M (= MS(+, ·)) a non-trivial right S-semi-module. For x, y ∈ M , let Px,y = z ∈ M | z ≤ x, z ≤ y . If this set is non-emptythen it is a subsemilattice of M(+). Anyway, we put W1 = (x, y) | x, y ∈ M,Px,y = ∅ and W2 = (x, y) ∈ W1 | oP ∈ P = Px,y . Clearly, the ordered setM(≤) is a lattice iff W2 = M ×M .

Now, we define a (possibly partial) binary operation ∗ on M by x ∗ y = oP ,P = Px,y, for every pair (x, y) ∈ W2. Some easy observations follow:

2.1 Lemma.

(i) (x, x) ∈ W2 and x ∗ x = x for every x ∈ M .

(ii) x ∗ y = y ∗ x for every (x, y) ∈ W2.

(iii) (x, y) ∈ W2 and x ∗ y = x iff x ≤ y.

(iv) (x, x+ y) ∈ W2 and x ∗ (x+ y) = x for all x, y ∈ M .

(v) If (x, y) ∈ W2 then x+ (x ∗ y) = x.

(vi) If (x, y) ∈ W2, (y, z) ∈ W2 and (x ∗ y, z) ∈ W2 then (x, y ∗ z) ∈ W2 andx ∗ (y ∗ z) = (x ∗ y) ∗ z.

(vii) If oM ∈ M then (x, oM) ∈ W2 and x ∗ oM = x for every x ∈ M .

(viii) If 0M ∈ M then W1 = M × M , (x, 0M) ∈ W2 and x ∗ 0M = 0M for everyx ∈ M .

on how to construct left semimodules from the right ones 563

For a ∈ S and x ∈ M , let Qx,a = y ∈ M | ya ≤ x . If this set is non-emptythen it is a subsemilattice of M(+). Anyway, we put W3 = (x, a) |Qx,a = ∅ and W4 = (x, a) ∈ W3 | oQ ∈ Q = Qx,a .

Now, we define a (possibly partial) left S-scalar multiplication on M bya x = oQ, Q = Qx,a, (x, a) ∈ W4. The following observations are quite easyto check:

2.2 Lemma.

(i) (xa, a) ∈ W3 for all a ∈ S and x ∈ M .

(ii) If (xa, a) ∈ W4 then x ≤ a (xa) and xa = (a xa)a.(iii) If oM ∈ M then (oM , a) ∈ W4 for every a ∈ S and a oM = oM .

(iv) If oM ∈ M then (oMa, a) ∈ W4 for every a ∈ S and a (oMa) = oM .

2.3 Lemma. Let a ∈ S and x, y ∈ M .

(i) If (x, y) ∈ W2 then Qx∗y,a = Qx,a ∩Qy,a.

(ii) If (x, y) ∈ W2 and (x ∗ y, a) ∈ W3 then Qx,a ∩Qy,a = ∅.(iii) If (x, a), (y, a) ∈ W4 and (a x, a y) ∈ W1 then Qx,a ∩Qy,a = ∅.(iv) If Qx,a ∩Qy,a = ∅ then (x, y) ∈ W1 and (x, a), (y, a) ∈ W3.

(v) If Qx,a ∩Qy,a = ∅ and (x, y) ∈ W2 then (x ∗ y, a) ∈ W3.

(vi) If (x, y) ∈ W2 and (x, a), (y, a), (x ∗ y, a) ∈ W4 then (a x, a y) ∈ W2 anda (x ∗ y) = (a x) ∗ (a y).

2.4 Lemma. Let a, b ∈ S and x ∈ M .

(i) Qx,a+b = Qx,a ∩Qx,b.

(ii) If (x, a), (x, b) ∈ W4 and (a x, b x) ∈ W1 then Qx,a+b = ∅.(iii) If Qx,a+b = ∅ then (x, a), (x, b) ∈ W3.

(iv) If (x, a), (x, b), (x, a + b) ∈ W4 then (a x, b x) ∈ W2 and (a + b) x =(a x) ∗ (b x).

2.5 Lemma. Let a, b ∈ S and x ∈ M .

(i) If (x, b) ∈ W4 then Qbx,a = Qx,ab.

(ii) If (x, b), (b x, a) ∈ W4 then (x, ab) ∈ W4 and a (b x) = (ab) x.(iii) If (x, ab) ∈ W3 then (x, b) ∈ W3.

(iv) If (x, b), (x, ab) ∈ W4 then (b x, a) ∈ W4 and a (b x) = (ab) x.(v) If (x, b) /∈ W3 then (x, ab) /∈ W3.

2.6 Remark. Assume that W3 = W4 and the (partial) operation can becompleted to a full one (denoted again by ) in such a way that a(bx) = (ab)xfor all a, b ∈ S and x ∈ M . Furthermore, assume that for every x ∈ M there isax ∈ S with Max = x.(i) We have ax x = oM ∈ M for every x ∈ M . Moreover, if the right semi-

module MS is faithful then ax ∈ R(S).


(ii) Assume that the semiring S is simple. If MS is not faithful then xa = xbfor all a, b ∈ S and x ∈ M , and hence xa = xax = x and Max = M , acontradiction. Consequently, MS is faithful.

(iii) Assume that ax ∈ R(S) for every x ∈ M (see (i) and (ii)). Then b oM =b (ax x) = (bax) x = ax x = oM for every b ∈ S. If a ∈ R(S), x ∈ Mand y = a x then oM = ay y = ay(a x) = (aya) x = a x. ThusR(S) M = oM.

(iv) Assume, finally, that the semiring S is simple and there is a binary operation defined on M such that M(, ) is a left S-semimodule. Combining (i),(ii) and (iii), we see that R(S)M = oM, and hence the left S-semimodule

SM = M(, ) is not faithful. Since S is simple, it follows that ax = bx forall a, b ∈ S and x ∈ M . In particular, ax = axx = oM , SM = oM anda (xa) = oM . The latter equality implies xa = oMa. Thus xa = oMa = xbafor all a, b ∈ and x ∈ M . Since MS is faithful, a = ba and, S being simple,|S| = 2.

2.7 Remark. Assume that W1 = W2 = M × M , choose α ∈ M and putx ∗ y = α for all (x, y) ∈ (M ×M) \W1. Now, the binary operation ∗ is definedon M and it is idempotent and commutative. If u < α and (x, α) /∈ W1 then(x ∗ α) ∗ u = α ∗ u = u = α = x ∗ u = x ∗ (α ∗ u). If (x, y) /∈ W1 then either α xor α y. If α x and (x, α) ∈ W1 then (y∗x)∗x = α∗x = α = y∗x = y∗(x∗x).Now, it is easy to see thatM(∗) is a semilattice if and only if α is a minimal elementof the ordered set M(≤). Anyway, if α is minimal then α v for at least onev ∈ M (otherwise α = 0M ∈ M and W1 = M ×M) and (v ∗ α) + v = α + v = v.It means that M(+, ∗) is not a lattice.

2.8 Remark. If 0M ∈ M then W1 = M ×M and W3 = (x, a) | 0Ma ≤ x .

3. From the right to the left (b)

The foregoing section is immediately continued. Here, we assume that W1 = W2

and W3 = W4. Choose and fix an element ω such that ω /∈ S ∪ M , put M+ =M∪ω and extend the (partial) operations ∗ and defined on M in the followingway: x ∗ u = z ∗ ω = ω ∗ z = ω ∗ ω = ω for all x, y, z ∈ M , (x, y) /∈ W1, anda u = b ω = ω for all a, b ∈ S, u ∈ M , (u, a) /∈ W3. Furthermore, putx+ ω = ω + x = x for every x ∈ M , ω + ω = ω and ωa = ω for every a ∈ S.

3.1 Proposition. The algebraic structure M+(+, ∗) is a lattice.

Proof. First, it follows from 2.1(i),(ii) that the binary structure M+(∗) is bothidempotent and commutative. If α, β, γ ∈ M+ and ω ∈ α, β, γ then (α∗β)∗γ =ω = α ∗ (β ∗ γ) (use 2.1(vi) and the equality W1 = W2). We have proved thatM+(∗) is a semilattice. Using 2.1(iv),(v), we see that M+(+, ∗) is a lattice.

3.2 Theorem. The algebraic structure SM+ = M+(∗, ) is a left S-semimodule.

Proof. As we already know, M+(∗) is a semilattice (see 3.1). It remains to showthat a(α∗β) = (aα)∗(aβ), (a+b)α = (aα)∗(bα) and a(bα) = (ab)α


for all a, b ∈ S and α, β ∈ M+. The case α = ω (or β = ω) is clear and we canassume α = x, β = y, x, y ∈ M . Now, we distinguish the following cases:

(i) Let (x, y) ∈ W1 and (x ∗ y, a) ∈ W3. By 2.3(i),(ii), we have Qx∗y,a =Qx,a∩Qy,a = ∅ and, by 2.3(iv) we have (x, a), (y, a) ∈ W3. Now, (ax, ay) ∈ W1

and a (x ∗ y) = (a x) ∗ (a y) by 2.3(vi).

(ii) Let (x, y) ∈ W1 and (x∗y, a) /∈ W3. By 2.3(v), Qx,a∩Qy,a = ∅, and hence(a x, a y) /∈ W1 follows from 2.3(ii). Thus a (x ∗ y) = ω = (a x) ∗ (a y).

(iii) Let (x, y) /∈ W1. Then a (x ∗ y) = a ω = ω. On the other hand,Qx,a ∩ Qy,a = ∅ by 2.3(iv), and hence, due to 2.3(iii), (a x, a y) /∈ W1. Then(a x) ∗ (a y) = ω.

(iv) Let (x, a + b) ∈ W3 and (x, a), (x, b) ∈ W3. By 2.4(iv), (a + b) x =(a x) ∗ (b x).

(v) Let (x, a+ b) ∈ W3 and (x, a) /∈ W3. Then Qx,a = ∅, and hence Qx,a+b =Qx,a ∩Qx,a = ∅, a contradiction.

(vi) Let (x, a + b) /∈ W3. Then (a + b) x = ω. We have Qx,a+b = ∅, andtherefore (a x, b x) /∈ W1 follows from 2.4(ii). Thus (a x) ∗ (b x) = ω.

(vii) Let (x, b) ∈ W3 and (b x, a) ∈ W3. Then a (b x) = (ab) x by 2.5(ii).

(viii) Let (x, b) ∈ W3 and (b x, a) /∈ W3. Then a (b x) = ω, (x, ab) /∈ W3

by 2.5(i) and (ab) x = ω.

(ix) Let (x, b) /∈ W3. Then a (b x) = ω = (ab) x by 2.5(v).

3.3 Proposition. The algebraic structureM+S = M+(+, ·) is a right S-semimodule.

Proof. It is easy.

