2 0 1 9 O M T S Abirimler.dpu.edu.tr/.../172/files/Abstract_Book.pdf · 2019-07-24 · The meeting will be devoted to various aspects of Theory Function Spaces, Opera-tor Theory of

OMTSA2019

E d i t o r s

V A G I F S . G U L I Y E V

I . E K I N C I O G L U

A B S T R A C T B O O K

O P E R A T O R S I N G E N E R A L M O R R E Y - T Y P E S P A C E S

A N D A P P L I C A T I O N S

[email protected] http://omtsa.dpu.edu.tr ISBN NO: 978-605-69425-0-1

Operators in General Morrey-Type Spaces andApplications (OMTSA 2019),

Kutahya Dumlupinar University,Kutahya, TURKEY,16-20 July, 2019

Editors:

Vagif S. GULIYEV and Ismail EKINCIOGLU

Associate Editor:

Esra KAYA

Copyright c© OMTSA 2019KUTAHYA DUMLUPINAR UNIVERSITY

Published 24 JULY 2019

Operators in General Morrey-Type Spaces and ApplicationsOMTSA 2019

Dumlupınar University

PREFACE

Operators in General Morrey-Type Spaces and Applications (OMTSA 2019) will beheld on July 16-20, 2019 in Kutahya, Turkey. Firstly, we would like to thank all theinvited speakers who have kindly accepted our invitation and have come to spend theirprecious time by sharing their ideas during the conference.

The "Operators in General Morrey-Type Spaces and Applications" conference washeld in 2011 and 2017 at Kirsehir Ahi Evran University. This year OMTSA 2019 hasbeen held at Kutahya Dumlupinar University, Kutahya, Turkey on 16-20 July 2019 asan international conference. Kutahya has been home to many civilizations with its7000 years of history and is an open air museum with its historical richness as the cityof establishment and salvation, and the city of history and culture. Also, because ofits location on the graben system of the Aegean Region and the cracks formed by thissystem Kutahya is one of the most important regions in terms of geothermal sources.Those sources have quite high thermal value and are important for health tourism.

OMTSA 2019 International Conference on Operator Theory and its Applicationsand Harmonic Analysis, Partial Differential Equations and to explore interactions withDifferential Geometry, Topology and Algebra aims to bring together leading academicscientists, researchers and research scholars to exchange and share their experiencesand research results on all aspects of Operator Theory and Harmonic Analysis. It alsoprovides a premier interdisciplinary platform for researchers, practitioners and educa-tors to present and discuss the most recent innovations, trends, and concerns as wellas practical challenges encountered and solutions adopted in the above area relatedareas.

The meeting will be devoted to various aspects of Theory Function Spaces, Opera-tor Theory of Function Spaces, Theory and applications of all function spaces relatedto Morrey spaces, Integral operators on Morrey type spaces, Regularity in Morrey typespaces of solutions to elliptic, parabolic and hypoelliptic equations, Approximation the-ory and Interpolation theory.

Finally, we would like to convey our heartiest welcome to each of you who havecome to attend this conference and we wish for an enjoyable high scientific level con-ference and hope to meet you again in the future.

Best wishes and regards,

On behalf of the Organizing CommitteeProf. Dr. Vagif S. GULIYEV

Prof. Dr. Ismail EKINCIOGLU

iii

OMTSA 2019Dumlupınar University Operators in General Morrey-Type Spaces and Applications

ORGANIZING COMMITTEE

• Vagif S. GULIYEV (Kutahya Dumlupinar University, TURKEY)

• Ismail EKINCIOGLU (Kutahya Dumlupinar University, TURKEY)

• Victor BURENKOV (Nikolskii Institute of Mathematics, RUSSIA)

• Maria Alessandra RAGUSA (Catania University & Accademia Gioenia Catania, ITALY)

• Amiran GOGATISHVILI (Institute of Mathematics AS CR, CZECH REPUBLIC)

• Salauddin UMARKHADZHIEV (Academy of Sciences of the Chechen Republic, RUSSIA)

• Rovshan A. BANDALIYEV (Institute of Mathematics and Mechanics, AZERBAIJAN)

• Przemyslaw GORKA (Warsaw University of Technology, POLAND)

• Ayhan SERBETCI (Cankiri Karatekin University, TURKEY)

• Ali S. NAZLIPINAR (Kutahya Dumlupinar University, TURKEY)

• Ali AKBULUT (Kirsehir Ahi Evran University, TURKEY)

• Cansu KESKIN (Kutahya Dumlupinar University, TURKEY)

• Mehriban N. OMAROVA (Baku State University, AZERBAIJAN)

• Ahmet BOZ (Kutahya Dumlupinar University, TURKEY)

• Ozgun GURMEN ALANSAL (Kutahya Dumlupinar University, TURKEY)

• Ilkem TURHAN CETINKAYA (Kutahya Dumlupinar University, TURKEY)

• Abdulhamit KUCUKARSLAN (Pamukkale University, TURKEY)

• Canay AYKOL YUCE(Ankara University, TURKEY)

• Fatih DERINGOZ (Kirsehir Ahi Evran University, TURKEY)

• Esra KAYA (Kutahya Dumlupinar University, TURKEY)

• Tugce Unver YILDIZ (Kırıkkale University, TURKEY)

iv



SCIENTIFIC COMMITTEE• Akbar B. ALIYEV (Institute of Mathematics and Mechanics, AZERBAIJAN)

• Alexey KARAPETYANTS (Southern Federal University, RUSSIA)

• Alik M. NAJAFOV (Azerbaijan University of Architecture and Construction,AZERBAIJAN)

• Amiran GOGATISHVILI (Institute of Mathematics AS CR, CZECH REPUBLIC)

• Arash GHORBANALIZADEH (Institute for Advanced Studies in Basic Sciences, IRAN)

• Ayhan SERBETCI (Cankiri Karatekin University, TURKEY)

• Daniyal ISRAFILZADE (Balıkesir University, TURKEY)

• Ekrem SAVAS (Usak University, TURKEY)

• Elcin YUSUFOGLU (Usak University, TURKEY)

• Fahreddin ABDULLAYEV (Mersin University, TURKEY)

• Ismail EKINCIOGLU (Kutahya Dumlupinar University, TURKEY)

• I. Naci CANGUL (Uludag University, TURKEY)

• Kemal AYDIN (Selcuk University, TURKEY)

• Lubomira SOFTOVA (Second University of Naples, ITALY)

• Maria Alessandra RAGUSA (Catania University & Accademia Gioenia Catania, ITALY)

• Michael RUZHANSKY (Imperial College, London, UK)

• Mikhail GOLDMAN (Peoples’ Friendship University of Russia, RUSSIA)

• Miloud ASSAL (Carthage University, TUNUSIA)

• Misir C. MARDANOV (Institute of Mathematics and Mechanics, AZERBAIJAN)

• Mubariz G. HACIBAYOV (National Academy, AZERBAIJAN)

• Natasha SAMKO (Luleå University of Technology, SWEDEN )

• Pankaj JAIN (South Asian University, INDIA)

• Przemyslaw GORKA (Warsaw University of Technology, POLAND)

• Rabil AYAZOGLU (MASHIYEV) (Bayburt University, TURKEY)

• Radouan DAHER (Faculty of Sciences Ain Chock University Hassan II, MOROCCO)

• Rovshan A. BANDALIYEV (Institute of Mathematics and Mechanics, AZERBAIJAN)

• Salauddin UMARKHADZHIEV (Academy of Sciences of the Chechen Republic, RUSSIA)

• Sergio POLIDORO (Univerita di Modena e Reggio Emilia, ITALY)

• Stefan SAMKO (Universidade do Algarve, PORTUGAL)

• Tahir S. HACIYEV (Institute of Mathematics and Mechanics, AZERBAIJAN)

• Vagif S. GULIYEV (Kutahya Dumlupinar University, TURKEY)

• Victor BURENKOV (Nikolskii Institute of Mathematics, RUSSIA)

• Yagub Y. MAMMADOV (Nakhchivan Teacher-Training Institute, AZERBAIJAN)

• Yoshihiro SAWANO (Tokyo Metropolitan University, JAPAN)

v

Contents

Interpolation Theory and Local Morrey-Type Spaces (Victor I. BURENKOV) . . 2Relatively Compact Sets in Variable Exponent Morrey Spaces on Metric Spaces

(Rovshan A. BANDALIYEV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Omega Invariant and Its Applications on Topological Graph Theory (Ismail

Naci CANGUL1, Sadik DELEN1, Aysun YURTTAS1, Muge TOGAN1) . . . . . 4Boundedness of Classical Operators in Morrey Type Spaces and Hardy Inequal-

ities (Amiran GOGATISHVILI) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Sobolev Embeddings and Regularity of the Space (Przemyslaw GORKA) . . . . 6Riesz Potential and Its Commutators in Variable Exponent Generalized Weighted

Morrey Spaces (Vagif S. GULIYEV) . . . . . . . . . . . . . . . . . . . . . . . . 7Recent Trends in Bilinear Hardy-Type Inequalities (Pankaj JAIN) . . . . . . . . 9Moser’s Estimates for Degenerate Kolmogorov Equations (Francesca ANCESCHI1,

Sergio POLIDORO1, Maria Alessandra RAGUSA2) . . . . . . . . . . . . . . . 10On Partial Regularity of Minimizers of Functionals with Discontinuous Coeffi-

cients (Maria Alessandra RAGUSA) . . . . . . . . . . . . . . . . . . . . . . . 11Strongly Lacunary Invariant Summable Sequence Spaces (Ekrem SAVAS) . . . 12On Smoothness of Solution of One Class Differential Equations Higher Order

(Alik M. NAJAFOV1,2, Sain T. ALEKBERLI3) . . . . . . . . . . . . . . . . . . . 15New Ring of Ponderation Functions and Applications to Solve Cauchy-Euler

Equation for Several Variables (Miloud ASSAL) . . . . . . . . . . . . . . . . 17Regularity Estimates in Weighted Generalized Morrey Spaces for Quasilinear

Parabolic Equations (Tahir S. GADJIEV1, Faig M. NAMAZOV2) . . . . . . . 18The Uniformly Elliptic and Parabolic Equations of Higher Order with Dis-

countinuous Data in Generalized Morrey Spaces and Elliptic Equations inUnbounded Domains (Tahir S. GADJIEV1, Shahla GALANDAROVA1, KonulSULYEMANOVA1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Estimations of the Norm of Functions from Lizorkin-Triebel-Morrey Type SpacesReduced by Polynomials (Azizgul M. GASUMOVA) . . . . . . . . . . . . . . 20

Problems of Approximation in Generalized Sobolev Spaces (Leyla Sh. KADI-MOVA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Interpolation Theorems for Type Spaces Besov-Morrey with Dominant MixedDerivatives (Nilufer R. RUSTAMOVA) . . . . . . . . . . . . . . . . . . . . . . 23

vi



Characterizations for the Fractional Maximal Commutator Operator in Gen-eralized Morrey Spaces on Carnot Group (Ismail EKINCIOGLU) . . . . . . 25

Weighted Norm Inequalities for Bn-maximal Operators and the High OrderRiesz-Bessel Transforms (Esra KAYA1, Ismail EKINCIOGLU1) . . . . . . . . 26

Poincare Type Inequality in Generalized Sobolev-Morrey Type Spaces (RovshanF. BABAYEV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Approximate Solutions of Nonlinear Bernoulli Equation by Using BernsteinPolynomials with Error Analysis (Fadil JARADAT1, M.H.T. ALSHBOOL1) . 29

The Regularity of Solutions Elliptic Equations of Higher Order with Discon-tinuous Data in Generalized Orlicz-Morrey Spaces (Tahir S. GADJIEV1,Konul YASINLI2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Poincare Type Inequality in Lizorkin-Triebel-Morrey Type Spaces (Firida F.MUSTAFAYEVA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Abstract Korovkin Theorems via Relative Summation Process for Double Se-quences on Modular Spaces (Sevda YILDIZ) . . . . . . . . . . . . . . . . . . 33

Approximation in Weighted Morrey Spaces by Trigonometric Polynomials (CanayAYKOL1, Ayhan SERBETCI2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

On φ-Recurrent Kenmotsu Finsler Manifolds (A. Funda SAGLAMER1, NurtenKILIC1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Monic Torsion Precovers Relative to A Torsion Theory (Mustafa Kemal BERKTAS1,Semra DOGRUOZ2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

An Application of Gauss-Chebyshev Quadrature Formulas to the GeneralizedFrictionless Contact Problem (Elcin YUSUFOGLU1, llkem TURHAN CETINKAYA2) 37

Multivalent Harmonic Starlike Functions Defined by Subordination (Sibel YAL-CIN TOKGOZ1, Sahsene ALTINKAYA1) . . . . . . . . . . . . . . . . . . . . . . 39

Some New Beta-Fractional Integral Inequalities (Deniz UCAR1, Veysel FuatHATIPOGLU2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

A New Local Smoothing Technique for Non-smooth Functions (Nurullah YIL-MAZ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

The (p, q)-Lucas Polynomial Coefficient Estimates of a Bi-univalent FunctionClass with Respect to Symmetric Points (Sahsene ALTINKAYA1, Sibel YAL-CIN TOKGOZ1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Approximation of the Reachable Sets of an SEIR Control System and Compar-ison with Optimal Control Solution (A. Serdar NAZLIPINAR1, BarbarosBASTURK1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

On Totally Asymtotically Nonexpansive Mappings in Cat(0) Spaces (A. ABKAR1,Mojtaba RASTGOO1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Integral Operators in Anisotropic Morrey Spaces (Andrea SCAPELLATO) . . . . 46Lipschitz Estimates for Rough Fractional Multilinear Integral Operators on

Local Generalized Morrey Spaces (S. Elifnur EKINCIOGLU1, A. SerdarNAZLIPINAR1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

vii


On the Explicit Solution of A Coupled System of Fractional Partial DifferentialEquations Using Symmetry (Nisrine MAAROUF1, Khalid HILAL1) . . . . . 48

A Numerical Approach to Equal Width Equation Using B-Spline Functions (Ah-met BOZ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Inclusion Relations Between Weighted Local Morrey-type Spaces (Tugce UnverYILDIZ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

On Potential Wells and Global Solvability of Cauchy Problem for System ofSemi-linear Klein-Gordon Equations (Akbar B. ALIEV1,2, Gulshan Kh. SHAFIYEVA2,3) 51

On Higher Dimensional Bilinear Hardy Inequalities (Saikat KANJILAL) . . . . 52An Extension of Muckenhoupt-Wheeden Theorem to Generalized Weighted

Morrey Spaces (Abdulhamit KUCUKASLAN) . . . . . . . . . . . . . . . . . . 53On Properties of Weighted Lebesgue Modulus of Continuity (Aynur N. MAMMADOVA1,

Dunya R. ALIYEVA1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Approximation of Functions by Linear Positive Operators in Variable Exponent

Lebesgue Spaces (Aytekin E. ABDULLAYEVA1, Lale R. ALIYEVA1) . . . . . . 55Different Approach to the Decomposition Theory of Hardy-Morrey Spaces

(Cansu KESKIN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56On Multidimensional Hausdorff Operator on Variable Lebesgue Spaces (Ka-

mala H. SAFAROVA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57A General Version of Neubauer’s Lemma and Closed Range Almost Closed

Operators (Abdellah GHERBI) . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Potential Operators on Carleson Curves in Morrey Spaces (Ahmet EROGLU) . 59Boundary Value Problems for Convolution Differential Operator Equations on

the Half Line (Hummet K. MUSAEV) . . . . . . . . . . . . . . . . . . . . . . 60Maximal Operator on Carleson Curves in Orlicz-Morrey Spaces (Hatice AR-

MUTCU) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Parabolic-Fractional Integral Operators with Rough Kernels in Parabolic Local

Generalized Morrey Spaces (Shemsiyye A. MURADOVA) . . . . . . . . . . . 62Simplicial Algebr(oid)s and Internal Categories (Ozgun GURMEN ALANSAL1,

Erdal ULUALAN1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Fractional Integral Associated with Schrödinger Operator on Generalized Mor-

rey Spaces (Ali AKBULUT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Fractional Integral Associated with Schrödinger Operator on Vanishing Gen-

eralized Morrey Spaces (Mehriban N. OMAROVA) . . . . . . . . . . . . . . 65New Characterizations of Lipschitz Space via Commutators on Orlicz Spaces

(Fatih DERINGOZ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Boundedness of Variable-order Fractional Operators in Variable Exponent Lebesgue

Spaces Lp(·) [0, l] (Rabil AYAZOGLU(MASHIYEV)) . . . . . . . . . . . . . . . 68Uniform Boundedness of Kantorovich Operators in Lebesgue Spaces with Vari-

able Exponent (Rabil AYAZOGLU(MASHIYEV)) . . . . . . . . . . . . . . . . 69The Welland Inequality on Hypergroups (Mubariz G. HAJIBAYOV) . . . . . . . 70Weighted Inequality for (p, q)-admissible B-potential Operators (Fatai A. ISAYEV) 72

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On q-Meyer König-Zeller-Durrmeyer Operators (Dilek SOYLEMEZ) . . . . . . . 73Oscillatory Integrals with Variable Calderon-Zygmund Kernel on Vanishing

Generalized Morrey Spaces (Aysel A. AHMADLI) . . . . . . . . . . . . . . . 74Lipschitz Estimates for Rough Fractional Multilinear Integral Operators on

Vanishing Generalized Morrey Spaces (Ramin V. GULIYEV) . . . . . . . . . 75Approximation by Nörlund Means in Weighted Generalized Grand Smirnov

Classes (Ahmet TESTICI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Characterization of Commutators of Parabolic Fractional Integral Operator in

Parabolic Generalized Orlicz-Morrey Spaces (Gulnara A. ABASOVA) . . . 78On the Equivalence of the K-functional and the Modulus of Continuity on the

Variable Exponent Morrey Spaces (Arash GHORBANALIZADEH) . . . . . . 79Commutators of Maximal Operator Associated with the Dunkl Operators on

Orlicz Spaces (Yagub Y. MAMMADOV1, Fatma A. MUSLIMOVA1) . . . . . . 80Fractional Maximal Operator Associated with the Dunkl Operators on Orlicz

Spaces (Fatma A. MUSLIMOVA) . . . . . . . . . . . . . . . . . . . . . . . . . 82Approximation Properties of Faber Series (Daniyal ISRAFILOV1, Emine KIRHAN1) 83Sublinear Operators with Rough Kernel Generated by Calderon-Zygmund Op-

erators on Generalized Weighted Morrey Spaces (Vugar H. HAMZAYEV) 84Fractional Multilinear Integrals with Rough Kernels on Generalized Weighted

Morrey Spaces (Amil A. HASANOV) . . . . . . . . . . . . . . . . . . . . . . . 86New Characterizations of BMO Space via Commutators on Orlicz Spaces (Sabir

G. HASANOV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88On Embeddings of Grand Sobolev-Morrey Spaces with Dominant Mixed Deriva-

tives (Alik M. NAJAFOV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Some Results on Hausdoff Operators (Radouan DAHER) . . . . . . . . . . . . . 91Continuous State Space Branching Process as Limit of Sequence of Bellman-

Harris Processes (Soltan A. ALIYEV1, Sahib A. ALIYEV2) . . . . . . . . . . . 92The Determination of the Right-hand Side in a Semilinear Heat Conduction

Problem with the Nonlocal Dirichlet Boundary Condition (Arzu ERDEMCOSKUN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Relativistic Derivation Formalism of Electromagnetic Field Equations in Quater-nion Algebra (Mustafa Emre KANSU1, Murat TANISLI2, Suleyman DEMIR2) 94

Formulation of Maxwell-type Gravity Equations with Dual Quaternions (IsmailAYMAZ1, Mustafa Emre KANSU1) . . . . . . . . . . . . . . . . . . . . . . . . . 95

A Necessary Condition and Sufficient Condition for the Summability of theDiscrete Hilbert Transform (Aynur F. HUSEYNLI) . . . . . . . . . . . . . . . 96

Approximation of Hypersingular Integral Operators with Hilbert Kernel onHölders Spaces (Chinara A. GADJIYEVA) . . . . . . . . . . . . . . . . . . . . 98

On Singular Operators in Vanishing Generalized Variable Exponent MorreySpaces and Applications to Bergman Type Spaces (Alexey KARAPETYANS) 100

Some Relations Between Partially q-Poly-Euler Polynomials of the Second Kind(Burak KURT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

ix


Grand Lebesgue Spaces on Sets of Infinite Measure (Salauddin UMARKHADZHIEV)102Condition for The Affilation of the Function to Genarilized Hölders Class Hγ

αβ(R+m+k,k)

(Asim A. AKBAROV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103The Solution of a Class of Nonlinear Integral Equations by Newton-Kantorovich

Method (Fuad A. ABDULLAYEV) . . . . . . . . . . . . . . . . . . . . . . . . . 105Some Progress on Reversible Rings (Handan KOSE) . . . . . . . . . . . . . . . . 107Constructing a System of First Kind Chebyshev Polynomials (Sahib A. ALIYEV) 108

Author Index 109

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INVITED SPEAKERS

1


Interpolation Theory and Local Morrey-Type SpacesVictor I. BURENKOV

Nikolskii Institute of Mathematics, RUSSIA

Abstract. Local Morrey-type spaces, in contrast to global ones, appeared to be verygood from the point of view of the interpolation theory: the scale of local Morrey-typespaces is closed under the procedure of real interpolation. Moreover, application oflocal Morrey-type spaces and their generalisations allows us to obtain a very generalinterpolation theorem which contains as particular cases all main classical interpolationtheorems and their new variants and generalisations.

A survey of results in this direction will be given including also Marcinkiewcz-typetheorems for general Morrey-type spaces and their applications.

Open problems will be formulated.

2



Relatively Compact Sets in Variable Exponent Morrey Spaces onMetric Spaces

Rovshan A. BANDALIYEVInstitute of Mathematics and Mechanics of ANAS, AZERBAIJAN

S.M. Nikolskii Institute of Mathematics at RUDN University, RUSSIA

Abstract. We study totally bounded sets in variable exponent Morrey spaces. Somecharacterization of this kind of sets is given for the case of variable exponent Morreyspace on metric measure spaces. Furthermore, the sufficient conditions for compact-ness are shown without assuming log-Hölder continuity of the exponent.

This is jointly work with Przemysław Górka and Vagif S. Guliyev.

References

[1] Bandaliyev R.A., Górka P., Relatively compact sets in variable-exponent Lebesgue spaces,Banach J. Math. Anal., 2(12), 2018, 331-346.[2] Bokayev N., Burenkov V.I., Dauren M., On precompactness of a set in general localand global Morrey-type spaces, Eurasian Math.J., 8(3), 2017, 109-115.[3] Wang W., Xu J., Precompact sets, boundedness, and compactness of commutatorsfor singular integrals in variable Morrey spaces, J. Function Spaces, 2017, Article ID3764142, 9 p.

3


Omega Invariant and Its Applications on Topological Graph TheoryIsmail Naci CANGUL1, Sadik DELEN1, Aysun YURTTAS1, Muge TOGAN1

1Bursa Uludag University, Turkey

Abstract. Realizability of a degree sequence is an important property due to its ap-plications. Delen and Cangul recently defined a new graph topological invariant fora given degree sequence which shows properties similar to those obtained by Eulercharacteristic and cyclomatic number. This invariant can also be calculated for a givengraph. This new invariant called omega invariant helps us to determine many algebraic,topological, combinatorial, graph theoretical, number theoretical properties of all therealizations of the given degree sequence including connectivity, numbers of compo-nents, cycles, loops, chords, multiple edges, pendant and support vertices, etc. Since ithas been defined in 2018, many different applications of the omega invariant has beengiven. In this talk, we introduce the omega invariant and give several of its numerousapplications to topological graph indices, matching number and connectivity.

Keywords: Connectedness, Degree sequence, Matching number, Omega invariant, Re-alizability.

References

[1] Delen S., Cangul I.N., A New Graph Invariant, Turkish Journal of Analysis and Num-ber Theory, 6(1), 2018, 30-33.[2] Delen S., Cangul I.N., Extremal Problems on Components and Loops in Graphs, ActaMathematica Sinica, English Series, 35(2), 2019, 161-171.

4



Boundedness of Classical Operators in Morrey Type Spaces andHardy InequalitiesAmiran GOGATISHVILI

Institute of Mathematics of the Czech Academy of Sciences, CZECH REPUBLIC

Abstract. In this talk, I will present main tools which allowed us to reduce the problemof the boundedness of many operators from Harmonic analysis in Morrey type spacesto the boundedness of Hardy operators and its iteration. We give characterization suchHardy inequalities.

Keywords: Hardy operator, Morrey type space.

5


Sobolev Embeddings and Regularity of the SpacePrzemyslaw GORKA

Warsaw University of Technology, POLAND

Abstract. We shall discuss various type of Sobolev inequalities in Euclidean space aswell as in the setting of metric measure spaces. In particular, we shall describe therelation between Sobolev embeddings and lower bound for the measure.

The talk will be based on joint works with Adimurthi, Ryan Alvarado, Piotr Hajłasz,Nijjwal Karak and Daniel Pons.

6



Riesz Potential and Its Commutators in Variable ExponentGeneralized Weighted Morrey Spaces

Vagif S. GULIYEV

Kutahya Dumlupinar University, TURKEYInstitute of Mathematics and Mechanics, Baku, AZERBAIJAN

Abstract. This talk is based on joint research with J.J. Hasanov and X.A. Badalov.The classical Morrey spaces were introduced by Morrey [1] to study the local be-

havior of solutions to second-order elliptic partial differential equations. Guliyev [2,3],Mizuhara [4] and Nakai [5] introduced generalized Morrey spaces M p,ϕ(Rn). Recently,Komori and Shirai [6] considered weighted Morrey spaces Lp,κ

w and studied the bound-edness of some classical operators such as Hardy-Littlewood maximal operator andCalderón-Zygmund operators on these spaces. Guliyev [7] gave a concept of general-ized weighted Morrey space M p,ϕ

w which could be viewed as extension of both general-ized Morrey space M p,ϕ and weighted Morrey space Lp,κ

w .The variable exponent generalized Morrey spacesM p(·),ϕ(Ω) over an open set Ω ⊂

Rn was introduced and the boundedness of the Hardy-Littlewood maximal operator, theRiesz potential, the singular integral operators and their commutators on these spaceswas proved in [9] (bounded set) and [10] (unbounded set), respectively. The variableexponent generalized weighted Morrey spacesM p(·),ϕ

w (Ω) over an open set Ω ⊂ Rn wasintroduced and the boundedness of the Hardy-Littlewood maximal operator, the Rieszpotential, the singular integral operators and their commutators on these spaces wasproved in [11].