3.4 Remark. Clearly, (a x)b ≤ a (xb) for all a, b ∈ S and x ∈ M+.

3.5 Lemma.

(i) S ω = ω.(ii) If oM ∈ M then S oM = oM.(iii) If 0M ∈ M then S 0M = 0M iff 0MS = 0M and xa = 0M for all a ∈ S

and x ∈ M \ 0M.(iv) If x ∈ M then S x = x iff xS ≤ x and ya x for all a ∈ S, y ∈ M ,

y x.

3.6 Lemma. The following conditions are equivalent for a, b ∈ S:

(i) a x = b x for every x ∈ M .

(ii) xa = xb for every x ∈ M .

3.7 Corollary. The left semimodule SM+ is faithful if and only if the right

semimodule MS is so.

3.8 Lemma. Let a ∈ S and w ∈ M be such that Ma ≤ w. Then aw = oM ∈ M .

3.9 Proposition. Assume that for every x ∈ M there is ax ∈ S with Max = x.Then:


(i) ax y = oM ∈ M for every y ∈ M+, x ≤ y (or x ∗ y = x).

(ii) ax z = ω for every z ∈ M+, x z (or x ∗ z = x).

(iii) ax ∗M+ = oM , ω.(iv) The semimodule SM

+ is simple.

Proof. Only (iv) needs a short proof. Let ϱ = id be a congruence of SM+. We

have oM ∈ M by 3.8 and if (ω, oM) ∈ ϱ then ϱ = M ×M . If (ω, x) ∈ ϱ for somex ∈ M then (ω, oM) = (ax ω, ax x) ∈ ϱ. If (x, y) ∈ ϱ, where x, y ∈ M , x y,then (oM , ω) = (ax x, ax y) ∈ ϱ. Thus ϱ = M ×M anyway.

The following assertions are easy.

3.10 Proposition. Assume that 0M ∈ M and put L+ = M+ \ 0M. ThenW1 = W2 = M ×M and L+ is a subsemimodule of the left S-semimodule SM

+ ifand only if the following three conditions are satisfied:

1. For all x, y ∈ L = M \ oM there is at least one z ∈ L with z ≤ x andz ≤ y;

2. If a ∈ S is such that 0Ma = 0M then for every x ∈ L there is at least oney ∈ L with ya ≤ x;

3. If a ∈ S is such that 0Ma = 0M then there is at least one v ∈ L with0Ma = va.

3.11 Lemma. Assume that 0M ∈ M and that the set L = M \ 0M has thesmallest element w. Let a, b ∈ S be such that Ma = 0M and Mb = w. Then:(i) a = b and a x = b x for every x ∈ L+ = L ∪ w.(ii) If L+ is a subsemimodule of SM

+ (see 3.10) then L+ is not faithful.

3.12 Remark.

(i) We have P ′x,y=z∈M+ |x ≤ z, y ≤ z=z ∈ M |x+y ≤ z, P ′

x,ω=z ∈ M+|x ≤ z, ω ≤ z = z ∈ M |x ≤ z and P ′

ω,ω = z ∈ M+ |ω ≤ z = M+ forall x, y ∈ M .

(ii) Q′x,a = y ∈ M+ |x ≤ a y = y ∈ M |xa ≤ y and Q′

ω,a = y ∈ M+ |ya ≤ ω = ω for all a ∈ S and x ∈ M .

3.13 Remark. M is a subsemimodule of SM+ iff the following two conditions

are satisfied:

1. For all x, y ∈ M there is z ∈ M with z ≤ x and z ≤ y;

2. For all a ∈ S and x ∈ M there is y ∈ M with ya ≤ x.

If the condition (2) is true then the set Ma is downwards cofinal in M(≤) (cf. 3.9).

4. From the right to the left (c)

The second section is continued. We will assume here that (W1 =) W2 = M ×M(cf. 2.6 and 2.7),


4.1 Proposition. The algebraic structure M(+, ∗) is a lattice.

Proof. Use 2.1.

Now, choose α ∈ M and put W ′3 = (x, a) ∈ W3 |x = α . Furthermore,

assume that W3 = W4 and put a x = a x for every pair (x, a) ∈ W ′3 and

b y = α for every pair (b, y) ∈ (S ×M) \W ′3.

4.2 Lemma. a (α ∗ x) = (a α) ∗ (a x) for all a ∈ S and x ∈ M iff thefollowing two conditions are satisfied:

1. If a ∈ S and x ∈ M are such that αa ≤ x, α x and (α, a) ∈ W3 then(a α) ∗ (a x) = α;

2. If a ∈ S and x ∈ M are such that x = α, αa x and (x, a) ∈ W3 then(α, a) ∈ W3, α x and (a α) ∗ (a x) = α ∗ (a x).

Proof. (i) Let a ∈ S and x ∈ M be such that αx ≤ x, α x and (α, a) ∈ W3.We have α ∈ Rx,a and we get ax = ax. Now, (aα)∗(ax) = α∗(ax) = α,since α ≤ a x. On the other hand, (α, a), (x, a) ∈ W3, α ∗ x = α, and hence(α ∗ x, a) ∈ W3 and a (α ∗ x) = (a α) ∗ (a x) by 2.3(iii),(v). Of course,a (α ∗ x) = a (α ∗ x).

(ii) Let a ∈ S and x ∈ M \α be such that αa x and (x, a) ∈ W3. We have(aα)∗(ax) = α∗(ax) = α, since αa x. Now, if a(α∗x) = (aα)∗(ax)then a(α∗x) = α, and hence a(α∗x) = a(α∗x), (α∗x, a) ∈ W ′

3. Consequently,(α, a) ∈ W3 and a (α ∗ x) = (a α) ∗ (a x) by 2.3(vi). If α ≤ x then α ∗ x = αand a (α ∗ x) = α.

(iii) Assume that both conditions (1) and (2) are satisfied. We wish to showthat y = z, where y = a (α ∗ x) and z = (a α) ∗ (a x) = α ∗ (a x).

If x = α then y = α = z. If (x, a) /∈ W3 then (α ∗ x, a) /∈ W3 and y = α = zagain. Consequently, assume that x = α and (x, a) ∈ W3, so that (x, a) ∈ W ′

3 anda x = a x, z = α ∗ (a x).

Let αa ≤ x. That is, α ≤ a x and z = α. If α ≤ x or (α ∗ x, a) /∈ W3

then y = α. If α x and (α ∗ x, a) ∈ W3 then (α, a) ∈ W3, α ∗ x = α andy = a (α ∗ x) = (a α) ∗ (a x) = α by (1) and 2.3(vi).

Let αa x. Then α a x and z = α ∗ (a x) = α. By (2), (α, a) ∈ W3,α x and z = α ∗ (a x) = (a α) ∗ (a x). On the other hand, (α ∗ x, a) ∈ W3

by 2.3(iii),(v) and y = a (α ∗ x) = a (α ∗ x) = (a α) ∗ (a x) = z by 2.3(vi).

4.3 Lemma. a (x ∗ y) = (a x) ∗ (a y) for all a ∈ S and x, y ∈ M \ αsuch that x ∗ y ∈ M \ α iff the following condition is satisfied:

1. If (z, b) ∈ W ′3 is such that (u, b) /∈ W3 for some u ∈ M \ α, z ∗ u = α,

then αb ≤ z.

Proof. (i) As concerns (1), we have (z ∗ u, b) /∈ W3, and hence α = b (z ∗ u).On the other hand, (b z) ∗ (b u) = (b z) ∗ α.

(ii) Assume that (1) is true and put u = a (x ∗ y) and v = (ax) ∗ (a y).If (x, a) /∈ W3 and (y, a) /∈ W3 then (x ∗ y, a) /∈ W3 and we get u = α = v.If (x, a), (y, a) ∈ W3 then (x ∗ y, a) ∈ W3 (use 2.3) and u = a (x ∗ y) = (a x) ∗


(a y) = v (use 2.3 again). On the other hand, if (x, a) ∈ W3 and (y, a) /∈ W3

then (x ∗ y, a) /∈ W3 and u = α, v = (a x) ∗ α. By (1), αa ≤ x, α ≤ a x andv = α.

4.4 Lemma. (a + b) x = (a x) ∗ (b x) for all a, b ∈ S and x ∈ M iff thefollowing condition is satisfied:

1. If (x, a) ∈ W3, x = α, and (x, b) /∈ W3 then αa ≤ x.

Proof. (I) As concerns (1), we have (x, a + b) /∈ W3 by 2.4(i),(iii), and hence(a + b) x = α. On the other hand, (a x) ∗ (b x) = (a x) ∗ α and if(a x) ∗ α = α then αa ≤ x.

(ii) Assume that (1) is true and put y = (a+ b)x and z = (ax)∗ (bx).If x = α then y = α = z, so that we will assume that x = α. If (x, a + b) /∈ W3

then y = α and, by 2.4(ii), either (x, a) /∈ W3 or (x, b) /∈ W3. Assume the formercase, the latter one being symmetric. If (x, b) ∈ W3 then z = α ∗ (b x) αb ≤ xby (1), α ≤ b x and z = α. If (x, b) /∈ W3 then z = α ∗ α = α. Finally, let(x, a+ b) ∈ W3, so that y = (a+ b) x. By 2.4(i),(iii), we have (x, a), (x, b) ∈ W3,and hence z = (a x) ∗ (a y) = (a+ b) x = y by 2.4(iv).

4.5 Lemma. (ab) x = a (b x) for all a, b ∈ S and x ∈ M iff the followingcondition is satisfied:

1. If (α, c) ∈ W3, (x, d) ∈ W3, x = α and d x = α then c α = α.

Proof. (i) As concerns (1), we have (x, cd) ∈ W4 and cα = c(dx) = (cd)x =(cd) x by 2.5(ii). On the other hand, c (d x) = c (d x) = c α = α.

(ii) Assume that (1) is true. We wish to show that y = z, where y = (ab)xand z = a (b x). If x = α then y = α = z, and hence we assume x = α.If (x, b) /∈ W3 then (x, ab) /∈ W3 and y = α = z again. If (x, b) ∈ W3 and(b x, a) /∈ W3 then the same is true. If (x, b) ∈ W3, (b x, a) ∈ W3 and b x = αthen (x, ab) ∈ W3 and y = (ab) x = a (b x) = a (b x) = a (b x) = zby 2.5(ii). Finally, if (x, b), (b x, a) ∈ W3 and b x = α then y = (ab) x =a (b x) = a α = α = a α = a (b x) = a (b x) = z by (1).

4.6 Lemma. Assume that 4.5(1) is true. Let (x, a) ∈ W3, x = α = a x. Then:

(i) If b ∈ S and y ∈ M are such that yb ≤ α then y ≤ α and ya ≤ x.