Let Ω ⊂ Rn be an open unbounded set. We consider generalized weighted Morreyspaces M p(·),ϕ

ω(Ω) and vanishing generalized weighted Morrey spaces V M p(·),ϕ

ω(Ω) with

variable exponent p(x) and a general function ϕ(x , r) defining the Morrey-type norm.The main result of this study are to prove the boundedness of Riesz potential and itscommutators on the spaces M p(·),ϕ

ω(Ω) and V M p(·),ϕ

ω(Ω) (see [12]). This result gener-

alizes several existing results for Riesz potential and its commutators on Morrey typespaces. Especially, it gives a unified result for generalized Morrey spaces and variableMorrey spaces which currently gained a lot of attentions from researchers in the theoryof function spaces.

7


References

[1] Morrey C.B., On the solutions of quasi-linear elliptic partial differential equations,Trans. Amer. Math. Soc., 43, 1938, 126-166.[2] Guliyev V.S., Integral operators on function spaces on the homogeneous groups and ondomains in Rn, Doctor’s degree dissertation, Moscow, Mat. Inst. Steklov, 1994, 1-329.(Russian)[3] Guliyev V.S., Boundedness of the maximal, potential and singular operators in thegeneralized Morrey spaces, J. Inequal. Appl. Art., ID 503948, 2009, 20 pp.[4] Mizuhara T., Boundedness of some classical operators on generalized Morrey spaces,Harmonic Analysis (S. Igari, Editor), ICM 90 Satellite Proceedings, Springer-Verlag,Tokyo, 1991, 183-189.[5] Nakai E., Hardy-Littlewood maximal operator, singular integral operators and Rieszpotentials on generalized Morrey spaces, Math. Nachr., 166 (1994), 95-103.[6] Komori Y., Shirai S., Weighted Morrey spaces and a singular integral operator, Math.Nachr., 282 (2) (2009), 219-231.[7] Guliyev V.S., Generalized weighted Morrey spaces and higher order commutators ofsublinear operators, Eurasian Math. J., 3(3), 2012, 33-61.[8] Guliyev V.S., Karaman T., Mustafayev R.Ch., and Serbetci A., Commutators of sublin-ear operators generated by Calderón-Zygmund operator on generalized weighted Morreyspaces, Czechoslovak Math. J., 64(139)(2), 2014, 365-386.[9] Guliyev V.S., Hasanov J.J., Samko S.G., Boundedness of the maximal, potential andsingular operators in the generalized variable exponent Morrey spaces, Math. Scand., 107,2010, 285-304.[10] Guliev V.S., Samko S.G., Maximal, potential and singular operators in the general-ized variable exponent Morrey spaces on unbounded sets, J. Math. Sci. (N. Y.), 193(2),2013, 228-248.[11] Guliyev V.S., Hasanov J.J., Badalov X.A., Maximal and singular integral operatorsand their commutators on generalized weighted Morrey spaces with variable exponent,Math. Ineq. Appl., 21(1), 2018, 41-61.[12] Guliyev V.S., Hasanov J.J., Badalov X.A., Commutators of Riesz potential in thevanishing generalized weighted Morrey spaces with variable exponent, Math. Ineq. Appl.,22(1), 2019, 331-351.

8



Recent Trends in Bilinear Hardy-Type InequalitiesPankaj JAIN

South Asian University, INDIA

Abstract. In this talk, we shall be concerned about the bilinear Hardy inequality

∫ ∞

0

[H2( f , g)(x)]qw(x) d x

1/q

≤∫ ∞

0

f p1(x)v1(x) d x

1/p1 ∫ ∞

0

g p2(x)v2(x) d x

1/p2

,

(1)where

H2( f , g)(x) =

∫ x

0

f

∫ x

0

g

.

is the bilinear Hardy operator. We shall discuss necessary and sufficient conditions for(1) to hold for various combinations of the indices p1, p2, q. We shall also discuss somevery recent results concerning the discrete version of the above inequality. Moreover,we shall derive inequality (1) in the framework of quantum-calculus.

9


Moser’s Estimates for Degenerate Kolmogorov EquationsFrancesca ANCESCHI1, Sergio POLIDORO1, Maria Alessandra RAGUSA2

1Università di Modena e Reggio Emilia, ITALY2Università degli Studi di Catania, ITALY

Abstract. I present a recent regularity result for a family of degenerate second orderlinear Partial Differential Equations, tha arise in Stochastic Theory and in its applica-tions to several research areas, including the Kinetic Theory and the Finance. As ameaningful prototype of this family we consider the following Kolmogorov equation indivergence form

∂tu(x , y, t) =n∑

j,k=1

∂x j

a jk(x , y, t)∂xku(x , y, t)

+n∑

j=1

x j∂y ju(x , y, t)+

n∑

i=1

bi(x , y, t)∂iu(x , y, t)−n∑

i=1

∂x i(ai(x , y, t)u(x , y, t)) + c(x , y, t)u(x , y, t)

where (x , y, t) ∈ R2n+1, and A(x , y, t) :=

a jk(x , y, t)

j,k=1,...,nis a symmetric, uniformly

positive matrix, and the coefficients a jk’s are bounded measurable functions. We applythe Moser’s iteration method to prove the local boundedness of the solution u underminimal integrability assumption on the coefficients a1, . . . , an, b1, . . . , bn and c.

Keywords: Divergence form Kolmogorov equations, Moser’s iteration, Weak solutions.

10



On Partial Regularity of Minimizers of Functionals withDiscontinuous Coefficients

Maria Alessandra RAGUSACatania University & Academia Gioenia Catania, ITALY

Abstract. Let Ω ⊂ Rm (m ≥ 2) be a bounded open set. For maps u : Ω → Rn weconsider the p(x)-energy functional defined as

E (u;Ω) :=

∫

Ω

gαβ(x)Gi j(u)Dαui(x)Dβu j(x)

p(x)/2d x ,

where (gαβ(x)) and (Gi j(u)) are symmetric positive definite matrices whose entries arecontinuous functions defined on Ω and Rn respectively, and p(x) a continuous functionon Ω with p(x)≥ 2.

Main goal is the study of regularity properties, interior and up to the boundary, ofthe minimizers u of E and recent developments in this direction.

Keywords: Minimizer, Partial and boundary regularity.

11


Strongly Lacunary Invariant Summable Sequence SpacesEkrem SAVAS

Usak University, TURKEY

Abstract. Let σ be a one-to-one mapping of the set of positive integers into itself. Acontinuous linear functional ϕ on l∞ is said to be an invariant mean or a σ- mean ifand only if

1. ϕ ≥ 0 when the sequence x = (xn) has xn ≥ 0 for all n.

2. ϕ(e) = 1, where e = (1, 1, . . .) and

3. ϕ

xσ(n)

= ϕ(x) for all x ∈ l∞.

For a certain kinds of mapping σ every invariant mean ϕ extends the limit func-tional on space c, in the sense that ϕ(x) = lim x for all x ∈ c. Consequently, c ⊂ Vσwhere Vσ is the bounded sequences all of whose σ-means are equal, (see, [8]).

If x = (xk), set T x = (T xk) =

xσ(k)

it can be shown that (see, Schaefer [8]) that

Vσ =n

x ∈ l∞ : limk

tkm (x) = Le uniformly in m for some L = σ− lim xo

(2)

where

tkm(x) =xm + T xm + . . .+ T k xm

k+ 1and t−1,m = 0

We say that a bounded sequence x = (xk) is σ-convergent if and only if x ∈ Vσ suchthat σk(n) 6= n for all n≥ 0, k ≥ 1.

Just as the concept of almost convergence lead naturally to the concept of strong almostconvergence, σ- convergence leads naturally to the concept of strong σ-convergence.A sequence x = (xk) is said to be strongly σ-convergent (see Mursaleen [5]) if thereexists a number L such that

1k

k∑

i=1

xσi(m) − L

→ 0 (3)

as k → ∞ uniformly in m. We write [Vσ] as the set of all strong σ- convergent se-quences. When (0.2) holds we write [Vσ]− lim x = `. Taking σ(m) = m+1, we obtain[Vσ] = [c] so strong σ- convergence generalizes the concept of strong almost conver-gence. Note that [Vσ] ⊂ Vσ ⊂ l∞.

By a lacunary θ = (kr); r = 0, 1,2, ... where k0 = 0, we shall mean an increasingsequence of non-negative integers with kr − kr−1 → ∞ as r → ∞. The intervals

12



determined by θ will be denoted by Ir = (kr−1, kr] and hr = kr − kr−1. The ratio krkr−1

will be denoted by qr . The space of lacunary strongly convergent sequences Nθ wasdefined by Freedman at al [1] as follows:

Nθ =

(

x = (xk) : limr

1hr

∑

k∈Ir

|xk − L|) = 0, for some L

)

.

More results on this convergence can be seen from [2, 3, 6, 7].

There is a strong connection between Nθ and the space of strongly Cesàro summablesequences which is defined by,

[c] =

¨

x = (xk) : limn

1n

n∑

k=0

|xk − L|) = 0, for some l L

«

.

In the special case where θ = (2r), we have Nθ = [c], which is defined by Maddox [4].The goal of this paper is to investigate some new sequence spaces which arise from thenotation of lacunary sequence and introduce the spaces of strongly lacunary invariantsummable sequences which happen to be complete paranormed spaces under certainconditions. Some topological results, characterization of strongly lacunary invariantregular matrices, uniqueness of generalized limits and inclusion relations of such se-quences have been discussed.

References

[1] Freedman A.R., Sember J.J., Rapheal M., Some Cesaro-type summability spaces, Proc.London Math. Soc., 37(3), 1973, 508-520.[2] Karakaya V., Savas E., On almost θ -bounded variation of lacunary sequences, Com-puters & Math. Appl., 61(6), 2011, 1502-1506[3] Li J., Lacunary statistical convergence and inclusion properties between lacunary meth-ods, Int. J. Math. Math. Sc., 23(3), 2000, 175-180.[4] Maddox I.J., Spaces of strongly summable sequences, Quart. J. Math. Oxford Ser.,18(2), 1967, 345-55.[5] Mursaleen, On some new invariant matrix methods of summability, Q.J. Math., 34,1983, 77-86.[6] Savas E., Karakaya V., Some new sequence spaces defined by lacunary sequences, Math-ematica Slovaca, 57(4), 2007, 393-399.[7] Savas E., On lacunary strong σ -convergence, Indian J. Pure Appl. Math., 21(4),1990, 359-365.[8] Schaefe r P., Infinite matrices and invariant means, Proc. Amer. Math. Soc., 36,1972, 104-110.

13


ABSTRACTS

14



On Smoothness of Solution of One Class Differential EquationsHigher Order

Alik M. NAJAFOV1,2, Sain T. ALEKBERLI3

1Institute of Mathematics and Mechanics of ANAS, AZERBAIJAN,2Azerbaijan University of Architecture and Construction, AZERBAIJAN,

3Baku Engineering University, AZERBAIJAN

Abstract. In this paper, a quasi-elliptic equation of the form is investigated∑

(α, 1l )≤1,

(β , 1l )≤1

Dα(aαβ(x)Dβu(x)) =

∑

(α, 1l )≤1

Dα fα, (4)

Dνu(x)|∂ G = ϕν|∂ G, (ν,1l)< 1, (5)

where G ⊂ Rn, α = (α1, . . . ,αn), β = (β1, . . . ,βn), α j > 0, β j > 0 are integers ( j =

1,2. . . . , n); l ∈ N n; (α, 1l ) =

n∑

j=1

α j

l j. Suppose that the coefficients aαβ(x) ≡ aαβ(x), and

aαβ(x) ∈ L 2−ε1−ε(G)(0< ε < 1) for all (α, 1

l )≤ 1,(β , 1l )≤ 1, and such, that

∑

(α, 1l )=(β , 1

l )=1

(−1)|α|aαβ(x)ξαξβ ≥ c0

∑

(α, 1l )=1

|ξα|2−ε, c0 = const > 0,

where ξ ∈ Rn and for all 0< ε < 1.First, a solution generalized is sought in the grand Sobolev space W l

2)(G), then,using the proven embedding theorem in the grand Sobolev-Morrey space W l

2),a,c(G) in[1] smoothness of the solution of problem (4)-(5) is studied. Note that in this paper thecoefficients of the equation to (4) aαβ(x) belongs to a broader class and "smoothnessindex" more than previous works.

A generalized solution to (4)-(5) in G a function u(x) ∈ W l2)(G), if Dνu − ϕν ∈

L2−ε(G)(0< ε < 1) such that

∑

(α, 1l )≤1,

(β , 1l )≤1

∫

G

(−1)|α|aαβ(x)Dβu(x)Dαv(x)d x =

∑

(α, 1l )≤1

∫

G

(−1)|α| fαDαv(x)d x

For every function v(x) ∈0

W l2(G).

Keywords: Generalized solution, Grand Sobobev-Morrey spaces, Smoothness solu-tion.

15


References

[1] Najafov, A.M., Rustamova, N.R., On properties of functions from Sobolev-Morrey typespaces with dominant mixed derivatives, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech.Math. Sci. Mathematics, 37(4), 2017, 132-141.

16



New Ring of Ponderation Functions and Applications to SolveCauchy-Euler Equation for Several Variables

Miloud ASSALCarthage University, TUNUSIA

Abstract. In this paper we introduce a new ring R of ponderation functions we charac-terize the projective modules and simple modules and we prove that the socle of thisring is not an injective module. As an application we use the ring properties to give anew method to solve equations of the form Div(X f ) = g in several variables. Further-more, we give a general solution of the Cauchy-Euler Equations in high dimensions andthe general solution of some kind of discrete equations.

Keywords: Differential equations, Discrete equations, Ideals, Rings.

References

[1] Assal M., Zeyada N., New ring of a class of Bessel integral operators, Integral Trans-forms Spec Funct., 27(8), 2016, 611-619.[2] Nguon D., Bessel integral operators, Integral Transforms Spec Funct., 25(8), 2014,647-662.[3] Nguon D., Covariant symbolic calculus for Toeplitz operators on the sphere, IntegralTransforms Spec. Funct., 18(4), 2007, 255-269.[4] Atiyah, MF, Macdonald, IG., Introduction to Commutative Algebra, Westview Press:New York, 1969.[5] Pierce R.S., Associative Algebras, Graduate Texts in Mathematics 88, Springer-Verlag,1982.

17


Regularity Estimates in Weighted Generalized Morrey Spaces forQuasilinear Parabolic EquationsTahir S. GADJIEV1, Faig M. NAMAZOV2

1 Institute of Mathematics and Mechanics of ANAS, AZERBAIJAN2 Baku State University, AZERBAIJAN

Abstract. We study regularity for solution of quasilinear parabolic equations in boundeddomain in Rn with a very general irregular domain. The nonlinearity is assumed tobe merely measurable only in the time variable t and belongs to the small boundedmean oscillation (BMO) class as functions of the spatial variable x . We establish gra-dient estimates in weighted generalized Morrey spaces for weak solutions equation.As a consequence, we obtain that gradient of solution in the generalized Morrey spaceor weighted generalized Morrey space whenever right-hand is respectively from thisspaces. A weight is any weight in the Muckenhoupt class. In addition, our two-weightestimate allows the possibility to acquire the regularity for gradient of solution in aweighted generalized Morrey space that is different from the functional space that thedata of right-hand belongs to. The results are new also in case when absence of weight.

Keywords: Calderón-Zygmund estimates, Gradient estimates, Muckenhoupt class,Parabolic equations, p-Laplacian, Regularity, Two-weight inequality, Weightedgeneralized Morrey space.

18



The Uniformly Elliptic and Parabolic Equations of Higher Orderwith Discountinuous Data in Generalized Morrey Spaces and

Elliptic Equations in Unbounded DomainsTahir S. GADJIEV1, Shahla GALANDAROVA1, Konul SULYEMANOVA1

1Institute of Mathematics and Mechanics of ANAS, AZERBAIJAN

Abstract. We study the regularity of the solutions of Cauchy-Dirichlet problem forlinear uniformly parabolic equations of higher order with vanishing mean oscillation(V MO) coefficients. We prove continuity in generalized parabolic Morrey spaces Mp,ϕ

of sublinear operators generated by the parabolic Calderón-Zygmund operator and bythe commutator of this operator with bounded mean oscillation (BMO) functions. We

obtain strong solution belongs to the generalized Sobolev-Morrey space

Wm,1

p,ϕ(Q) . Alsowe consider elliptic equation in unbounded domains.

Keywords: Calderón-Zygmund integrals, Cauchy-Dirichlet problem, Elliptic equation,Generalized Morrey space, Higher order parabolic equations, Reifenberg flat domain,Sublinear operators, Unbounded domain, V MO.

19


Estimations of the Norm of Functions from Lizorkin-Triebel-MorreyType Spaces Reduced by Polynomials

Azizgul M. GASUMOVASumgait State University, AZERBAIJAN

Abstract. In this abstract we give a conditions for the validity of estimation in the Lq(G)norm functions

Dν( f (x)− Pl−1)

where f ∈ F lp,θ ,ϕ,β(G) the spaces type Lizorkin-Triebel-Morrey is defined the finite norm

[1] (mi > li > ki ≥ 0, i = 1,2, . . . , n):

‖ f ‖F lp,θ ,ϕ,β (G)

= ‖ f ‖p,ϕ,β;G +n∑

i=1

1∫

0

δmii (ϕi(t))D

kii f (·)

(ϕi(t))li−ki

θ

dϕi(t)ϕi(t)

1θ

p,ϕ,β

,

δmii (ϕi(t)) =

1∫

−1

∆mii (ϕi(t), Gϕ(t)) f (x)

d t,

‖ f ‖p,ϕ,β;G = ‖ f ‖Lp,θ ,ϕ,β (G) = supx∈Gt>0

|ϕ([t]1)|−β‖ f ‖p,Gϕ(t)(G)

,

Gϕ(t)(x) = G ∩ Iϕ(t)(x) = G ∩§

y : |y j − x j|<12ϕ j(t)( j = 1,2, ·, n)

ª

,

G ∈ Rn, 1 ≤ p <∞, 1 < θ <∞, mi ∈ N , ki ∈ N0, ϕ(t) = (ϕ1(t), . . . ,ϕn(t)), ϕ j(t) >0(t > 0) is continuously differentiable functions; lim

t→+0ϕ j(t) = 0, lim

t→+∞ϕ j(t) = L ≤∞,

|ϕ([t]1)|−β =n∏

j=1(ϕ([t]1))−β j , β j ∈ [0, 1], [t]1 =min1, t.

Main object is method the integralrepresentation of functions from F lp,θ ,ϕ,β(G) Lizorkin-

Triebel-Morrey type spaces defined on dimensional domains satisfy the flexible ϕ horncondition.

We obtained the inequality of the type

‖Dν( f (x)− Pl−1)‖q,G ≤ Cn∑

i=1

1∫

0

δmii (ϕi(t))D

kii f (·)

(ϕi(t))li−ki

θ

dϕi(t)ϕi(t)

1θ

p,ϕ,β

Keywords: Estimates of Lq norm of functions integral representations, Lizorkin-Triebel-Morrey spaces.

20



References

[1] Najafov A.M., Gasumova A.M., On properties of functions from Lizorkin-Triebel-Morrey type spaces, Ukrainian Mathematical Bulletin, 15(2), 2018, 295-297.

21


Problems of Approximation in Generalized Sobolev SpacesLeyla Sh. KADIMOVA

Institute of Mathematics and Mechanics ANAS, AZERBAIJAN

Abstract. In this abstract we study the approximation problems in the generalized

Sobolevn⋂

i=0L l i

pi(G) spaces in [1] with a finite norm:

‖ f ‖ n⋂

i=0L l i

pi (G)=

n∑

i=0

‖Dl if ‖pi(G),

where G ⊂ Rn, l i ∈ N n, 1 ≤ pi <∞ (i = 0, 1, . . . , n), Dl if is generalized mixed deriva-

tives.

In cases l0 = (0, 0, . . . , n), l i = (0, . . . , li, . . . , 0) and pi = p the spacesn⋂

i=0L l i

pi(G)

coincides anisotropic Sobolev spaces W lp(G).

The following theorem is proved.

Theorem. Let 1 < p < ∞, and f ∈n⋂

i=0L l i

pi(G). Then there exist the functions

hs = hs(x) (s = 1,2, . . .) infinitely differentiable finite in Rn such that

lims→∞‖ f − hs‖ n

⋂

i=0L l i

pi (G)= 0.

Keywords: Approximation problems, Generalized Sobolev spaces.

References

[1] Jabrayilov A.J., Investigation of differential-difference properties of functions definedon n-dimensional domains. Doct. thesis., 1972, 385 p.

22



Interpolation Theorems for Type Spaces Besov-Morrey withDominant Mixed Derivatives

Nilufer R. RUSTAMOVAInstitute of Mathematics and Mechanics ANAS, AZERBAIJAN

Abstract. In the abstract we study a differential and differential-difference proper-ties of functions from intersection of type spaces Besov-Morrey with dominant mixedderivatives S lµ

pµ,θµ,ϕ,βB(G) (µ = 1,2, . . . , N) , where G ⊂ Rn, 1 ≤ pµ <∞, 1 ≤ θµ ≤∞,

lµ = (lµ1 , lµ2 , . . . , lµn ), lµj > 0 ( j ∈ en = 1, 2, . . . , n), and lµe= (lµ

e

1 , lµe

2 , . . . , lµe

n ), lµe

j = lµj( j ∈ e ⊆ en), lµ

e

j = 0( j ∈ en\e); and vector-functions ϕ(t) = (ϕ1(t1), . . . ,ϕn(tn)),ϕ j(t j) > 0 and ϕ′j(t j) > 0 (t > 0, j ∈ en) is continuously differentable functions;limt→+0

ϕ j(t j) = 0, limt→+∞

ϕ j(t j) = L ≤∞, and β ∈ [0, 1]n.

The spaces S lp,θ ,ϕ,βB(G) is defined [1] as a linear normed space of functions f , on

G, with the finite norm (mi > li − ki > 0 j ∈ en) :

‖ f ‖S lp,θ ,ϕ,βB(G) =

=∑

e⊆en

t e0∫

0e

∆me(ϕ(t), Gϕ(t))Dke

f

p,ϕ,β∏

j∈e(ϕ j(t j))l j−k j

θ

∏

j∈e

dϕ j(t j)

ϕ j(t j)

1θ

,

where‖ f ‖p,ϕ,β;G = ‖ f ‖Lp,ϕ,β (G) = sup

x∈G,t>0,j∈en

|ϕ j([t j]1)|−β j‖ f ‖p,Gϕ(t)(x)

here m j ∈ N , k j ∈ N0; [t]1 = min1, t; j ∈ en; t0 = (t01, . . . , t0n) fixed positivevector, t e

0 = (te01, t e

02, . . . , t e0n), t e

0 j = t0 j ( j ∈ e), t e0 j = 0( j ∈ en\e) and Gϕ(t) (x) =

G ∩

y :

y j − x j

< 12ϕ j(t), j ∈ en

.By the method of integral representations is proved embedding theorems of the

type:

1. Dν :N⋂

µ=1S lµ

pµ,θµ,ϕ,βB(G) ,→ Lq,ψ,β(G) (C(G))

2. Dν :N⋂

µ=1S lµ

pµ,θµ,ϕ,βB(G) ,→ S l1

q,θ1,ψ,β1B(G) (l1 ∈ (0,∞)n,θµ < θ

1,µ = 1,2, . . . , N) is

holds;

3. it is also proved that for the function from spaceN⋂

µ=1S lµ

pµ,θµ,ϕ,βB(G) the generalized

derivatives Dν f satisfy the Hölder condition in the metric Lq(G) and C(G).

23


Keywords: Besov-Morrey type spaces, Interpolation theorems, Mixed derivatives.

References

[1] Rustamova N.R., The embbeding theorems for type spaces Besov-Morrey with domi-nant mixed derivatives, International Conference MADEA-8, June 2018, 102-103, Bishkek,Kyrgyz Republic.

24



Characterizations for the Fractional Maximal CommutatorOperator in Generalized Morrey Spaces on Carnot Group

Ismail EKINCIOGLUKutahya Dumlupinar University, TURKEY

Abstract. The boundedness of the fractional maximal operator and its commutatorsplays an important role in harmonic analysis. In recent decades, many authors haveproved the boundedness of the commutators with BMO functions of fractional maxi-mal operator in Rn. In this study generalizes the certain known results in generalizedMorrey spaces on nilpotent (stratified) Lie groups.

Carnot groups appear in quantum physics and many parts of mathematics, includingFourier analysis, several complex variables, geometry and topology. Analysis on thegroups is also motivated by their role as the simplest and the most important model inthe general theory of vector fields satisfying Hörmander’s condition (see [1]).

In this work, we will obtain the new results of the strong and weak Spanne-Guliyev,Adams-Guliyev and Adams-Gunawan type boundedness characterization of the frac-tional maximal operator Mα in generalized Morrey spaces, respectively. Moreover, wewill give a characterization for the boundedness of the fractional maximal commutatoroperator Mb,α in generalized Morrey spaces (see [2]).

Keywords: BMO, Commutator, Carnot group, Fractional maximal operator, General-ized Morrey space.

References

[1] Bonfiglioli A., Lanconelli E, Uguzzoni F., Stratified Lie groups and potential theoryfor their sub-Laplacians, Springer Monographs in Mathematics, Springer: Berlin, 2007,xxvi+802.[2] Guliyev V.S., Ekincioglu I., Kaya E., Safarov Z., Characterizations for the fractionalmaximal operator and its commutators in generalized Morrey spaces on Carnot groups,Integral Transforms and Special Functions, 30(6), 2019, 453-470.

25


Weighted Norm Inequalities for Bn-maximal Operators and theHigh Order Riesz-Bessel Transforms

Esra KAYA1, Ismail EKINCIOGLU1

1Kutahya Dumlupinar University, TURKEY

Abstract. In this paper, we investigate the weighted norm inequalities for Bn-singularintegral operators associated with the ∆Bn

Laplace-Bessel differential operator

∆Bn=

n−1∑

i=1

∂ 2

∂ x2i

+∂ 2

∂ x2n

+γ

xn

∂

∂ xn, γ > 0,

with smoothness conditions assumed on the kernels.We will concentrate on studying Coifman’s inequality:

∫

Rn+

|R(k)γ

f (x)|pω(x)xγnd x ≤ C

∫

Rn+

Mγ f (x)pω(x)xγnd x

for any 0 < p < ∞ and ω ∈ A∞,γ, where R(k)γ

denotes the high order Riesz-Besseltransforms and Mγ is the Bn-maximal operator generated by generalized shift operator,and ω satisfies necessary and sufficient conditions.

Keywords: Bn-maximal operators, Generalized shift operator, Riesz-Bessel transforms.