(ii) If c ∈ S is such that Mc ≤ α then α = oM and Ma ≤ x.

Proof. It is easy.

4.7 Lemma. Let a ∈ S and x, y ∈ M \ α be such that x ∗ y = α anda (x ∗ y) = (a x) ∗ (a y). Then:

(i) If z ∈ M is such that za ≤ α then z ≤ α and a α ≤ α.

(ii) va α for at least one v ∈ M .


Proof. (i) First, (ax)∗ (ay) = a (x∗y) = aα = α. Then za ≤ α < x, y,so that z ≤ a x and z ≤ a y. We conclude that z ≤ (a x) ∗ (a y) =(a x) ∗ (a y) = α. If Ma ≤ α then M ≤ α by (i), and hence oM = α < x, acontradiction.

In the remaining part of this section, assume that α = 0M ∈ M .

4.8 Lemma. a (x ∗ y) = (a x) ∗ (a y) for all a ∈ S and x, y ∈ M iff thefollowing condition is satisfied:

1. If a ∈ S and x, y, w ∈ M \ 0M are such that wa ≤ x and wa ≤ y thenz ≤ x and z ≤ y for at least one z ∈ M \ 0M.

Proof. First, 0M ≤ x for every x ∈ M , and hence the condition 4.2(1) is satisfied.Further, if (x, a) ∈ W3 then 0Ma ≤ x and 4.2(2) is true as well. By 4.2, a (x∗y)= (ax) ∗ (a y) whenever 0M ∈ x, y. Similarly, 4.3(1) is true and a (x ∗ y)= (a x) ∗ (a y) whenever x ∗ y = 0M .

Now, let x, y ∈ M \ 0M and x ∗ y = 0M . Then a (x ∗ y) = 0M and,if (x, a) /∈ W3 or (y, a) /∈ W3 then (a x) ∗ (a y) = 0M . Assume, therefore,that (x, a), (y, a) ∈ W3. Then (a x) ∗ (a y) = (a x) ∗ (a y), and so(a x) ∗ (a y) = 0M iff wa ≤ x and wa ≤ y implies w = 0M .

4.9 Lemma. Assume that wa0 = 0M for some a0 ∈ S and w ∈ M , w = 0M .Then a (x ∗ y) = (a x) ∗ (a y) for all a ∈ S and x, y ∈ M iff the followingcondition is satisfied:

1. For all x, y ∈ M \ 0M there is z ∈ M \ 0M with z ≤ x and z ≤ y.

Proof. See 4.8 (and (4.7).

4.10 Lemma. (a+ b) x = (a x) ∗ (b x) for all a, b ∈ S and x ∈ M .

Proof. Apparently, the condition 4.4(1) is true.

4.11 Lemma. (ab)x = a (bx) for all a, b ∈ S and x ∈ M iff the followingcondition is satisfied:

1. If xa = 0M for some a ∈ S and x ∈ M \ 0M then for all b ∈ S andy ∈ M \ 0M such that 0Mb ≤ y there is z ∈ M \ 0M with zb ≤ y.

Proof. Use 4.5, where α = 0M (see also 4.6).

4.12 Theorem. The algebraic structure M(∗,) is a left S-semimodule if andonly if the conditions 4.8(1) and 4.11(1) are satisfied.

Proof. See 4.8, 4.10 and 4.11.

4.13 Proposition. (cf. 3.10) Assume that w0a0 = 0M for some a0 ∈ S andw0 ∈ M \ 0M. Then SM = M(∗,) is a left S-semimodule if and only if thefollowing three conditions are satisfied:

1. For all x, y ∈ M \ 0M there is z ∈ M \ 0M such that z ≤ x and z ≤ y;

2. If a ∈ S is such that 0Ma = 0M then, for every x ∈ M \ 0M, there is atleast one y ∈ M \ 0M such that ya ≤ x;


3. If a ∈ S is such that 0Ma = 0M then there is at least one v ∈ M \ 0Msuch that 0Ma = va.

Proof. See 4.9, 4.10 and 4.11 (clearly, (2) and (3) are equivalent to 4.11(1) underour assumptions).

4.14 Proposition. Assume that xa = 0M for all a ∈ S and x ∈ M \ 0M (i.e.,L = M \0M is a subsemimodule of MS). Then M(∗,) is a left S-semimodule.

Proof. See 4.8, 4.10 and 4.11.

4.15 Lemma.

(i) S 0M = 0M.(ii) If oM ∈ M then S oM = oM.(iii) If x ∈ M \ 0M then S x = x iff xS ≤ x and yz x for all a ∈ S,

y ∈ M , y x.

4.16 Lemma. Let a, b ∈ S be such that a x = b x for every x ∈ M . Ify ∈ M \ 0M is such that ya = 0M = yb then ya = yb.

Proof. We have a ya = a ya = b ya and a ya ≥ y > 0M . Thusb ya = b ya ≥ y and yb ≥ ya. Symmetrically, ya ≥ yb, so that ya = yb.

4.17 Corollary. Assume that SM is a semimodule and for all a, b ∈ S, a = b,there is x ∈ M \0M such that 0M = xa = xb = 0M . Then the left S-semimodule

SM is faithful.

4.18 Lemma. Let a, b ∈ S be such that a x = b x for every x ∈ M . If0Ma = 0M = a 0Ma and 0Mb = 0M = b 0Mb then 0Ma = 0Mb.

Proof. We can proceed similarly as in the proof of 4.16.

4.19 Lemma. Let a ∈ S and x ∈ M \ 0M be such that Ma ≤ x. Then:

(i) a x = oM ∈ M .

(ii) If S x = oM then oMS ≤ x.

Proof. It is easy.

4.20 Corollary. Assume that the semiring S is simple, SM is a semimodule andthere are a, b ∈ S and x ∈ M \ 0M such that Ma ≤ x and oMb x. Then theleft S-semimodule SM is faithful.

4.21 Lemma. Let x ∈ M \ 0M be such that Max = x for some ax ∈ S.Then:

(i) ax y = oM ∈ M for every y ∈ M , x ≤ y (or x ∗ y = x).

(ii) ax z = 0M for every z ∈ M , x z (or x ∗ z = x).

(iii) ax M = oM , 0M.

4.22 Lemma. Let a ∈ S be such that Ma = 0M. Then:(i) a x = oM ∈ M for every x ∈ M \ 0M.(ii) a 0M = 0M .


4.23 Proposition. Assume that SM = M(∗,) is a left S-semimodule and thatfor every x ∈ M \0M there is ax ∈ S with Max = x. Then the left semimodule

SM is simple.

Proof. We can proceed similarly as in the proof of 3.9(iv) (where ω is replacedby 0M).

4.24 Lemma. Assume that the set L = M \ 0M has the smallest element w.Let a, b ∈ S be such that Ma = 0M and Mb = w. Then:(i) a = b and a x = b x for every x ∈ M .

(ii) If SM is a left S-semimodule then it is not faithful.

4.25 Proposition. Assume that SM is a faithful left S-semimodule and that forevery x ∈ M there is ax ∈ S with Max = x. Then both S and M are infiniteand the set L = M \ 0M has no minimal element.

Proof. Let w ∈ L be minimal in L. We have wa0 = 0 (= 0M) and it followsfrom 4.13(1) that w is the smallest element of L. Now, SM is not faithful due to4.24(ii).

4.26 Remark.

(i) Assume that the conditions 4.13(1),(2),(3) are satisfied. Then SM = M(∗,)is a left S-semimodule (see 4.13, 4.14) and, by 3.10, the set L+ =(M \ 0M)∪ω is a subsemimodule of the left S-semimodule SM

+. Now,it is easy to see that the mapping x 7→ x, x ∈ L = M \ 0M and 0M 7→ ωis an isomorphism of the semimodule SM onto the semimodule SL

+.

(ii) Assume that SM is a (left S-)semimodule. Then either all the three condi-tions 4.13(1),(2),(3) are true or 0M /∈ LS and L = M \ 0M is a subsemi-module of the right semimodule MS.

5. From the right to the left (d)

The second section is continued. Here, we assume that 0M ∈ M (then W1 =M ×M) and that W1 = W2 (then W2 = M ×M and M(+, ∗) is a lattice – see4.1). Furthermore, assume that W ′

3 = (x, a) ∈ W3 |x = 0M = (x, a) |x = 0M ,0Ma ≤ x ⊆ W4. Similarly as in the fourth section, we put a x = a x for(x, a) ∈ W ′

3 and a x = 0M for (x, a) ∈ (S ×M) \W ′3.

5.1 Remark. Let (x, a) ∈ W3\W ′3. Then x = 0M , Q = Q0M ,a=y | ya = 0M = ∅

and we get 0M ∈ Q. If Q = 0M then oQ = 0M and (0M , a) ∈ W4. Of course,Q ⊆ Qz,a for every z ∈ M .

If oQ /∈ Q (i.e., (0M , a) /∈ W4) then Q = Qu,a for every u ∈ M \ 0M andthere is vu ∈ Qu,a such that 0M = vua ≤ u, vu = 0M . We have shown that forevery u ∈ L = M \ 0M there is v ∈ L such that va ≤ u, va ∈ L.

5.2 Lemma. a (x ∗ y) = (a x) ∗ (a y) for all a ∈ S and x, y ∈ M iff thecondition 4.8(1) is satisfied.


Proof. If 0M ∈ x, y then a (x ∗ y) = a 0M = 0M = (a x) ∗ (a y).Assume, therefore, that x = 0M = y.

If wa ≤ x and wa ≤ y for some w ∈ M \ 0M then wa ≤ x ∗ y, w ≤ a x,w ≤ a y and w ≤ (a x) ∗ (a y). Moreover, (a x)a ≤ x, (a y)a ≤ y, andhence ((ax)∗(ay))a ≤ x∗y. Then (ax)∗(ay) = a(x∗y), provided thatx∗y = 0M . On the other hand, if x∗y = 0M then ax)∗y) = 0M = (ax)∗(ay).

If wa x∗y for every w ∈ M \0M then a(x∗y) = 0M = (ax)∗(ay).

5.3 Lemma. (a+ b) x = (a x) ∗ (b x) for all a.b ∈ S and x ∈ M .

Proof. Easy to check.

5.4 Lemma. (ab) x = a (b x) for all a, b ∈ S and x ∈ M iff the condition4.11(1) is satisfied.

Proof. Easy to check.

5.5 Theorem. The algebraic structure SM = M(∗,) is a left S-semimodule ifand only if the conditions 4.8(1) and 4.11(1) are satisfied.

Proof. Combine 5.2, 5.3 and 5.4.