References

[1] Ekincioglu I., The boundedness of high order Riesz-Bessel transformations generatedby the generalized shift operator in weighted Lp,ω,γ-spaces with general weights, Acta Appl.Math., 109, 2010, 591-598.[2] Guliyev V.S., On maximal function and fractional integral, associated with the Besseldifferential operator, Math. Inequal. Appl., 6, 2003, 317-330.

26



Poincare Type Inequality in Generalized Sobolev-Morrey TypeSpaces

Rovshan F. BABAYEV

Mingachevir State University, AZERBAIJAN

Abstract. In this paper, by means of integral representation method, we estimate the

norms of functions from generalized Sobolev-Morrey type spacesn⋂

i=0L⟨l

i⟩pi ,ϕ,β(G) intro-

duced in [1], reduced by polynomials, defined in n-dimensional domaing satisfying theflexible ϕ-horn condition.

Let G ⊂ Rn, l i = (l i1, . . . , l i

n), l ij ≥ 0 (i 6= j = 1,2, . . . , n) are integers, l i

i > 0 beintegers, 1 ≤ pi < ∞, ϕ(t) = (ϕ1(t), . . . ,ϕn(t)), ϕ j(t) > 0 (t > 0) by Lebesguemeasurable functions; lim

t→+0ϕ j(t) = 0, and lim

t→+∞ϕ j(t) = L ≤∞, β ∈ [0, 1]n.

Norm in spacesn⋂

i=0L⟨l

i⟩pi ,ϕ,β(G)

‖ f ‖ n⋂

i=0L⟨l

i ⟩pi ,ϕ,β

(G)=

n∑

i=0

‖Dl if ‖pi ,ϕ,β;G,

where‖ f ‖p,ϕ,β;G = ‖ f ‖Lp,ϕ,β (G) = sup

x∈G,t>0

|ϕ([t]1)|−β‖ f ‖p,Gϕ(t)(x)

.

Gϕ(t)(x) = G ∩ Iϕ(t)(x) = G ∩

y : |y j − x j|<12ϕ j(t), j = 1, . . . , n

.

[t]1 =min1, t.

The spacesn⋂

i=0L⟨l

i⟩pi ,ϕ,β(G) in the case l0 = (0, . . . , 0), l i = (0, . . . , 0, l i, 0, . . . , 0), pi = p,

(i = 0, 1, . . . , n) coincides spaces type W lp,ϕ,β(G). In other words, we prove the inequal-

ity‖Dν( f (x)− Pl−1)‖q,G ≤ C‖ f ‖ n

⋂

i=0L⟨l

i ⟩pi ,ϕ,β

(G).

Keywords: Generalized Sobolev-Morrey spaces, Poincare type inequality.

27


References

[1] Najafov A.M., Babayev R.F., Some properties of functions from generalized Sobolev-Morrey type spaces, Mathematica Aeterna, 7(3), 2017, 301-311.

28



Approximate Solutions of Nonlinear Bernoulli Equation by UsingBernstein Polynomials with Error Analysis

Fadil JARADAT1, M.H.T. ALSHBOOL1

1Abu Dhabi University, UNITED ARAB EMIRATES

Abstract. We present an approximate solution depending on collocation method andBernstein polynomials for numerical solution of nonlinear Bernoulli equation. Themethod is given with two different priori error estimates. By using the residual cor-rection procedure, the absolute error might be estimated and obtained more accurateresults.

29


The Regularity of Solutions Elliptic Equations of Higher Order withDiscontinuous Data in Generalized Orlicz-Morrey Spaces

Tahir S. GADJIEV1, Konul YASINLI2

1Institute of Mathematics and Mechanics of ANAS, AZERBAIJAN2Nakhchivan State University, AZERBAIJAN

Abstract. We study the regularity of the solutions of Cauchy-Dirichlet problem for lin-ear uniformly elliptic equations of higher order with vanishing mean oscillation (V MO)coefficients. We prove continuity in generalized Orlicz-Morrey spaces of sublinear op-erators generated by the Calderón-Zygmund operator and by the commutator of thisoperator with bounded mean oscillation (BMO) functions. We obtain strong solutionbelongs to the generalized Sobolev-Orlicz-Morrey space.

Keywords: Calderón-Zygmund integrals, Cauchy-Dirichlet problem, Elliptic equation,Higher order elliptic equations, Orlicz-Morrey space, Reifenberg flat domain, Sublinearoperators, V MO.

30



Poincare Type Inequality in Lizorkin-Triebel-Morrey Type Spaces

Firida F. MUSTAFAYEVAShamakhi Branch of Azerbaijan State Pedagogigal University, AZERBAIJAN

Abstract. In this abstract we give a conditions for the validity of estimation in the Lq(G)norm functions

Dν( f (x)− Pr0−1)

where f ∈ F rp,θ ,a,c,τ(G), G ∈ Rn,r ∈ (0,∞)n, 1 ≤ p <∞, 1 < θ <∞,a ∈ [0, 1]n,

c ∈ (0,∞)n, 1≤ τ≤∞.By help is integral representations of functions from Lizorkin-Triebel F r

p,θ (G) spaces,defined on n-dimensional domains satisfy the flexible horn condition.

We obtained the inequality of the type

‖Dν( f (x)− Pr−1)‖q,G ≤ C‖ f ‖F rp,θ ,a,c,τ(G)

,

ν = (ν1, . . . ,νn),ν j ≥ 0 be integers ( j = 1,2, . . . , n). The spaces F rp,θ ,a,c,τ(G) is defined

[1] with the finite norm (mi > ri > ki ≥ 0, i = 1, 2, . . . , n):

‖ f ‖F rp,θ ,a,c,τ(G)

= ‖ f ‖p,a,c,τ;G +n∑

i=1

1∫

0

δmii (t)D

kii f (·)

tλi(ri−ki)i

θ

d tt

1θ

p,a,c,τ

,

‖ f ‖p,a,c,τ;G = ‖ f ‖Lp,a,c,τ(G) = supx∈G

∞∫

0

[t]− (c,a)p

1 ‖ f ‖p,Gtc (x)

τ d tt

1τ

,

δmii (t) f (x) =

1∫

−1

∆mii (t, Gtc) f (x)

du,

mi ∈ N , ki ∈ N0, [t]1 =min1; t;

Gtc(x) = G ∩ Itc(x) = G ∩§

y : |y j − x j|<12

tc j( j = 1,2, ·, n)ª

, for all Rn.

Keywords: Estimates of Lq norm of functions integral representations, Lizorkin-Triebel-Morrey type spaces.

31


References

[1] Guliyev V.S., Najafov A.M., The imbedding theorems on the Lizorkin-Triebel-Morreytype spaces, Progress in Analysis, 1, 2003, 23-30.

32



Abstract Korovkin Theorems via Relative Summation Process forDouble Sequences on Modular Spaces

Sevda YILDIZSinop University, TURKEY

Abstract. We obtain an abstract version of the Korovkin type approximation theoremsvia statistical relative A -Summation process in modular spaces for double sequencesof positive linear operators. Hence, we changed classical test functions of Korovkintheorem. Then, we give an application showing that our results are stronger. Finally,we study an extension to non-positive linear operators.

Keywords: Abstract Korovkin theorem, Double sequence, Matrix summability, Modularspaces, Statistical convergence.

References

[1] Bardaro C., Mantellini I., Approximation properties in abstract modular spaces for aclass of general sampling-type operators, Appl. Anal., 85, 2006, 383-413.[2] Bardaro C., Mantellini I., Korovkin’s theorem in modular spaces, CommentationesMath., 47, 2007, 239-253.[3] Bardaro C., Mantellini I., A Korovkin theorem in multivariate modular function spaces,Journal of Function Spaces, 7(2), 2009, 105-120.[4] Bardaro C., Boccuto A., Dimitriou X., Mantellini I., Abstract Korovkin-type theo-rems in modular spaces and applications, Central European Journal of Mathematics,11(10),2013, 1774-1784.[5] Demirci K., Orhan S., Statistical relative approximation on modular spaces, Resultsin Mathematics, 71(3-4), 2017, 1167-1184.[6] Demirci K., Kolay, B., A -statistical relative modular convergence of positive linearoperators, Positivity, 21(3), 2017, 847863.[7] Demirci K.,Orhan S., Kolay B., Statistical RelativeA -Summation Process for DoubleSequences on Modular Spaces, Revista de la Real Academia de Ciencias Exactas, Fisicasy Naturales. Serie A. Matematicas, 2018, 1-16.[8] Karakus S., Demirci K., Duman O., Statistical approximation by positive linear oper-ators on modular spaces, Positivity, 14(2), 2010, 321-334.[9] Karakus S., Demirci K.,A -summation process and Korovkin-type approximation the-orem for double sequences of positive linear operators, Math. Slovaca, 62, 2012, 281-292.[10] Kolay B., Orhan S., Demirci K., Statistical Relative A -Summation Process andKorovkin-Type pproximation Theorem on Modular Spaces, Iranian Journal of Science andTechnology, Transactions A: Science, 2016, 1-10.

33


Approximation in Weighted Morrey Spaces by TrigonometricPolynomials

Canay AYKOL1, Ayhan SERBETCI2

1Ankara University, TURKEY,2Cankiri Karatekin University, TURKEY

Abstract. In this presentation we investigate the best approximation in weighted Mor-rey spaces Mp,λ(I0, w) by trigonometric polynomials, where the weight function wis in the Muckenhoupt class Ap(I0) with 1 < p < ∞ and I0 = [0,2π]. We provethe direct and inverse theorems of approximation by trigonometric polynomials in thespaces ÝMp,λ(I0, w) the closure of C∞(I0) inMp,λ(I0, w). We give the characterizationof K−functionals in terms of the modulus of smoothness and obtain the Bernstein typeinequality for trigonometric polynomials in the spacesMp,λ(I0, w).

Keywords: Bernstein inequality, Best approximation, Muckenhoupt class, Trigonomet-ric polynomials, Weighted Morrey space.

References

[1] Cakir Z., Aykol C., Soylemez D., Serbetci A., Approximation by trigonometric poly-nomials in Morrey spaces, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci.,39(1), 2019, 24-37.

34



On φ-Recurrent Kenmotsu Finsler Manifolds

A. Funda SAGLAMER1, Nurten KILIC1


Abstract. The objective of this study is to define φ-Recurrent Kenmotsu structures onFinsler manifolds. These structures are established on the (M0)h and (M0)v vector sub-bundles where M is an (2n+1)-dimensional C∞ manifold, M0 is a non-empty open sub-manifold of T M . F is fundamental Finsler function and F2n+1 = (M , M0, F) is a Finslermanifold. Firstly, we present φ-Recurrent Kenmotsu Finsler manifolds and give somedefinitions and lemmas. Then, we prove thatφ-Recurrent Kenmotsu manifold is an Ein-stein Finsler manifold and φ-Recurrent Kenmotsu manifolds ((M0)h,φH ,ξH ,ηH , GH)and ((M0)v,φV ,ξV ,ηV , GV ) are the manifolds of constant curvatures −1

4 , i.e, they arelocally hiperbolic spaces. Also, we study three-dimensional Kenmotsu Finsler mani-folds and obtain some results for curvatures of these manifolds. Finally, φ-RecurrentKenmotsu manifolds are compared to φ-Recurrent Kenmotsu structure on Riemannianmanifold.

Keywords: Finsler metric, Kenmotsu manifolds, Locally φ-recurrent Kenmotsu mani-folds, φ-recurrent Kenmotsu manifolds.

References

[1] Caliskan N., Saglamer A. F. and Kılıç N., Kenmotsu Finsler Structures on Vector Bun-dles (submitted).[2] De U. C., Shaikh A. A. and Biswas S., On φ-Recurrent Sasakian Manifolds, Novi SadJ. Math., 3, 2003, 13-48.[3] De U.C. and Pathak G., On 3-dimensional Kenmotsu Manifolds, Indian J. Pure Ap-plied Math., 35, 2004, 159-165.[4] De U. C., Yıldız A. and Yalınız A.F., On φ-Recurrent Kenmotsu Manifolds, Turk J.Math., 33, 2009, 17-25.[5] Dileo G., Pastore A. M., Almost Kenmotsu manifolds and local symmetry, Bulletin ofthe Belgian Mathematical Society-Simon Stevin, 14(2), 2007, 343-354.[6] Jun J-B, De U. C. and Pathak G., On Kenmotsu Manifolds, J. Korean Math. Soc., 42,2005, 435-445.[7] Kenmotsu K., A class of almost contact Riemannian manifolds, Tohoku MathematicalJournal, Second Series, 24(1), 1972, 93-103.[8] Prasad K. L., Kenmotsu and P-Kenmotsu Finsler structures and connections on vectorbundle, International Mathematical Forum, 3(17), 2008.[9] Sinha B. B., Yadav R. K., On almost contact Finsler structures on vector bundle, IndianJ. pure appl. Math., 19(1), 1988, 27-35.

35


Monic Torsion Precovers Relative to A Torsion TheoryMustafa Kemal BERKTAS1, Semra DOGRUOZ2

1Usak University, TURKEY2Adnan Menderes University, TURKEY

Abstract. Let R be any associative ring with identity and let τ be a torsion theory onmod-R. For an R-module M , a submodule N of M is called τ-pure in M if M/N isτ-torsionfree. In this talk, first we introduce a τ-torsion precover and some of its ap-plications, and then show the existence of τ-pure submodules relative to the existenceof monic τ-torsion precovers.

Keywords: Torsion theory, τ-pure submodule, τ-torsion precover.

References

[1] Berktas M. K., Dogruoz S., A relative extending module and torsion precovers. Bull.Iranian Math. Soc., 41(5), 2015, 1249-1257.[2] Crivei S., Injective modules relative to torsion theories, Efes Publishing House, 2004.

36



An Application of Gauss-Chebyshev Quadrature Formulas to theGeneralized Frictionless Contact Problem

Elcin YUSUFOGLU1, llkem TURHAN CETINKAYA2

1Usak University, TURKEY,2Kutahya Dumlupinar University, TURKEY

Abstract. In this study, the generalization of a frictionless contact problem in case ofshearing deformation for an elastic inhomogeneous half space is presented. The basicequations of the elasticity theory and Fourier transform technique are applied to theproblem to derive the system of singular integral equations. The obtained system ofsingular integral equations is approached by Gauss-Chebyshev quadrature formulas.The numerical results are presented for the case of N=1, N=2, N=3, where N denotesthe number of the punches whose base are flat.

Keywords: Contact problems, Elasticity theory, Integral equations.

References

[1] Aizikovich S.M., Aleksandrov V.M., Belokon A.V. , Krenev L. I., Trubchik I. S., Con-tact problems of the theory of elasticity for non-homogeneous medium, Fizmatli, 2006 (InRussian).[2] Vasiliev A., Volkov S., Aizikovich S., Jeng Y.R., Axisymmetric contact problems of thetheory of elasticity for inhomogeneous layers, Journal of Applied Mathematics and Me-chanics, 94(9), 2014, 705-712.[3] Generalova N.V., Kovalenko Ye.V., The effect of a strip-shaped punch on a linearlydeformable foundation strengthened by a thin covering, Journal of Applied Mathematicsand Mechanics, 59(5), 1995, 789-795.[4] Singh B.M., Rokne J., Dhaliwal R.S., Vrbik J., Contact problem for bonded nonho-mogeneous materials under shear loading, International Journal of Mathematics andMathematical Sciences, 29, 2003, 1821-1832.[5] Aleksandrov V.M., Smetanin B.I., Sobol B.V., Thin Stress Concentrators in ElasticSolids, Nauka, Moskow, 1993.[6]Muskheleshvili N.I., Singular Integral Equations, Edited by J.R.M. Rodok, NoordhoffInternational publishing Leyden, 1997.[7] Babeshko V.A., Glushkov E.V., Glushkova N.V., Methods for constructing the Greenfunction of a stratified elastic half-space, Zh. Vychisl. Mat. Mat. Fiz., 27(1), 1987,93–101.[8] Erdogan F., Gupta G.D., Cook T.S., Numerical solution of singular integral equations,In: Sih GC, editor. Method of analysis and solution of crack problems, Leyden: NoordhoffInternational Publishing, 1973.

37


[9] Kahya V., Birinci A., Erdol R., Frictionless Contact Problem Between Two OrthotropicElastic Layers, World Academy of Science, Engineering and Technology InternationalJournal of Civil, Architectural Science and Engineering, 1(1), 2007.

38



Multivalent Harmonic Starlike Functions Defined by SubordinationSibel YALCIN TOKGOZ1, Sahsene ALTINKAYA1

1Bursa Uludag University, TURKEY

Abstract. We have introduced a generalized class of complex-valued multivalent har-monic starlike functions defined by subordination. We study some properties of ourclass. The results obtained here include a number of known and new results as theirspecial cases.

Keywords: Harmonic functions, Multivalent functions, Starlike functions, Subordina-tion.

References

[1] Ahuja, O. P., Jahangiri J. M., Multivalent harmonic starlike functions, Ann. Univ.Mariae Cruie Sklod. Sec. A., 55(1), 2001, 1-13.[2] Ahuja, O. P., Jahangiri J. M., On a linear combination of classes of multivalently har-monic functions, Kyungpook Math. J., 42(1), 2002, 61-70.[3] Clunie, J., Sheil-Small, T., Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser.A I Math., 9, 1984, 3-25.[4] Dziok, J., Jahangiri, J. M., Silverman, H., Harmonic functions with varying coeffi-cients, Journal of Inequalities and Applications, 139, 2016, 1-12.[5] Jahangiri, J. M., Harmonic functions starlike in the unit disk, J. Math. Anal. Appl.,235, 1999, 470-477.[6] Jahangiri, J. M., Seker, B., Sumer Eker, S., Salagean-type harmonic multivalent func-tions, Acta Univ. Apulensis, 18, 2016, 233-244.[7] Yasar, E., Yalcin, S., On a new subclass of Ruscheweyh-type harmonic multivalent func-tions, Journal of Inequalities and Applications, 271, 2013, 1-15.[8] Yasar, E., Yalcin, S., Partial sums of starlike harmonic multivalent functions, AfrikaMathematica, 26(1-2), 2015, 53-63.[9] Yasar, E., Yalcin, S., Neighborhoods of a new class of harmonic multivalent functions,Comput. Math. Appl., 62(1), 2011, 462-473.

39


Some New Beta-Fractional Integral InequalitiesDeniz UCAR1, Veysel Fuat HATIPOGLU2

1Usak University, TURKEY2Mugla University, TURKEY

Abstract. Several definitions of fractional derivative have been proposed in the liter-ature, but some basic properties of derivative cannot be satisfied. In this paper, weintroduce a fractional derivative called beta-derivative which is easier and respects ba-sic properties. We investigate some useful inequalities for this new simple interestingfractional calculus. We also obtain some results close to the results in classical calculususing a new parameter.

Keywords: Beta-fractional calculus, Fractional integral inequalities.

References

[1] Atangana A., Derivative with a New Parameter, Theory, Methods and Applications,Elsevier, 2016.[2] Ross B., Fractional Calculus and Its Applications, Springer-Verlag, 1975.[3] Anderson D.R., Taylor’s formula and integral inequalities for conformable fractionalderivatives, Contributions in Mathematics and Engineering, Springer International Pub-lishing, 2016, 25-43.[4] Khalil R., Horani M.A., Yousef A., Sababheh M., A new definition of fractional deriva-tive, Journal of Computational and Appl. Math., 264, 2014, 65-70.

40



A New Local Smoothing Technique for Non-smooth FunctionsNurullah YILMAZ

Suleyman Demirel University, TURKEY

Abstract. In this study, we propose a new local smoothing technique for some sub-classes of the non-smooth functions. We introduce useful properties of this new smooth-ing technique. Finally, the application of the technique is illustrated numerically onnon-smooth test problems for global optimization.

Keywords: Non-smooth analysis, Optimization, Smoothing technique.

References

[1] Bagirov A., Hyperbolic smoothing functions for nonsmooth minimization, Optimiza-tion, 62(6), 2013, 759-782.[2] Chen X., Smoothing methods for nonsmooth, nonconvex minimization, Math. Pro-gram., Ser. (B), 134, 2012, 71-99.[3] Bertsekas D., Nondifferentiable optimization via approximation, Mathematical Pro-gramming Study, 3, 1975, 1-25.[4] Zang I., A smooting out technique for min-max optimization, Math. Program, 19,1980, 61-77.[5] Xavier A.E., The hyperbolic smoothing clustering method, Patt. Recog., 43, 2010,731-737.[6] Grossmann C., Smoothing techniques for exact penalty function methods, Contempo-rary Mathematics, In book: Panorama of Mathematics: Pure and Applied, 658, 2016,249-265.

41


The (p, q)-Lucas Polynomial Coefficient Estimates of a Bi-univalentFunction Class with Respect to Symmetric Points

Sahsene ALTINKAYA1, Sibel YALCIN TOKGOZ1

1Bursa Uludag University, TURKEY

Abstract. The main idea of this current paper stems from the work of Lee and Asci [8].By using the (p, q)-Lucas polynomials, we aim to build a bridge between the theoryof geometric functions and special functions, which are usually considered as differentfields.

Keywords: Bi-univalent functions, Coefficient bounds, (p, q)-Lucas polynomials.

References

[1]Altinkaya S., Yalcin S., Faber polynomial coefficient bounds for f subclass of bi-univalentfunctions, C. R. Acad. Sci. Paris, Ser. I, 353(12), 2015, 1075-1080.[2] Altinkaya S., Yalcin S., Faber polynomial coefficient estimates for bi-univalent func-tions of complex order based on subordinate conditions involving of the Jackson (p, q)-derivative, Miskolc Mathematical Notes, 17(2), 2017, 1075-1080.[3] Brannan D.A., Clunie J.G., Aspects of contemporary Complex Analysis, Proceedingsof the NATO Advanced Study Institute Held at University of Durham, 1979.[4] Brannan D.A., Taha T.S., On some classes of bi-univalent functions, Studia Universi-tatis Babes-Bolyai Mathematica, 31(2), 1986, 70-77.[5] Duren P.L., Univalent functions, Grundlehren der Mathematischen Wissenschaften,Springer, 1983.[6] Fekete M., Szego G., Eine Bemerkung Uber ungerade Schlichte funktionen, J. LondonMath. Soc., 1-8(2), 1983, 85-89.[7] Lewin M., On a coefficient problem for bi-univalent functions, Proc. Amer. Math.Soc., 18(1), 1967, 63-68.[8] Lee G.Y., Asci M., Some properties of the (p, q)-Fibonacci and (p, q)-Lucas polynomi-als, Journal of Applied Mathematics, 2012, 1-18.[9] Lupas A., A guide Of Fibonacciand Lucas polynomials, Octagon Mathematics Maga-zine, 7, 1999, 2-12.[10]Ma R., Zhang W., Several identities involving the Fibonacci numbers and Lucas num-bers, Fibonacci Q, 45, 2007, 164-170.[11]Ma W.C.,Minda D., A unified treatment of some special classes of univalent functions,Conf. Proc. Lecture Notes Anal. I, Int. Press, Cambridge, MA, 1994.[12] Netahyahu E., The minimal distance of the image boundary from the origin and thesecond coefficient of a univalent function in |z| < 1, Archive for Rational Mechanics andAnalysis, 32, 1969, 100-112.

42



[13] Filipponi P., Horadam A.F., Derivative sequences of Fibonacci and Lucas polynomials,Applications of Fibonacci Numbers, 4, 1991, 99-108.[14] Ravichandran V., Starlike and convex functions with respect to conjugatepoints, ActaMath. Acad. Paedagog. Nyireg., 20, 2004, 31-37.[15] Sakaguchi K., On a certain univalent mapping, J. Math. Soc. Japan, 11, 1959,72-75.[16] Srivastava H.M., Mishra A.K., Gochhayat P., Certain subclasses of analytic and bi-univalent functions, Applied Mathematics Letters, 23, 2010, 1188-1192.[17] Srivastava H.M., Murugusundaramoorthy G., Magesh N., Certain subclasses of bi-univalent functions associated with the Hohlov Operator, Applied Mathematics Letters,1, 2013, 67-73.[18]Wang T., Zhang W., Some identities involving Fibonacci, Lucas polynomials and theirapplications, Bull. Math. Soc. Sci. Math. Roum., 55, 2012, 95-103.[19] Wang Z.G. Gao C.Y., Yuan S.M., On certain subclasses of close-to-convex and quasi-convex functions with respect to k-symmetric points, J. Math. Anal. Appl., 322, 2006,97-106.[20] Zireh A., Adegani E.A., Bulut S., Faber polynomial coefficient estimatesfor a compre-hensive subclass of analytic bi-univalent functions defined by subordination, Bull. Belg.Math. Soc. Simon Stevin, 23, 2016, 487-504.

43


Approximation of the Reachable Sets of an SEIR Control Systemand Comparison with Optimal Control Solution

A. Serdar NAZLIPINAR1, Barbaros BASTURK1


Abstract. The most commenly implemented models in epidemiology are the SIR andSEIR models. In the SIR model population is divided into three groups as suscepti-ple, infectious and recovered individuals. But many diseases have a latent phase dur-ing which the individual is infected but not yet infectious. SEIR model is obtained byadding an extra compartment, the so-called exposed class(E). In the study, we considerthe reachable sets of an SEIR control system with integral constraints on control func-tions. This kind of control functions mean that the whole control stock is limited andexhausted by using. Because of the similiarities with optimal control problems, it isbeneficial to investigate the optimal control solution of the system with an appropriatecost function. At the end of the study, we make a comparision beetween the optimalcontrol solution and the reachable sets of the system.

Keywords: Optimal control, Reachable sets, SEIR model.

References

[1] Guseinov Kh.G., Nazlipinar A.S., On the continuity property of Lp balls and an appli-cation, J.Math. Anal. Appl., 335, 2007, 1347-1359.[2] Guseinov Kh.G., Nazlipinar A.S., On the continuity properties of attainable sets ofnonlinear control systems with integral constraint on controls, Abstr. Appl. Anal., 2008,p.14.[3] Guseinov KH.G., Approximation of the attainable sets of the nonlinear control systemswith integral constraints on control, Nonlinear Analysis, TMA, 71, 2009, 622-645.[4] Guseinov Kh.G., Nazlipinar A.S., An algorithm for approximate calculation of the at-tainable sets of the nonlinear control systems with integral constraint on controls, Comp.Math. Appl., 62(4), 2011, 1887-1895.

44



On Totally Asymtotically Nonexpansive Mappings in Cat(0) SpacesA. ABKAR1, Mojtaba RASTGOO1

1Imam Khomeini International University, IRAN

Abstract. In this article, we prove the existence of fixed points for totally asymptoti-cally quasi-nonexpansive mappings on CAT(0) spaces. We prove a strong convergencetheorem under suitable conditions. The result we obtain improves and extends severalrecent results stated by many others; they also complement many known recent resultsin the literature.