5.6 Proposition. Assume that w0a0 = 0M for some a0 ∈ S and w0 ∈ L =M \ 0M (equivalently, L is not a subsemimodule of MS). Then SM = M(∗,)is a left S-semimodule if and only if the conditions 4.13(1),(2),(3) are satisfied.

Proof. See the proof of 4.13.

5.7 Lemma.

(i) S 0M = 0M.(ii) If oM ∈ M then S oM = oM.

5.8 Proposition. Assume that the semiring S is simple, SM is a semimoduleand that there are a, b ∈ S and x ∈ M \ 0M such that Ma ≤ x and oMb x.Then the left S-semimodule SM is faithful.

Proof. See 4.19 and 4.20.

5.9 Lemma. Let x ∈ M \ 0M be such that Max = x for some ax ∈ S. Then:

(i) ax y = oM ∈ M for every y ∈ M , x ≤ y (or x ∗ y = x).

(ii) ax z = 0M for every z ∈ M , x z) (or x ∗ z = x).

5.10 Lemma. Let a ∈ S be such that Ma = 0M. Then:(i) a x = oM ∈ M for every x ∈ M \ 0M.(ii) a 0M = 0M .

5.11 Proposition. Assume that SM = M(∗,) is a left S-semimodule andthat for every x ∈ M \ 0M there is ax ∈ S with Max = x. Then the leftS-semimodule SM is simple.



5.12 Proposition. Assume that SM is a faithful left S-semimodule and that forevery x ∈ M there is ax ∈ S with Max = x. Then both S and M are infiniteand the set M \ 0M has no minimal element.


6. A few conditions

Let M (= M(+)) be a non-trivial semilattice and let x ≤ y iff x + y = y. LetN = M \ oM (N = M iff oM /∈ M) and N ′ = N \ oN (N ′ = N iff oN /∈ N).Consider the following conditions:(C1) If x1 < x2 < x3 < . . . is an infinite strictly increasing sequence of elements

from M then for every x ∈ N ′ there is i ≥ 1 with x ≤ xi;

(C2) If x1 < x2 < x3 < . . . is an infinite strictly increasing sequence of elementsfrom M then for every x ∈ N there is i ≥ 1 with x ≤ xi;

(C3) If x1 < x2 < x3 < . . . is an infinite strictly increasing sequence of elementsfrom M then for every x ∈ M there is i ≥ 1 with x ≤ xi;

(C4) There is no infinite strictly increasing sequence of elements from M .Clearly, (C4) implies (C3), (C3) implies (C2) and (C2) implies (C1). If M is

finite then (C4) is true. If oM /∈ M then (C1), (C2) and (C3) are equivalent. IfoN /∈ N then (C1) and (C2) are equivalent. If oM ∈ M then (C3) and (C4) areequivalent. Finally, if oN ∈ N then (C2), (C3) and (C4) are equivalent.

6.1 Lemma. Let MS be a right S-semimodule satisfying (C1). Then W1 = W2.

Proof. Let x, y ∈ M and P = Px,y. If x ≤ y or y ≤ x then P = ∅ and oP ∈ Ptrivially. Assume, therefore, that x y and y x. Then x, y ∈ N ′. Now, ifz1 < z2 < z3 < . . . is an infinite strictly increasing sequence of elements fromP then x ≤ zi for some i ≥ 1, so that x = zi and zi = zi+1, a contradiction.Consequently, P satisfies (C4) and for every z ∈ P there is v ∈ P such that z ≤ vand v is maximal in P . But v + P ⊆ P , and so v = oP ∈ P .

6.2 Lemma. Let MS be a right S-semimodule satisfying (C1) and let a ∈ S andx ∈ M be such that Q = Qx,a = ∅ and oQ ∈ Q. Then:(i) Q = N ′ (i.e., N ′a ≤ x).

(ii) The set N ′ has no maximal element.

(iii) N ′ +N ′ = N ′ and N +N = N .

(iv) If oN ∈ N then M does not satisfy (C2).

(v) If oM ∈ M then M does not satisfy (C3).

(vi) M does not satisfy (C4).

Proof. If v ∈ Q is maximal in Q then v + Q ⊆ Q implies v = oQ ∈ Q, acontradiction. Therefore, the set Q has no maximal element and (C1) impliesN ′ ⊆ Q. Of course, oM /∈ Q (otherwise oM = oQ), and hence oN /∈ Q either. ThusQ = N ′ and the rest is clear.

6.3 Lemma. Let MS be a right S-semimodule satisfying (C1) and 0M ∈ M . Leta ∈ S be such that 0Ma = 0M and oQ /∈ Q = Q0M ,a. Then:


(i) N ′a = 0M.(ii) If oM ∈ M and oN ∈ N then Ma = 0M , oNa, oMa.(iii) If oM ∈ M and oN /∈ N then Na = 0M and Ma = 0M , oMa.(iv) If oM /∈ M then Ma = 0M.Proof. Use 6.2.

6.4 Proposition. Let MS be a right S-semimodule satisfying (C1) and oM /∈ M .Then:

(i) M satisfies (C3).

(ii) M does not satisfy (C4).

(iii) W1 = W2.

(iv) W3 = W4 if and only if for all a ∈ S and x ∈ M there is at least one y ∈ Mwith ya x.

(v) If 0M ∈ M and W ′3 ⊆ W4 then W3 = W4.

Proof. Combine 6.1 and 6.2.

6.5 Proposition. Let MS be a right S-semimodule satisfying (C1) and such thatoM ∈ M and oN /∈ N . Then:

(i) M satisfies (C2).

(ii) M satisfies (C3) if and only if M satisfies (C4).

(iii) W1 = W2.

(iv) W3 = W4, provided that either x + y = oM for some x, y ∈ N or the set Nhas at least one maximal element.

(v) If 0M ∈ M and W ′3 ⊆ W4 then W3 = W4.


6.6 Proposition. Let MS be a right S-semimodule satisfying (C1) and such thatoM ∈ M and oN ∈ N . Then:

(i) M satisfies (C2) iff M satisfies (C3) iff M satisfies (C4).

(ii) W1 = W2.

(iii) W3 = W4, provided that either x + y = oN for some x, y ∈ N or the set N ′

has at least one maximal element.

(iv) If 0M ∈ M and W ′3 ⊆ W4 then W3 = W4.


7. One example

In this section, let S be a semiring such that |R(S)| ≥ 2. Then R(S)S is a non-trivial right S-semimodule and the semiring S is non-trivial. In fact, R(S) is thesmallest (right) ideal of S.


7.1 Proposition. If the semiring S is simple then the right semimodule R(S)Sis faithful.

Proof. If R(S)S is not faithful then, using the fact that S is simple, we seethat ab = ac for all a ∈ R(S) ans b, c ∈ S. In particular, if b, c ∈ R(S) thenb = bb = bc = c, a contradiction.

If a, b ∈ R(S) then Pa,b=c ∈ R(S) | a+c=a, b+c=b. If a ∈ S and b ∈ R(S)then Qb,a= c ∈ R(S) | ca+b=b. As we have already defined in the second partof this note, we put W1=(a, b) |Pa,b = ∅, W2=(a, b) ∈ W1 | oP ∈ P = Pa,b,W3 = (b, a) |Qb,a = ∅ and W4 = (ba) ∈ W3 | oQ ∈ Q = Qba.7.2 Assume that W1 = W2 and W3 = W4.

7.2.1 Theorem. The algebraic structure 1SR(S) = R(S)+(∗, ) is a simple left

S-semimodule.

Proof. See 3.2 and 3.9(iv).

7.2.2 Proposition.

(i) The left semimodule 1SR(S) is faithful if and only if the right semimodule

R(S)S is so.

(ii) If the semiring S is simple then the left semimodule 1SR(S) is faithful.


7.2.3 Proposition. o = oR(S) ∈ R(S).

Proof. a a = oR(S) for every a ∈ R(S).

7.2.4 Proposition.

(i) S ω = ω.(ii) S o = o.(ii) If a ∈ R(S) \ o then S a = a.

Proof. See 3.5.

7.2.5 Proposition. Let a ∈ R(S). Then:

(i) a b = o for every b ∈ R(S) such that a+ b = b.

(ii) a a = o.

(iii) a c = ω for every c ∈ R(S) such that a+ c = c.

(iv) a ω = ω.

(v) a R(S)+ = o, ω.Proof. See 3.9.

7.2.6 Proposition. Assume that 0 = 0R(S) ∈ R(S) and put L = R(S) \ 0.Then:

(i) W1 = W2 = R(S)×R(S).

(ii) W3 = W4 = (b, a) | a ∈ S, b ∈ R(S), 0a+ b = b .


(iii) L+ = L ∪ ω is a subsemimodule of the left semimodule 1SR(S) iff the

following three conditions are satisfied:

1. For all a, b ∈ L there is at least one c ∈ L with c+ a = a and c+ b = b;2. If a ∈ S is such that 0a = 0 then for every b ∈ L there is at least one

c ∈ L with ca+ b = B;3. If a ∈ S is such that 0a = 0 then there is at least one b ∈ L with

0a = ba.

(iv) If L+ is a faithful semimodule then the set L has no minimal element.


7.3 Assume that 0 = 0R(S) ∈ R(S) and that (W1 =) W2 = R(S) × R(S). Fur-thermore, put α = 0 and assume that W ′

3 ⊆ W4 (see 7.2.6 and the fifth section ofthis note).

7.3.1 Theorem. The algebraic structure 2SR(S) = R(S)(∗,) is a left S-semi-

module if and only if the three conditions 7.2.6(iii)(1),(2),(3) are satisfied.

Proof. See 5.6 and 7.2.6(iii).

In the rest of 7.3, assume that 2SR(S) is a left semimodule (see 7.3.1).

7.3.2 Proposition. The left semimodule 2SR(S) is simple.

Proof. See 4.23.

7.3.3 Proposition. The semimodule 2SR(S) is faithful in each of the following

two cases:

(1) For all a, b ∈ S, a = b, there is at least one c ∈ R(S) with c = 0 = ca =cb = 0.

(2) |R(S)| ≥ 3 and the semiring S is simple.


7.3.4 Proposition. If the left semimodule 2SR(S) is faithful then R(S) is infinite

and the set L = R(S) \ 0 has no minimal element.

Proof. See 4.25.

7.3.5 Proposition. Let a ∈ R(S), a = 0. Then:

(i) a b = o = oR(S) ∈ R(S) for every b ∈ R(S) such that a+ b = b.

(ii) a a = o.

(iii) a c = 0 for every c ∈ R(S) such that a+ c = c.

(iv) aR(S) = 0, o.

Proof. See 4.21.


7.3.6 Proposition.

(i) 0 a = o for every a ∈ R(S) \ 0.(ii) 0 0 = 0.

Proof. See 4.22.