Keywords: Iterative algorithm, Totally asymptotically quasi-nonexpansive mapping,∆-convergence, CAT (0) space.

45


Integral Operators in Anisotropic Morrey SpacesAndrea SCAPELLATO

Università degli Studi di Catania, ITALY

Abstract. The aim of this study is to investigate the behavior of fractional integrals asso-ciated to operators with Gaussian kernel bounds in the framework of some anisotropicMorrey spaces. Precisely, we obtain boundedness properties for the above operators andtheir commutators with functions having bounded mean oscillation in Morrey spaceswith iterated norm, recently defined in [1].

Following [1,2], if 1 < p, q <∞, 0 < λ < n, 0 < µ < 1, T > 0, let us definethe mixed Morrey space Lq,µ(0, T, Lp,λ(Rn)) as the class of functions f such that thefollowing norm is finite:

‖ f ‖Lq,µ(0,T,Lp,λ(Rn)) :=

supt0,t∈(0,T )ρ>0

1ρµ

∫

(0,T )∩(t0−ρ,t0+ρ)

supx∈Rnρ>0

1ρλ

∫

Bρ(x)

| f (y, t)|p dy

qp

dt

1q

,

where Bρ(x) is the ball centered in x ∈ Rn and with radius ρ.

Keywords: Bounded mean oscillation, Commutators, Gaussian bounds, Integral oper-ators, Mixed Morrey spaces.

References

[1] Ragusa M.A., Scapellato A., Mixed Morrey spaces and their applications to partialdifferential equations, Nonlinear Anal.-Theory Methods Appl., 151, 2017, 51-65.[2] Scapellato A., New perspectives in the theory of some function spaces and their appli-cations, AIP Conference Proceedings, 1978, 2018.

46



Lipschitz Estimates for Rough Fractional Multilinear IntegralOperators on Local Generalized Morrey Spaces

S. Elifnur EKINCIOGLU1, A. Serdar NAZLIPINAR1


Abstract. Let γ = (γ1,γ2, . . . ,γn), and γi (i = 1, 2, . . . , n) be nonnegative integers.Denote |γ|=

∑ni=1 γi and

γ!= γ1!γ2! . . .γn!, xγ = xγ11 xγ2

2 . . . xγnn ,

Dγ =∂ |γ|

∂ γ1 x1 ∂ γ2 x2 . . . ∂ γn xn.

Suppose that Ω ∈ Ls(Sn−1) (s > 1) is homogeneous of degree zero on Rn withzero means value on Sn−1, A is a function defined on Rn. Rough fractional multilinearintegral operator IA,m

Ω,α , is defined by

IA,mΩ,α f (x) =

∫

Rn

Rm(A; x , y)|x − y|n−α+m−1

Ω(x − y) f (y)d y,

where 0< α < n, and Rm(A; x , y) is the m-th remainder of Taylor series of A at x abouty . More precisely,

Rm(A; x , y) = A(x)−∑

|γ|<m

1γ!

DγA(y)(x − y)γ.

The main goal of the work is to prove the Lipschitz boundedness for a class of frac-tional multilinear operators IA,m

Ω,α with rough kernels Ω ∈ Ls(Sn−1), s > n/(n − α) onthe local generalized Morrey spaces LM x0

p,ϕ , where the functions A belong to homo-geneous Lipschitz space Λβ , 0 < β < 1. We find the sufficient conditions on the pair(ϕ1,ϕ2)which ensures the boundedness of the operators IA,m

Ω,α from LM x0p,ϕ1

to LM x0q,ϕ2

for1 < p < q <∞ and 1/p− 1/q = (α+ β)/n. In all cases the conditions for the bound-edness of the operator IA,m

Ω,α is given in terms of Zygmund-type integral inequalities on(ϕ1,ϕ2), which do not assume any assumption on monotonicity of ϕ1(x , r), ϕ2(x , r) inr.

This contribution is based on recent joint works with V.S. Guliyev and I. Ekincioglu.

Keywords: Fractional multilinear integral, Lipshitz function, Local generalized Morreyspace, Rough kernel.

47


On the Explicit Solution of A Coupled System of Fractional PartialDifferential Equations Using Symmetry

Nisrine MAAROUF1, Khalid HILAL1

1University Sultan Moulay Slimane, MOROCCO

Abstract. The method of Lie group analysis can be successfully extended to the in-vestigation of symmetry proprieties of Fractional Differential Equations, and can beeffectively used for constructing exact solutions of these Equations. We have adaptedthe methods of Lie continuous groups for symmetry analysis of a system of FractionalDifferential Equations and proposed prolongation formulae for fractional derivatives.To illustrate our results ,some examples are given to construct the explicit solution byusing Lie Symmetry Techniques.

Keywords: Fractional differential equations, Transformation, Lie symmetry analysis,Riemann-Liouville fractional derivative.

References

[1] Bluman G.W., Anco S., Symmetry and Integration Methods for Differential Equations,Springer-Verlag, Heidelburg, 2002.[2] Buckwar E., Luchko Y., Invariance of a partial differential equation of fractional orderunder the Lie group of scaling transformations, J. Math. Anal.Appl., 227, 1998, 81-97.[3] Gazizov R.K., Kasatkin A.A., Lukashchuk S.Yu., Symmetry properties of fractionaldiffusion equations, Phys. Scr., T136, 2009, 014016.[4] Olver P.J., Applications of Lie Groups to Differential Equations, Springer-Verlag, Hei-delberg, 1986.[5] Podlubny I., Fractional Differential Equations, Academic Press, San Diego, CA, 1999.[6] Sahadevan R., Bakkyaraj T., Invariant analysis of time fractional generalized Burgersand Korteweg–de Vries equation, J. Math. Anal. Appl., 393, 2012, 341-347.

48



A Numerical Approach to Equal Width Equation Using B-SplineFunctionsAhmet BOZ

Kutahya Dumlupinar University, TURKEY

Abstract. In this paper a numerical solution of the equal width wave(EW) equation, hasbeen obtained by a numerical technique based on subdomain finite element Galerkinmethod with quadratic B-Spline. A linear recurrence relationship for the numericalsolution of resulting system of ordinary differential equations is found employing theCrank-Nicolson approach including a product approximate. Test problems includingthe motion of solitary wave , wave undulation and wave generation are studied tovalidate the proposed method. The three invariants of the motion are calculated todetermine the conservation properties of the system. L2 and L∞ error norms are usedto measure differences between the analytical and numerical solution.

Keywords: Equal Width equation, Finite element methods, Galerkin, Quadratic B-splines , Solitary waves, Crank-Nicolson, Solitary waves, Splines.

References

[1] Prenter P.M., Splines and Variational Methods, John Wiley, New York, 1975.[2] Olver P.J., Euler operators and conservation laws of the BBM equation, Math. Proc.Camb. Philos. Soc., 85, 1979, 143-159.[3]Morrison P.J., Meiss J.D., Carey J.R., Scattering of RLW solitary waves, Physica, 11D,1984, 324-336.[4] Gardner L.R.T., Gardner G.A., Solitary waves of the equal width wave equation, J.Comput. Phys., 101, 1992, 218-223.[5] Gardner L.R.T., Gardner G.A., Ayoub F.A., Amein N.K., Simulations of the EW undu-lar bore, Commun. Numer. Meth. Eng., 13, 1997, 583-592.[6] Zaki S.I., A least-squares finite element scheme for the EW equation, Comput. Meth-ods Appl. Mech. Eng., 189, 2000, 587-594.[7] Saka B., Irk D., Dag I., A numerical study of the equal width equation, Hadronic J.Suppl., 18, 2003, 99-116.[8] Irk D., Saka B., Dag I., Cubic spline collocation method for the equal width equation,Hadronic J. Suppl., 18, 2003, 201-214.[9] Dag I., SakaB., A cubic B-spline collocation method for the EW equation, Math. Com-put. Appl., 90, 2004, 381-392.[10] Dogan A., Application of Galerkins method to equal width wave equation, Appl.Math. Comput., 160, 2005, 65-76.

49


Inclusion Relations Between Weighted Local Morrey-type SpacesTugce Unver YILDIZ

Kirikkale University, TURKEY

Abstract. The aim of this presentation is to give the characterizations of the bounded-ness of the identity operator between weighted local Morrey-type spaces. The solutionsof the embeddings between weighted local Morrey-type spaces and weighted comple-mentary local Morrey-type spaces are known for the range of parameters where theSawyer duality works. The same method which reduces the problem to the solutionsof some new iterated Hardy-type inequalities is valid for investigating the solutions ofthe problem we consider here. However, we will use a different approach and thistime we will investigate the boundedness of the identity operator between weightedlocal Morrey-type spaces with the characterizations of the embeddings of some otherfunction spaces.

Keywords: Copson operator, Embeddings, Hardy operator, Iterated inequalities, LocalMorrey-type spaces.

References

[1] Burenkov V.I., Recent progress in studying the boundedness of classical operators ofreal analysis in general Morrey-type spaces, I. Eurasian Mathematical Journal, 33, 2012,11-32.[2] Gogatishvili A., Mustafayev R.Ch., Unver T., Embedding relations between weightedcomplementary local Morrey-type spaces and weighted local Morrey-type spaces, EurasianMathematical Journal, 8(1), 2017, 34-49.[3] Unver T., Embeddings Between Weighted Cesàro Function Spaces, Submitted.

50



On Potential Wells and Global Solvability of Cauchy Problem forSystem of Semi-linear Klein-Gordon Equations

Akbar B. ALIEV1,2, Gulshan Kh. SHAFIYEVA2,3

1Azerbaijan Technical University, AZERBAIJAN2Institute of Mathematics and Mechanics of ANAS, AZERBAIJAN

3Baku State University, AZERBAIJAN

Abstract. We study the Cauchy problem for the system consisting of severalnonlinearKlein-Gordon equations with weak coupling and dissipative term

ui t t −∆ui +miui + γiui t =m∑

j=1

ai j

u j

p+1 |ui|p−1 ui, i = 1, ..., m (6)

ui(0, x) = ui0(x), ui t(0, x) = ui1(x), i = 1, ..., m, (7)

where u1, ..., um are real functions depending on t ∈ [0,∞), x ∈ Rn.We suppose that

i) n≥ 2, p ≥ 0, j = 1, ..., m and in addition 0< p ≤ 1n−2 , i, j = 1, ..., m if n≥ 3;

ii) ai j ∈ R, ai j = a ji , i, j = 1,2, ..., m ;∑m

i, j=1 ai jξiξ j > 0 ∀ (ξ1, ...,ξm) ∈ Rm/ (0, ..., 0).Through introducing a family of potential wells and investigation of the invariant

sets, we prove the global existence and blow up ofsolution in finite time.In case m= 2 aii = 0, ai j = −1,i 6= j problem (6), (7) was studied in [1, 2].

Keywords: Blow-up, Global existence, Potential wells, System of nonlinear Klein-Gordonequations.

References

[1] Aliev A.B., Kazimov A.A., The existence and nonexistence of global solutions of theCauchy problem for Klein-Gordom systems, Doklady Math., 87(1), 2013, 39-41.[2] Liu W., Global existence, asympthotic behavior and blow-up of solutions for coupledKlein-Gordon equations with damping terms, Nonlinear Anal., 73, 2010, 244-255.

51


On Higher Dimensional Bilinear Hardy InequalitiesSaikat KANJILAL

South Asian University, INDIA

Abstract. We consider the weighted bilinear Hardy inequality

∫ ∞

0

∫ x

0

F(t) d t

q ∫ x

0

G(t) d t

q

W (x)d x

1q

≤ C

∫ ∞

0

F p1(x)V1(x) d t

1p1∫ ∞

0

Gp2(x)V2(x)

1p2

.

(8)

The weight characterization of (8) has been obtained by Cañestro et al. [1] and Krepela[4].The N-dimensional version of the inequality (8) reads as

∫

RN

∫

B(0,|x |)f (t) d t

q∫

B(0,|x |)g(t) d t

q

w(x)d x

1q

≤ C

∫

RN

f p1(x)v1(x) d t

1p1∫

RN

g p2(x)v2(x)

1p2

.

(9)

The aim of this presentation is to provide the weight characterizations for (9) and itsequivalence with (8), see [2,3].

Keywords: Bilinear Hardy inequalities, Characterizations, Equivalence conditions, Hardyinequalities, higher dimensional Hardy type inequalities, Inequalities, Weights.

References

[1] Aguilar Cañestro M.I., Ortega SalvadorP., Ramirez Torreblanca C., Weighted bilinearHardy inequalities, J. Math. Analysis Applic., 387(1), 2012, 320-334.[2] Jain P., Kanjilal S., Persson L.E., Hardy-type inequalities over balls in RN for somebilinear and iterated operators, J. Inequal. Spl. Funct., to appear.[3] Kanjilal S., Persson L.E., Shambilova G.E., Equivalent integral conditions related tobilinear Hardy-type inequalities, Math. Ineqal. Appl., to appear.[4] Krepela M., Iterating bilinear Hardy inequalities, Proc. Edin. Math. Soc., 60(4),2017, 955-971.

52



An Extension of Muckenhoupt-Wheeden Theorem to GeneralizedWeighted Morrey Spaces

Abdulhamit KUCUKASLANPamukkale University, TURKEY

Abstract. In this talk, we find the condition on the function ω and the weight vwhich ensures the equivalency of norms of the Riesz potential Iα and the fractionalmaximal operator Mα in generalized weighted Morrey spacesMp,ω(Rn, v) and gener-alized weighted central Morrey spaces Mp,ω(Rn, v), when v belongs to MuckenhouptA∞ class.