7.3.7 Remark. By 7.2.6(iii), if the conditions (1),(2),(3) hold then the set L+ =L∪ω, L = R(S)\0, is a subsemimodule of the left semimodule 1

SR(S). Now,the mapping a 7→ a for a ∈ L and 0 7→ ω is an isomorphism of 2

SR(S) onto SL+.

7.4 Proposition. Assume that oR(S) /∈ R(S) and that the right semimoduleR(S)S satisfies (C1). Then:

(i) R(S) satisfies (C2) and (C3) and does not satisfy (C4).

(ii) W1 = W2.

(iii) W3 = W4.

Proof. See 6.4 (we have (a, b) ∈ W3 \W4 for all a, b ∈ R(S), a ≤ b).

7.5 Proposition. Assume that o = oR(S) ∈ R(S), oT /∈ T , T = R(S) \ o andthat R(S)S satisfies (C1). Then:

(i) R(S) satisfies (C2).

(ii) W1 = W2.

(iii) W3 = W4, provided that either a+ b = o for some a, b ∈ T or the set T hasat least one maximal element.

Proof. See 6.5.

7.6 Proposition. Assume that o = oR(S) ∈ R(S), oT ∈ T = R(S) \ oS andthat R(S)S satisfies (C1). Then:

(i) W1 = W2.

(ii) W3 = W4, provided that either a+ b = oT for some a, b ∈ T \ o or the setT \ oT has at least one maximal element.

Proof. See 6.6.

References

[1] El Bashir, R., Hurt, J., Jancarık, A., Kepka, T., Simple commutativesemirings, J. Algebra, 263 (2001), 277–306.

[2] El Bashir, R., Kepka, T., Congruence-simple semirings, Semigroup Fo-rum, 75 (2007), 588–608.

[3] Hebisch, V., Weinert, H.J., Halbringe – Algebraische Theorie und An-wendungen in der Informatik, Teubner, Stuttgart, 1993.


[4] Kendziorra, A., Zumbragel, J., Finite simple additively idempotentsemirings, J. Algebra, 388 (2013), 43–64.

[5] Mitchell, S.S., Fenoglio, P.B., Congruence-free commutative semirings,Semigroup Forum, 37 , (1988), 79–91.

[6] Monico, C., On finite congruence-simple semirings, J. Algebra, 271 (2004),846–854.

[7] Zumbragel, J., Classification of finite congruence-simple semirings withzero, J. Algebra Appl., 7 (2008), 363–377.



TOWARD A NEW ALGORITHM FOR SYSTEMS OF FRACTIONALDIFFERENTIAL-ALGEBRAIC EQUATIONS

H.M. Jaradat1

M. Zurigat

Safwan Al-Shara’

Department of MathematicsAl al-Bayt UniversityJordan

Qutaibeh Katatbeh

Department of Mathematics and StatisticsJordan University of Science & TechnologyJordan

Abstract. This paper is concerned with the development of an efficient algorithm forthe analytic solutions of systems of fractional differential -algebraic equations (FDAE).The proposed algorithm is an elegant combination of the Laplace transform method(LTM) with the homotopy analysis method (HAM). The biggest advantage of theLaplace homotopy analysis method (LHAM) over the existing standard analytical tech-niques is that it overcomes the difficulty arising in calculating complicated terms. Nume-rical examples are examined to highlight the significant features of this method.

Keywords: analytic solution; Laplace transform; HAM; fractional differential-algebraicequations.

Introduction

The fractional calculus has a long history from 30 September 1695, when thederivative of order α = 1/2 has been described by Leibniz [18], [21]. The theoryof derivatives and integrals of non-integer order goes back to Leibniz, Liouville,Grunwald, Letnikov and Riemann.There are many interesting books about frac-tional calculus and fractional differential equations [18], [21], [6], [22]. Differentialequations of fractional order have been found to be effective to describe somephysical phenomena such as rheology, damping laws, fluid flow and so on [15],

1Corresponding author. E-mail: [email protected]; Tel:+962777719675;Fax: +962(5)3903349.

580 h.m. jaradat, m. zurigat, safwan al-shara’, qutaibeh katatbeh

[11]. Recently, many important mathematical models can be expressed in termsof systems of algebraic differential equations of fractional order.

Derivatives of non-integer order can be defined in different ways, e.g. Riemann–Liouville, Grunwald–Letnikow, Caputo and Generalized Functions Approach [21].In this paper we focus attention on Caputo’s definition which turns out to bemore useful in real-life applications since it can be coupled with initial conditionshaving a clear physical meaning.

DAEs arise in the mathematical modeling of a wide variety of problems fromengineering and science such as in multibody and flexible body mechanics, elec-trical circuit design, optimal control, incompressible fluids, molecular dynamics,chemical kinetics (quasi steady state and partial equilibrium approximations), andchemical process control. The index concept plays an important role in the ana-lysis of DAEs. The index is a measure of the degree of difficulty in the numericalsolution. In general, the higher the index is, the more difficult it is to solve theDAE. While many different concepts exist to assign an index to a DAE such asthe differentiation index [3], [5], the perturbation index [4], the tractability in-dex [23], and the nilpotency index [4]. In the case of linear DAEs with constantcoefficients, all these indices are equal. In order to transform a DAE into an alter-native form easier to solve, some index reduction methods have been developed[10]. These methods introduce additional variables, which leads to a drawbackthat the resulting DAE is a larger system than the original one.

There are only a few techniques for the solution of fractional differential-algebraic equations, since it is relatively a new subject in mathematics. The so-lution of fractional differential equations is much involved. In general, there existsno method that yields exact solutions for fractional differential equations. Onlyapproximate solutions can be derived using linearization or perturbation method.In recent years, much research has been focused on the numerical solution of sys-tems of ordinary differential equations and algebraic differential equations. Somenumerical methods have been developed, such as implicit Runge Kutta method [1],Pade approximation method [12], [7], homotopy perturbation method (HPM) [16],[20], Adomian decomposition method (ADM) [8], [9], variation iteration method(VIM) [19], [17] and homotopy analysis method HAM [25].

The ADM and VIM are limited in that the former has complicated algorithmsin calculating Adomian polynomials for nonlinear problems, and the later has aninherent inaccuracy in identifying the Lagrange multiplier for fractional operators,which is necessary for constructing variational iteration formula. The HPM isindeed a special case of the HAM [13], [14], [2]. However, mostly, the resultsgiven by HPM converge to the corresponding numerical solutions in a rathersmall region. Although the HAM provides us with a simple way to adjust andcontrol the convergence region of solution series by choosing a proper value for theauxiliary parameters h, we face the difficulty in calculating complicated integralsthat arise when dealing with strongly nonlinear problems.

Therefore, in this work we will introduce a new alternative procedure to eli-minate these disadvantages in solving FDAE. The newly developed technique byno means depends on complicated tools from any field. This can be the most

toward a new algorithm for systems of fractional ... 581

important advantage over the other methods. It is worth mentioning that theproposed algorithm is an elegant combination of Laplace transform method andthe homotopy analysis method. Some FDAE are examined to illustrate the effec-tiveness, accuracy and convenience of this method. and in all cases, the presentedtechnique performed excellently.

1. Preliminaries and notations

In this section, let us recall essentials of fractional calculus. The fractional calcu-lus is a name for the theory of integrals and derivatives of arbitrary order, whichunifies and generalizes the notions of integer-order differentiation and n-fold in-tegration. For the purpose of this paper the Caputo’s definition of fractionaldifferentiation will be used, taking the advantage of Gaputo’s approach that theinitial conditions for fractional differential equations with Caputo’s derivativestake on the traditional form as for integer-order differential equations.

Definition 1.1 Caputo’s definition of the fractional-order derivative is defined as

Dαf(t) =1

Γ(n− α)

t∫

a

(t− τ)n−α−1f (n)(τ)dτ,

where n− 1 < α ≤ n, n ∈ N, α is the order of the derivative.

For the Caputo’s derivative we have:

DαC = 0, C is constant,

Dαtβ =

0 β ≤ α− 1,

Γ(β + 1)

Γ(β − α + 1)tβ−α β > α− 1.

Caputo’s fractional differentiation is a linear operation and if f(τ) is continuousin [a, t] and g(τ) has n + 1 continuous derivatives in [a, t], it satisfies the so-calledLeibnitz rule:

Dα(f(t)g(t)) =∞∑

k=0

(α

k

)g(k)(t)Dα−kf(t)

For establishing our results, we also necessarily introduce the following Riemann–Liouville fractional integral operator.

Definition 1.2 A real function f(x), x > 0, is said to be in the space Cµ, µ ∈ R ifthere exists a real number p > µ such that f(x) = xpf1(x),where f1(x) ∈ C[0,∞).Clearly Cµ ⊂ Cβ if β ≤ µ.


Definition 1.3 The Riemann–Liouville fractional integral operator of order α ≥ 0,of a function f ∈ Cµ, µ ≥ −1, is defined as

Jαf(t) =1

Γ(α)

t∫

a

(t− τ)α−1f(τ)dτ

We mention only some properties of the operator Jα:For f ∈ Cµ, µ, γ ≥ −1, α, β ≥ 0 :

1. J0(t) = f(t),

2. JαJβf(t) = Jα+βf(t) = JβJαf(t),

3. Jαtγ =Γ(γ + 1)

Γ(γ + α + 1)tγ+α, α > 0, γ > −1, t > 0.

Also, we need here two of its basic properties. If m − 1 < α ≤ m, m ∈ N, andf ∈ Cm

µ , µ ≥ −1, then

DαJαf(t) = f(t), JαDαf(t) = f(t)−m−1∑i=0

f (i)(0+)ti

i!, t > 0.

For more information on the mathematical properties of fractional derivativesand integrals, one can consult [22].

Lemma 1.4 If m− 1 < α ≤ m, m ∈ N, and f ∈ Cmµ , µ ≥ −1, then the Laplace

transform of the fractional derivative Dαf(t) is

£(Dαf(t)) = sαF (s)−m−1∑i=1

f (i)(0+)sα−i−1, t ≥ 0

Here £(f(t)) = F (s); for more details, see [26].

2. Laplace Homotopy analysis method

In this section, we employ the Laplace homotopy analysis method to the discussedproblem. To show the basic idea, let us consider the FDAEs

Dαiui(t) = fi(t, u1, u2, ..., un, u′1, u

′2, ..., u

′n)(2.1a)

0 = g(t, u1, u2, ..., un), i = 1, 2, ..., n− 1, 0 < αi ≤ 1,(2.1b)

subject to the initial conditions

ui(0) = ai, i = 1, 2, ..., n

where fi are known analytical functions.


Applying the Laplace transform to both sides of (2.1a) and using the linearityof Laplace transforms, we get

£[Dαiui(t)] = £[fi(t, u1, u2, ..., un, u′1, u

′2, ..., u

′n)], i = 1, 2, ..., n− 1, 0 < αi ≤ 1,

0 = g(t, u1, u2, ..., un)

Using Lemma 1.4, and applying the formulas of the Laplace transform, we get

Ui(s) =ai

s+

1

sαi£[fi(t, u1, u2, ..., un, u′1, u

′2, ..., u

′n)],

0 = g(t, u1, u2, ..., un), i = 1, 2, ..., n− 1, 0 < αi ≤ 1,(2.2)

where Ui(s) = £(ui(t)).The so-called zeroth-order deformation equations of equations (2.2) are

(2.3)

(1− q)Li[Φi(s, q)− Ui,0(s)] = qhi[Φi(s, q)− ai

s

− 1

sαi£[fi(t, φ1(t; q), ..., φn(t; q),

∂

∂tφ1(t; q), ...,

∂

∂tφn(t; q))],

i = 1, 2, ..., n− 1,

(1− q)Ln[φn(t; q)− un,0(t)] = −qhng(t, φ1(t; q), ..., φn(t; q)),

where q ∈ [0, 1] is an embedding parameter, when q = 0 and q = 1, we have

Φi(s, 0) = Ui,0(s), Φi(s, 1) = Ui(s), i = 1, 2, ..., n− 1,

φn(t; 0) = un,0(t), φn(t; 1) = un(t).

Expanding Φi(s, q), i = 1, 2, ..., n − 1 and φn(t; q) in Taylor series with respectto q, we get

(2.4)

Φi(s; q) = Ui,0(s) +∞∑

m=1

Ui,m(s)qm, i = 1, 2, ..., n− 1,

φn(t; q) = un,0(t) +∞∑

m=1

un,m(t)qm,

where

Ui,m(s) =1

m!

∂mΦi(s; q)

∂qm|q=0, i = 1, 2, ..., n− 1,

un,m(t) =1

m!

∂mφn(t; q)

∂qm|q=0.

If the initial guesses, the auxiliary linear operator Li, i = 1, 2, ..., n and the nonzeroauxiliary parameter hi are properly chosen so that the power series (2.4) convergesat q = 1.


Then, we have under these assumptions the solution series

Ui(s) = Φi(s; 1) = Ui,0(s) +∞∑

m=1

Ui,m(s), i = 1, 2, ..., n− 1,

un(t) = φn(t; 1) = un,0(t) +∞∑

m=1

ui,m(t)

For brevity, define the vector

−→U i,m(s) = Ui,0(s), Ui,1(s), Ui,2(s), ..., Ui,m(s), i = 1, 2, ..., n− 1,−→u n,m(t) = un,0(t), un,1(t), um,2(t), ..., un,m(t),

Differentiating the zero-order deformation equation (2.3) m times with respectiveto q and then dividing by m! and finally setting q = 0, we have the so-calledhigh-order deformation equation

Ui,m(s) = χmUi,m−1(s) + hi<i,m(−→U i,m−1(s)), i = 1, 2, ..., n− 1,(2.5)

un,m(t) = χmun,m−1(t) + hn<n,m(−→u n,m−1(t))

where

<i,m(−→U i,m−1(s)) = Ui,m−1(s)− 1

sαi

[1

(m− 1)!

∂m−1

∂qm−1(£[fi(t, φ1(t; q), ..., φn(t; q),

∂

∂tφ1(t; q), ...,

∂

∂tφn(t; q))])|q=0

]− ai

s(1− χm),

<n,m(−→u n,m−1(t)) =−1

(m− 1)!

∂m−1

∂qm−1[g(t, φ1(t; q), ..., φn(t; q))]|q=0, i = 1, 2, ..., n− 1,

and

χm =

0, m ≤ 1,

1, m > 1.

Finally, applying the inverse Laplace transforms of (2.5), then we have apower series solution

(2.6) ui(t) =∞∑

m=0

ui,m(t), i = 1, 2, ..., n.

Note that we have great freedom to choose the value of the auxiliary parameterhi. Mathematically, the value of ui(t) at any finite order of approximation isdependent upon the auxiliary parameter hi, because the zeroth and high orderdeformation equations contain hi. Let Rhi

denote the set of all values of hi whichensure the convergence of the LHAM series solution (2.6) of ui(t). Let hi be thevariable of the horizontal axis and the limit of the series solution (2.6) of ui(t)be the variable of vertical axis. Plot the curve ui(t) ∼ hi, where ui(t) denotesthe limit of the series (2.6). Because the limit of all convergent series solutions


(2.6) is the same for a given a, there exists a horizontal line segment above theregion h ∈ Rhi

. So, by plotting the curve ui(t) ∼ hi at a high enough orderapproximation, one can find an approximation of the set Rhi

[13].

3. Applications

In this part, we introduce some applications on LHAM to solve differential-algebraic equations with fractional derivatives.

Example 1. Consider the following fractional differential-algebraic equations

(3.1)Dαu(t) = −2tu2(t) + u(t)v(t)− 1, 0 < α ≤ 1,

v(t) = u(t)v(t) + t2, u(0) = v(0) = 1,

for α = 1, the exact solution u(t) =1

t2 + 1, v(t) = t2 + 1.

To derive the solution, take the Laplace transform of both sides of (3.1),we get

£[Dαu(t)] = −2£(tu2(t)) + £(u(t)v(t))− 1

s,

v(t) = u(t)v(t) + t2,

so

sαU(s)− sα−1 = −2£(tu2(t)) + £(u(t)v(t))− 1

s,

v(t) = u(t)v(t) + t2,

or

U(s) = − 2

sα£(tu2(t)) +

1

sα£(u(t)v(t)) +

(1

s− 1

sα+1

),

v(t) = u(t)v(t) + t2.

Hence, the mth-order deformation equations can be given by

Um(s) = χmUm−1(s) + h1<1,m(−→U m−1(s)),(3.2)

vm(t) = χmvm−1(t) + h2<2,m(−→v m−1(t)), m = 1, 2, 3, ...(3.3)

subject to the initial condition

um(0) = vm(0) = 0,m = 1, 2, 3, ....

where

<1,m(−→U m−1(s)) = Um−1(s) +

2

sα£

(t

m−1∑i=0

ui(t)um−i−1(t)

)

− 1

sα£

(m−1∑i=0

ui(t)vm−i−1(t)

)−

(1

s− 1

sα+1

)(1− χm),

<2,m(−→v m−1(t)) = vm−1(t)−m−1∑i=0

ui(t)vm−i−1(t))− t2(1− χm), m = 1, 2, 3, ...


According to the initial condition in (3.1), we can choose the initial guess ofU(s) and v(t) as follows:

U0(s) =1

s, v0(t) = 1

Using the mth-order deformation equations (3.2), (3.3)we can find that

U0(s) =1

s,

u0(t) = 1

U1(s) =2h1

sα+2,

u1(t) =2h1

Γ(2 + α)tα+1

U2(s) =2h1

sα+2+

2h21

sα+2+

8h21(2 + α)

s2α+3+

2h1h2

sα+3− 2h2

1

s2α+2,

u2(t) =2h1

Γ(2 + α)tα+1 +

2h21

Γ(2 + α)tα+1 +

8h21(2 + α)

Γ(3 + 2α)t2α+2 +

2h1h2

Γ(3 + α)tα+2

− 2h21

Γ(2 + 2α)t2α+1

...

v0(t) = 1

v1(t) = −h2t2

v2(t) = −h2t2 − 2h1h2

Γ(2 + α)tα+1

...

Then, using a mathematical software the solution, we successfully obtain

u(t) =∞∑

m=0

um(t), v(t) =∞∑

m=0

vm(t).

The convergence of these series is strongly depends upon the values of the auxiliaryparameters hi. In order to find the range of admissible values of hi, the hi−curvesare plotted in Fig.1 for the 8th-order of approximation. We can see that the rangefor values of h1, h2 are in the range −1.3 ≤ h1, h2 ≤ −0.4. Using h1 = h2 = −1,α = 1 we get

u0(t) = 1, u1(t) = −t2, u2(t) = t4, u3(t) = −t6, u4(t) = t8, ...

v0(t) = 1, v1(t) = t2, v2(t) = 0, v2(t) = 0, v2(t) = 0, ...

Proceeding in the same manner, we get that the solution u(t) and v(t) isgiven in series form by


u(t) = u0(t) +∞∑

m=1

um(t) = 1− t2 + t4 − t6 + t8 − t10 + · · ·

v(t) = v0(t) +∞∑

m=1

vm(t) = 1 + t2

and so in closed form

u(t) =1

t2 + 1, v(t) = t2 + 1

−2 −1.5 −1 −0.5 0

0.7

0.8

0.9

1

1.1

1.2

1.3

hi−2 −1.5 −1 −0.5 0 0.5

−6

−4

−2

0

2

4

6

8

10

12

hi

Fig.1. The hi-curve of 8th-order ap-proximation for u(0.3) and v(0.3),α = 1 of Example 1

Fig.2. The hi-curve of 8th-order ap-proximation for u(0.1) and v(0.1),α = 0.5 of Example 2

Example 2. Consider the following system of fractional differential-algebraicequations

Dαu(t) = tv′(t)− u(t) + (1 + t)v(t), 0 < α ≤ 1,(3.4)

v(t) = sin(t), u(0) = 1, v(0) = 0,

for α = 1, the exact solution u(t) = e−t + sin(t), v(t) = sin(t).Take the Laplace transform of both sides of (3.4) we get

U(s) =1

sα£(tv′(t))− 1

sαU(s) +

1

sV (s) +

1

sα£(tv(t)) +

1

s,

v(t) = sin(t),


Um(s) = χmUm−1(s) + h1<1,m(−→U m−1(s)),(3.5)

vm(t) = χmvm−1(t) + h2<2,m(−→v m−1(t)),m = 1, 2, 3, ...(3.6)


um(0) = vm(0) = 0.


where

<1,m(−→U m−1(s)) = Um−1(s)− 1

sα£(tv′m−1) +

1

sαUm−1(s)− 1

sVm−1(s)

− 1

sα£(tvm−1(t)) +

1

s(1− χm),

<2,m(−→v m−1(t)) = vm−1(t)− sin(t)(1− χm), m = 1, 2, 3, ...


U0(s) =1

s, v0(t) = 0

In order to find range of admissible values of hi, the hi-curve is plotted for 8th-order approximation when see Fig. 2. We can see that the range of values for hi

is between −1.3 ≤ hi ≤ −0.4, using the mth-order deformation equations (3.5),(3.6)and h1 = h2 = −1, we can find that

(3.7)

u(t) = u0(t) +∞∑

m=1

um(t) = 1 +1

Γ(1 + α)tα +

2

Γ(2 + α)tα+1

+

[ −2

Γ(1 + 2α)+

3.4−α√

π

Γ(1 + α)Γ(12

+ α)

]t2α

+2

Γ(3 + α)tα+2 − 2

Γ(2 + 2α)t2α+1 − 1

Γ(1 + 3α)t3α + · · ·

v(t) = v0(t) +∞∑

m=1

vm(t) = t− t3

6+

t5

120− t7

720+ · · ·

Setting α = 1 in (3.7), we obtain the following series solution

u(t) = u0(t) +∞∑

m=1

um(t) = 1− t +3t2

2− t3

3!− t4

8− t5

5!+

7t6

720− t7

7!− t8

5760

− t9

9!+

11t10

3628800− t11

11!+ · · ·

v(t) = v0(t) +∞∑

m=1

vm(t) = t− t3

3!+

t5

5!− t7

7!+

t9

9!− t11

11!+ · · ·

which is the exact solution for system (3.4), which better than solutions given byCelik, Bayram and Yeloglu [8] using Adomian decomposition method and thansolutions given by Zurigat , Momani, and Alawneh [25] using HAM. However, theresults given by the Adomian decomposition method converge to the correspon-ding numerical solutions in a rather small region. But in this method, the Laplacehomotopy analysis method gives a greater region of convergence with the exactsolution.


Fig. 3 shows the LHAM approximate solutions for various values of α whichhave the same trajectories.

0 2 4 6 8 10 12−12

−10

−8

−6

−4

−2

0

2

4

6

8

t

u(t)

Fig.3. Plots of solution of system (10) when h1 = h2 = −1.Solid line: α = 1, dotted line: α = 0.8, dash dotted line: α = 0.5

Example 3. Consider the following system of fractional differential-algebraicequations

Dαu1(t) = u1(t)− u2(t)v(t) + sin(t) + t cos(t), 0 < α ≤ 1,

Dαu2(t) = tv(t) + u21(t) + sec2(t)− t2(sin2(t) + cos(t))(3.8)

v(t) = u1(t) + t(cos(t)− sin(t)), u1(0) = u1(0) = v(0) = 0,

for α = 1, the exact solution u1(t) = t sin(t), u2(t) = tan(t), v(t) = t cos(t).

To derive the solution, we take the Laplace transform of both sides of (3.8)and we get

sαU1(s) = U1(s)−£(u2(t)v(t)) + £(sin(t) + t cos(t)),

sαU2(s) = £(tv(t)) + £(u21(t)) + £(sec2(t)− t2(sin2(t) + cos(t))),

v(t) = u1(t) + t(cos(t)− sin(t)),

or

U1(s) =1

sαU1(s)− 1

sα£(u2(t)v(t)) +

1

sα£(sin(t) + t cos(t)),

U2(s) =1

sα£(tv(t)) +

1

sα£(u2

1(t)) +1

sα£(sec2(t)− t2(sin2(t) + cos(t))),

v(t) = u1(t) + t(cos(t)− sin(t)).


Ui,m(s) = χmUi,m−1(s) + hi<i,m(−→U m−1(s)), i = 1, 2,

vm(t) = χmvm−1(t) + h3<3,m(−→v m−1(t)), m = 1, 2, 3, ...(3.9)



Ui,m(0) = vm(0) = 0, i = 1, 2,

where

<1,m(−→U 1,m−1(s)) = U1,m−1(s)− 1

sαU1,m−1(s)

+1

sα£

(m−1∑i=0

u2,i(t)vm−i−1(t)

)

− 1

sα£(sin(t) + t cos(t))(1− χm),

<2,m(−→U 2,m−1(s)) = U2,m−1(s)− 1

sα£(tvm−1(t))

− 1

sα£

(m−1∑i=0

u1,i(t)u1,m−i−1(t)

)

− 1

sα£(sec2(t)− t2(sin2(t) + cos(t)))(1− χm),

<3,m(−→v m−1(t)) = vm−1(t)− u1,m−1(t)− t(cos(t)− sin(t))(1− χm),

m = 1, 2, 3, ...


U1,0(s) = 0, U2,0(s) = 0, v0(t) = 0

The proper values of h1, h2, h3 found from the hi-curve shown in Fig.4, it is clearthat the series of ui(t), v(t) convergent when −1.4 ≤ hi ≤ −0.7, i = 1, 2, 3, if weset α = 1, h1 = h2 = h3 = −1 in (3.9), then we obtain the following series solution

u1(t) = u1,0(t) +∞∑

m=1

u1,m(t) = t2 − t4

6+

t6

120− t8

5040+

t10

362880+ · · ·

u2(t) = u2,0(t) +∞∑

m=1

um(t) = t +t3

3+

2t5

15+

17t7

315+

62t9

2835+ · · ·

v(t) = v0(t) +∞∑

m=1

vm(t) = t− t3

2+

t5

24− t7

720+

t9

40320+ · · ·

which the same solutions given by F. Soltanian, S.M. Karbassi, M.M. Hosseini[24] using He’s variational iteration method.


Figs. 5, 6 and 7 shows the LHAM approximate solutions for various valuesof α which have the same trajectories.

−2 −1.5 −1 −0.5 00

0.2

0.4

0.6

0.8

1

1.2

1.4

hi0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

u1(t

)

Fig.4. The hi-curve of 8th-order ap-proximation for u1(0.1), u2(0.4) andv(0.3), α = 0.75 of Example 3

Fig.5. Plots of solution of system(14) when h1 = h2 = −1. dottedline: exact solution when α = 1,solid line: α = 1, dash dotted line:α = 0.9, star line: α = 0.75

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

t0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t

Fig.6. Plots of solution of system(14) when h1 = h2 = −1. dotted-line: exact solution when α = 1,solid line: α = 1, dash dotted line:α = 0.9, star line: α = 0.75

Fig.7. Plots of solution of system(14) when h1 = h2 = −1. dotted-line: exact solution when α = 1,solid line: α = 1, dash dotted line:α = 0.9, star line: α = 0.75


4. Conclusion

A combined form of the Laplace transform method with Homotopy analysismethod is effectively used to handle linear and nonlinear fractional differential-algebraic equations. The main advantage of the method is its fast convergenceto the solution. Moreover, it avoids the volume of calculations that reguired byother existing analytical methods. In practice, the utilization of the method isstraightforward if some symbolic software as Matlab is used to implement thecalculations. The new method leads to higher accuracy and simplicity, and in allcases the solutions obtained are easily programmable approximants to the ana-lytic solutions of the original problems with the accuracy required. The proposedscheme can be applied for other nonlinear equations.

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A CHARACTERIZATION OF HIGHER DERIVATIONSON BANACH ALGEBRAS

T.L. Shatery

Department of Mathematics and Computer SciencesHakim Sabzevari UniversitySabzevarIrane-mail: [email protected]

S. Hejazian

Department of Pure MathematicsFerdowsi University of MashhadP.O.Box 1159, Mashhad 91775Irane-mail: [email protected]

Abstract. Let A be a Banach algebra and let every module-valued derivation from Ato any Banach A-bimodule be continuous. We show that if dm is a higher derivation

from A to a Banach algebra B with continuous d0, then there exist a continuous left

A-module homomorphism U : B(A1,B) → B and a sequence Dm of module-valued

derivations from A intoB(A1,B) such that dm = U Dm (m ≥ 1), and as a consequence

dm is automatically continuous. We also obtain a partial result concerning innerness

of higher derivations on W ∗-algebras.

Keywords: derivation, higher derivation, intertwining map, inner derivation.

2010 Mathematics Subject Classification: 46H40, 47B47.

1. Introduction

Let A and B be algebras. A family of linear mappings dmkm=0 (k might be ∞)from A into B is called a higher derivation of rank k if

dm(ab) =m∑j=0

dj(a)dm−j(b) (a, b ∈ A, m = 0, 1, 2, ..., k).

If there is no cause of ambiguity, a higher derivation will be simply denoted bydm. It is obvious that for a higher derivation dm, d0 is a homomorphism andd1 is a d0-derivation that is, d1(ab) = d0(a)d1(b)+d1(a)d0(b). A standard example

596 t.l. shatery, s. hejazian

of a higher derivation of rank k is Dm

m!km=0, where D : A −→ A is a derivation. A

higher derivation dm is said to be continuous if dm is continuous for all m ≥ 0.Higher derivations were introduced by Hasse and Schmidt [5] and algebraists

sometimes call them Hasse-Schmidt derivations. The reader may find more aboutthe algebraic properties of higher derivations in [1], [4], [5], [15], [18], [21], [19],[6], [12]. Loy [11], Jewell [9] and Villena [20] proved the automatic continuity ofhigher derivations in certain cases. In [7] and [8], the authors proved some resultsconcerning higher derivations on JB∗-algebras and Banach algebras. If dm is ahigher derivation from A to A such that d0 is the identity map on A, then d1 isa derivation and dm is called a strong higher derivation. In [10] Jun and Leeproved the Singer-Wrermer theorem for strong higher derivations. Mirzavaziri in[13] gives a characterization of a strong higher derivation defined on an algebra.

Let dm be a higher derivation from a Banach algebra A to a Banach algebraB. Define

a.x = d0(a)x, x.a = xd0(a) (a ∈ A, x ∈ B).(1.1)

Since d0 is a homomorphism, B is an A-bimodule with respect to the mappings

(a, x) → a.x, (a, x) → x.a, A× B → B.

It is easy to see that B is a Banach A-bimodule provided that d0 is continuous.In section 2 we give a characterization for higher derivations on certain Banachalgebras. We show that if every module-valued derivation on a Banach algebra Ais continuous, then each higher derivation dm fromA to a Banach algebra B withcontinuous d0, is of the form dm = U Dm (m ≥ 1), where U is a continuous leftA-module homomorphism and each Dm is a module-valued derivation. Thereforedm is continuous. As a consequence every higher derivation from a C∗-algebra,with continuous d0, is continuous. In section 3 we define an inner higher derivation.We show that if A is a commutative W ∗-subalgebra of a W ∗-algebra M, then eachstrong higher derivation from A to M is inner.

2. Characterization

LetA be a Banach algebra and X a BanachA-bimodule. A linear map S : A −→ Xis said to be left-intertwining if the map

b 7−→ aS(b)− S(ab), A −→ X ,

is continuous for each a ∈ A, and right-intertwining if the map

a 7−→ S(a)b− S(ab), A −→ X ,

is continuous for all b ∈ A. A linear map S : A −→ X is intertwining if it is bothleft- and right-intertwining. For more about these objects see [2, Section 2.7].

a characterization of higher derivations on banach algebras 597

Remark 2.1

(i) Let A and B be Banach algebras. Suppose that dm is a higher derivationfrom A into B for which d0 is continuous. Consider B as a Banach A-bimodule as in (1.1). Then it is easy to see that for every integer m ≥ 1,dm : A −→ B is an intertwining map whenever d0, ..., dm−1 are continuous.

(ii) Let A be a Banach algebra and X a Banach A-bimodule. Consider A1 =C ⊕ A to be the Banach algebra unitization of A. Even if A is unital,A1 = A and A1 is a unital Banach algebra containing A as a closed ideal.The identity (1, 0) of A1 will be denoted by 1. Set F = B(A1,X ), theBanach space of all bounded linear operators from A1 to X . For a ∈ A andT ∈ F , define

(a.T )(b) = aT (b), (T.a)(b) = T (ab) (b ∈ A1).

Then F is an A-bimodule with respect to the maps

(a, T ) −→ a.T, (a, T ) −→ T.a, A×F −→ F .

Now, the mapU : T 7−→ T (1), F −→ X

is a continuous linear operator and clearly

U(a.T ) = aU(T ) (a ∈ A, S ∈ F),

so that U is a left A-module homomorphism.

Dales and Villena in [3, Theorem 2.1 ] proved that F is a Banach A-bimodule. Also in the same theorem it has been shown that each left-intertwining map S : A −→ X is of the form S = U D, where U is definedas above and D : A −→ F = B(A1,X ) is a derivation defined by

D(a)(β, b) = S(βa+ ab)− a.S(b) (β ∈ C, a, b ∈ A).

Theorem 2.2 Let A be a Banach algebra for which every derivation from Ainto an arbitrary Banach A-bimodule is continuous. Suppose that dm is ahigher derivation from A to a Banach algebra B with a continuous d0. Thenthere exists a sequence Dmm≥1 of derivations from A to B(A1,B) such thatdm = U Dm (m ≥ 1), where U : B(A1,B) → B is the continuous left A-modulehomomorphism defined by U(T ) = T (1) for all T ∈ B(A1,B). Moreover, dm isautomatically continuous.

Proof. By continuity of d0, B is a Banach A-bimodule with module operationsdefined in (1.1). Therefore d1 is a module-valued derivation to a Banach A-bimodule and also an intertwining by Remark 2.1 (i). Now, by Remark 2.1 (ii)there exists a derivation D1 : A → B(A1,B) such that d1 = U D1, where


U : B(A1,B) → B is defined by U(T ) = T (1) (T ∈ B(A1,B)) which is a conti-nuous left A-module homomorphism. Continuity of d1 is obvious by the assump-tion. Now by induction assume that for i = 1, ...,m− 1, di = U Di, where eachDi is a derivation from A to B(A1,B) which is continuous by the hypothesis andU : B(A1,B) → B is as before. Now we have d0, d1, ..., dm−1 are continuous andhence dm is an intertwining map and again by Remark 2.1 (ii) it is of the formU Dm, where U is as before and Dm : A → B(A1,B) is a derivation. Since bythe assumption each Dm is continuous, the last assertion follows easily.

Corollary 2.3 Every higher derivation dm from a C∗-algebra to a Banach al-gebra, with continuous d0, is continuous.

Proof. Since every module-valued derivation from a C∗-algebra is continuous[14], then by Theorem 2.2 we have the result.

We recall that a derivation δ from a Banach algebra A to a Banach A-bimodule X is said to be inner if there exists x ∈ X such that δ(a) = ax − xa(a ∈ A). A Banach algebra A for which every bounded module-valued derivationto an arbitrary Banach A-bimodule is inner is called super-amenable [16].

Corollary 2.4 Let A be a super-amenable Banach algebra satisfying the hypoth-esis of Theorem 2.2. Then for every higher derivation dm from A to a Banachalgebra B, with continuous d0, we have dm = U δm (m ≥ 1), where each δm isan inner derivation from A to B(A1,B) and U is defined as in Remark 2.1 (ii).

3. Inner higher derivations

We recall the definition of an inner higher derivation from [15].

Definition 3.1 Let A and B be Banach algebras and let dm be a higher deriva-tion from A into B. Then dm is called inner if for each m ∈ N, there areu1, . . . , um ∈ B such that

dm(a) = d0(a)um −m−1∑i=0

um−idi(a) (a ∈ A, m ∈ N).

Note that if d0 is continuous, then the inner higher derivation dm is also conti-nuous.

Example 3.2 If dm is a higher derivation from a unital Banach algebra A to aBanach algebra B such that d0(A)B = 0, then dm is inner. To see this, supposea ∈ A and let e be the identity element of A. Then

dm(a) = dm(ea) =m∑i=0

di(e)dm−i(a) = d1(e)dm−1(a)+. . .+dm−1(e)d1(a)+dm(e)d0(a).


For 1 ≤ i ≤ m put ui = −di(e), then we have

dm(a) = d0(a)um − umd0(a)− um−1d1(a)− . . .− u1dm−1(a).

Therefore dm is inner.

It is well known that every derivation from a W ∗-algebra M to itself is inner[17, Theorem 2.5.3]. Also by [17, Corollary 2.5.4], if A is a C∗-subalgebra ofB(H) where B(H) is the C∗-algebra of all bounded linear operators on a Hilbertspace H, then every derivation δ : A −→ A is inner when we consider it as aderivation from A to B(H) . More precisely, there exists x ∈ B(H) such thatδ(a) = ax− xa (a ∈ A).

Proposition 3.3 Let M be a commutative W ∗-algebra and dm a strong higherderivation from M to M. Then each dm (m ≥ 1) is zero.

Proof. By [17, Theorem 2.5.3] the result is obvious.

We are far from a proof of Sakai’s result [17, Theorem 2.5.3] for higher deriva-tions, but we can prove some partial results concerning higher derivations onW ∗-algebras.

We recall the well known Markov-Kakutani theorem.

Theorem 3.4 Let K be a non-empty convex compact subset of a locally convexspace and let S be a commutative semigroup of continuous affine maps on K.Then S has a fixed point.

Theorem 3.5 Let M be a W ∗-algebra with identity element e and dm a stronghigher derivation from M to M. Let A be a commutative W ∗-subalgebra of Mcontaining e. Then for each m ∈ N there are u0 = e, u1, . . . , um in M such that

dm(a) = aum −m−1∑i=0

um−idi(a) for all a ∈ A and

∥um∥ ≤ ∥dm∥+ ∥um−1∥∥d1∥+ . . .+ ∥u1∥∥dm−1∥.

Proof. By [17, Lemma 2.5.1] the result holds for m = 1. Now suppose that foreach j ∈ 1, ...,m − 1 there exist u0 = e, u1, . . . , uj in M such that, dj(a) =

auj −j∑

i=0

uj−idi(a) for all a ∈ A and

∥um−1∥ ≤ ∥dm−1∥+ ∥um−2∥∥d1∥+ . . .+ ∥u1∥∥dm−2∥.

Let Au be the group of all unitary elements in A. Since each element of A isa finite linear combination of elements in Au, so it is enough to show that theresult holds for Au. For a ∈ Au, define Ta(x) = [ax− dm(a)− um−1d1(a)− . . .−u1dm−1(a)]a

−1 (x ∈ M). Then each Ta is an affine map. If a, b ∈ Au, then wehave


TaTb(x) = Ta

[(bx− dm(b)− um−1d1(b)− . . .− u1dm−1(b)

)b−1

]=

(abxb−1 − adm(b)b

−1 − aum−1d1(b)b−1 − . . .− au1dm−1(b)b

−1

− dm(a)− um−1d1(a)− . . .− u1dm−1(a))a−1

=(abx− dm(ab) + dm(a)b+ dm−1(a)d1(b) + . . .+ d1(a)dm−1(b)

)(ab)−1

− aum−1d1(b)(ab)−1 − aum−2d2(b)(ab)

−1 − . . .− au1dm−1(b)(ab)−1

−dm(a)a−1−um−1d1(a)a

−1−um−2d2(a)a−1− . . .−u2dm−2(a)a

−1−u1dm−1(a)a−1.

Now, by the induction hypothesis, we have

TaTb(x) = abx(ab)−1 − dm(ab)(ab)−1 −

(aum−1 − dm−1(a)

)d1(b)(ab)

−1

− . . .−(au1 − d1(a)

)dm−1(b)(ab)

−1 − um−1d1(a)a−1 − um−2d2(a)a

−1

− . . .− u2dm−2(a)a−1 − u1dm−1(a)a

−1 = abx(ab)−1 − dm(ab)(ab)−1

−(aum−1 − aum−1 + um−1a+ um−2d1(a) + . . .+ u1dm−2(a)

)d1(b)(ab)

−1

− . . .−(au1 − u1a+ u1a

)dm−1(b)(ab)

−1 − um−1d1(a)a−1

− um−2d2(a)a−1 − . . .− u2dm−2(a)a

−1 − u1dm−1(a)a−1

= abx(ab)−1 − dm(ab)(ab)−1 − um−1

(ad1(b) + d1(a)b

)(ab)−1

− um−2

(d1(a)d1(b) + d2(a)b+ ad2(b)

)(ab)−1

− . . .− u1

(adm−1(b) + d1(a)dm−2(b) + . . .+ dm−2(a)d1(b) + dm−1(a)b

)(ab)−1

=(abx− dm(ab)− um−1d1(ab)− um−2d2(ab)− . . .− u1dm−1(ab)

)(ab)−1

= Tab(x) = Tba(x) = TbTa(x).

Let σ denote the weak operator topology on M and let Km be the σ-closedconvex hull of Ta(0) : a ∈ Au. Since each Ta is σ-continuous and TaTb(x) =Tab(x) (a, b ∈ Au, x ∈ M), it is easy to show that Ta(Km) ⊆ Km. On the otherhand

∥Ta(0)∥ = ∥(−dm(a)− um−1d1(a)− . . .− u1dm−1(a)

)a−1∥

≤ ∥dm∥+ ∥um−1∥∥d1∥+ . . .+ ∥u1∥∥dm−1∥,

and it follows that

supx∈Km

∥x∥ ≤ ∥dm∥+ ∥um−1∥∥d1∥+ . . .+ ∥u1∥∥dm−1∥.

Therefore, Km is σ-compact and Ta : a ∈ Au is a commutative semigroup of σ-continuous affine maps on Km. Thus by Theorem 3.4 there is an element um ∈ Msuch that

Ta(um) = um (a ∈ Au).

Therefore

dm(a) = aum −m−1∑i=0

um−idi(a) (a ∈ Au),

and clearly∥um∥ ≤ ∥dm∥+ ∥um−1∥∥d1∥+ . . .+ ∥u1∥∥dm−1∥.

Corollary 3.6 If A is a commutative W ∗-subalgebra of a W ∗-algebra M con-taining the identity element, then each strong higher derivation from A to M isinner.


References

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