Keywords: Fractional maximal operator, Generalized weighted Morrey spaces, Muck-enhoupt weights, Riesz potential.

References

[1] Adams D.R., Xiao J., Nonlinear potential analysis on Morrey spaces and their capaci-ties, Indiana University Mathematics Journal, 53(6), 2004, 1629-1663.[2] Gogatishvili A., Mustafayev R., Equivalence of norms of Riesz potential and fractionalmaximal function on generalized Morrey spaces, Collect. Math., 63, 2012, 11-28.[3] Burenkov V.I., Gogatishvili A., Guliyev V.S., Mustafayev R., Boundedness of the Rieszpotential in the local Morrey-type spaces, Potential Anal., 2010.

53


On Properties of Weighted Lebesgue Modulus of ContinuityAynur N. MAMMADOVA1, Dunya R. ALIYEVA1

1Institute of Mathematics and Mechanics of ANAS, AZERBAIJAN

Abstract. In this abstract we consider a weighted Lebesgue modulus of continuity inweighted Lebesgue space and reduce some properties of modulus of continuity.

Let x ∈ (0,∞) and letρ(x) = 1+x2. Suppose that f Lebesgue measurable functionsdefined in (0,∞). The weighted Lebesgue space Lp,ρ(0,∞) is the collection of allLebesgue measurable functions, such that

‖ f ‖Lp,ρ(0,∞) =

∞∫

0

| f (x)|ρ(x)

p

d x

1p

, 1≤ p <∞.

We denote Lk fp,ρ(0,∞) =

n

f ∈ Lp,ρ(0,∞) : limx→∞

f (x)ρ(x) = k f

o

.

We consider a function as the form

Ω( f ,δ) := sup0<h<δ

∞∫

0

| f (x + h)− f (x)|ρ(x)ρ(h)

p

d x

1p

.

Note that the some properties of weighted modulus of continuity:1) Ω( f ,δ)≤ 5 · 2−

1p ‖ f ‖Lp,ρ(0,∞);

2) limδ→0Ω( f ,δ) = 0, f ∈ L

k fp,ρ(0,∞).

For Lebesgue modulus of continuity detail we refer to [1].

References

[1] Nikolskii S.M., Approximation of functions of several variables and imbedding theo-rems, Springer, Berlin-Heidelberg-New York, 1975.

54



Approximation of Functions by Linear Positive Operators inVariable Exponent Lebesgue SpacesAytekin E. ABDULLAYEVA1, Lale R. ALIYEVA1

1Institute of Mathematics and Mechanics of ANAS Academy of the Ministry of EmergencySituations, AZERBAIJAN

Abstract. In this abstract we consider analog of Korovkin type approximation theoremfor trigonometric polynomials in variable exponent Lebesgue spaces.

Let Ω = (0, 1), (0,∞) or (−∞,∞). The variable exponent Lp(x)(Ω) space is asso-ciated with a measurable function p : Ω→ [1,∞) called the exponent function. The

space Lp(x)(Ω) consists of all measurable function f onΩ such that

∫

Ω

| f (x)|λ0

p(x)

d x <∞

for some λ0 > 0. Its norm is defined by scaling as

‖ f ‖Lp(·)(Ω) := ‖ f ‖p(·) = inf

λ > 0 :

∫

Ω

| f (x)|λ

p(x)

d x ≤ 1

. (1)

Let p = p(x) be a Lebesgue measurable 2π-periodic function such that 1≤ p ≤ p <∞,where p = ess inf p(x) and p = ess sup p(x). Throughout the paper Lp(x) denotes thespace of all 2π periodic Lebesgue measurable functions f equipped with the norm (1).We introduce the following theorem.Theorem. Let 1 ≤ p ≤ p <∞ and let An be a sequence of linear positive operatorsacting boundedly from Lp(·) to Lp(·). In order to a sequence of linear positive operators An

for any function f ∈ Lp(x) converges in the Lp(x) metrics to this function, it is necessaryand sufficient that the following conditions are satisfied: 1) There exists a M > 0 suchthat for any f ∈ Lp(x) and n ∈ N

‖An f ‖Lp(·)≤ M‖ f ‖Lp(·)

;

2) For the sequence of operators An(1; x)∞n=1 , An(cos t; x)∞n=1 and An(sin t; x)∞n=1the following equalities hold

limn→∞

‖1− An(1; x)‖Lp(x)= lim

n→∞‖cos x − An(cos t; x)‖Lp(x)

= limn→∞

‖sin x − An(sin t; x)‖Lp(x)= 0.

55


Different Approach to the Decomposition Theory of Hardy-MorreySpaces

Cansu KESKINKutahya Dumlupinar University, TURKEY

Abstract. In this talk, we establish the decomposition of Hardy-Morrey spaces in termsof atoms and molecules concentrated on dyadic cubes, which has the same cancellationproperties of the classical Hardy spaces. After establishing the decomposition theory ofHardy-Morrey spaces, the HM p

q boundedness of Riesz transforms for 0 < q ≤ p <∞are also given through the molecular characterization and atomic decomposition.

This is a joint work with I. Ekincioglu.

56



On Multidimensional Hausdorff Operator on Variable LebesgueSpaces

Kamala H. SAFAROVAInstitute of Mathematics and Mechanics of ANAS, AZERBAIJAN

Abstract. Let Rn n-dimensional Euclidean space of points x = (x1, x2, . . . , xn) and letA :=

ai j

be an n × n matrix whose entries ai j : Rn 7→ R are Lebesgue measurablefunctions. Suppose f : Rn 7→ R is a Lebesgue measurable function. For a fixed ker-nel function φ ∈ L loc

1 (Rn) , the multidimensional Hausdorff operator is defined in the

integral form by

Hφ( f )(x) =

∫

Rn

φ(y) f (xA(y)) d y.

In this abstract the boundedness of multidimensional Hausdorff operator on vari-able Lebesgue spaces is studied.

References

[1] Lerner A., Liflyand E., Multidimensional Hausdorff operators on the real Hardy spaces,J. Austral. Math.Soc., 83, 2007, 79-86.[2] Bandaliyev R.A., Górka P., Hausdorff operator in Lebesgue spaces, Math. Inequal.Appl., 2(22), 2019, 657-676.

57


A General Version of Neubauer’s Lemma and Closed Range AlmostClosed Operators

Abdellah GHERBIOran’s High School of Electrical and Energetic Engineering, ALGERIA

Abstract. In this paper, almost closed subspaces and almost closed linear operatorsare described in a Hilbert space. We show Neubauer’s Lemma and we give necessaryand sufficient conditions for an almost closed operator to be with closed range and weexhibit sufficient conditions under which it is either closed or closable.

Keywords: Almost closed operators, Almost closed subspaces, Closed range, Neubauer’sLemma.

58



Potential Operators on Carleson Curves in Morrey SpacesAhmet EROGLU

Omer Halisdemir University, TURKEY

Abstract. The main goal of the work (see [1]) is to prove boundedness of the potentialoperator I α on Morrey spaces Lp,λ defined on Carleson curves Γ . To be precise, let Γbe a rectifiable Jordan curve in the complex plane C with arc-length measure ν(t) andΓ (t, r) = Γ ∩ B(t, r), t ∈ Γ , r > 0, where B(t, r) = z ∈ C : |z − t| < r. The curve Γ issaid to be a Carleson curve if the conditionνΓ (t, r) ≤ c0r holds for all t ∈ Γ and r > 0,where the constant c0 > 0 does not depend on t and r.

The Morrey space Lp,λ(Γ ) is the set of all locally integrable functions f on Γ withthe finite norm

‖ f ‖Lp,λ(Γ ) = supt∈Γ ,r>0

r−λp ‖ f ‖Lp(Γ (t,r))

for 1≤ p <∞, 0≤ λ≤ 1.The potential operator I α on Γ is defined as

I α f (t) =

∫

Γ

f (τ)dν(τ)|t −τ|1−α

for 0< α < 1.We prove Sobolev-Morrey inequalities for the operator I α. In particular, we get

the analog of the theorem by D.R. Adams [2] regarding the inequality for the Rieszpotentials in Morrey spaces defined on Carleson curves. We emphasize that in theinfinite case of Γ the derived conditions are necessary and sufficient for appropriateinequalities.

Keywords: Carleson curve, Potential operator, Sobolev-Morrey inequality.

References

[1] Eroglu A., Dadashova I.B., Potential Operators on Carleson Curves in Morrey spaces,Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 67(2), 2018, 88-194.[2] Adams D.R., A note on Riesz potentials, Duke Math., 42, 1975, 765-778.

59


Boundary Value Problems for Convolution Differential OperatorEquations on the Half Line

Hummet K. MUSAEVBaku State University, AZERBAIJAN

Abstract. This paper focuses uniform separability properties of parameter dependentconvolution-differential equations. The equations and boundary conditions containcertain small and spectral parameters. Here the explicit formula for the solution isgiven and behavior of solution is derived when the small parameter approaches zero.It used to obtain singular perturbation result for the convolution parabolic equation.

References

[1] Agarwal R., O’ Regan D., Shakhmurov V.B., Separable anisotropic differential oper-ators in weighted abstract spaces and applications, J. Math. Anal. Appl., 338, 2008,970-983.[2] Ashyralyev A., Akturk S., Positivity of a one-dimensional difference operator in thehalf-line and its applications, Appl. and Comput. Math., 14(2), 2015, 204-220.[3] Denk R., Hieber M., Prüss J., R-boundedness Fourier multipliers and problems of el-liptic and parabolic type, Mem. Amer. Math. Soc., 166(788), 2003.[4] Guliev V.S., To the theory of multipliers of Fourier integrals for functions with valuesin Banach spaces, Trudy Math. Inst., Steklov, 214(17), 1996, 164-181.[5] Favini A., Shakhmurov V., Yakubov Y., Regular boundary value problems for completesecond order elliptic differential operator equations in UMD Banach spaces, SemigroupForum, 79, 2009, 22-54.[6] Shakhmurov V.B., Ekincioglu I., Linear and nonlinear convolution-elliptic equations,Boundary Value Problems, 2013(211), 2013.[7] Shakhmurov V.B., Musaev H.K., Separability properties of convolution-differentialoperator equations in weighted Lp spaces, Appl. and Comput. Math., 14(2), 2015, pp.221-233.[8] Shakhmurov V.B., Shahmurov R., Sectorial operators with convolution term, Math.Inequal. Appl., 13(2), 2010, 387-404.[9] Weis L., Operator-valued Fourier multiplier theorems and maximal Lp regularity,Math. Ann., 319, 2001, 735-758.

60



Maximal Operator on Carleson Curves in Orlicz-Morrey SpacesHatice ARMUTCU

Gebze Technical University, TURKEY

Abstract. The main goal of the work is to prove boundedness of the maximal operatorM on Orlicz-Morrey spaces LΦ,ϕ(Γ ) defined on Carleson curves Γ . Moreover, we givesufficient conditions for the boundedness of maximal operator defined on Carlesoncurves in generalized Orlicz-Morrey spaces.

Let Γ be a rectifiable Jordan curve in the complex plane C with arc-length measureν(t) and Γ (t, r) = Γ ∩ B(t, r), t ∈ Γ , r > 0, where B(t, r) = z ∈ C : |z − t| < r. Thecurve Γ is said to be a Carleson curve if the conditionνΓ (t, r) ≤ c0r holds for all t ∈ Γand r > 0, where the constant c0 > 0 does not depend on t and r.

For a Young function Φ, the set

LΦ(Γ ) =

f ∈ Lloc1 (Γ ) :

∫

Γ

Φ(k| f (τ)|)dν(τ)<∞ for some k > 0

is called Orlicz space. If Φ(r) = r p, 1 ≤ p <∞, then LΦ(Γ ) = Lp(Γ ). If Φ(r) = 0, (0 ≤r ≤ 1) and Φ(r) =∞, (r > 1), then LΦ(Γ ) = L∞(Γ ).

Let ϕ(r) be a positive measurable function on (0,∞) and Φ any Young function.The generalized Orlicz-Morrey space MΦ,ϕ(Γ ) is the set of all locally integrable functionsf on Γ with the finite norm

‖ f ‖MΦ,ϕ(Γ ) = supt∈Γ ,r>0

ϕ(r)−1Φ−1(ν(Γ (t, r)))−1)‖ f ‖LΦ(Γ (t,r)).

Let f ∈ Lloc1 (Γ ). The maximal function MΓ f is defined by

MΓ f (t) = supt>0(ν(Γ (t, r))−1

∫

Γ (t,r)

| f (τ)|dν(τ).

Keywords: Carleson curves, Generalized Orlicz-Morrey space, Maximal operator.

61


Parabolic-Fractional Integral Operators with Rough Kernels inParabolic Local Generalized Morrey Spaces

Shemsiyye A. MURADOVABaku State University, AZERBAIJAN

ANAS Institute of Mathematics and Mechanics, AZERBAIJAN

Abstract. Let P be a real n× n matrix, whose all the eighenvalues have positive realpart, At = t P , t > 0, γ = t rP is the homogeneous dimension on Rn and Ω is an At-homogeneous of degree zero function, integrable to a power s > 1 on the unit spheregenerated by the corresponding parabolic metric. We study the parabolic fractionalintegral operator I P

Ω,α, 0 < α < γ with rough kernels in the parabolic local generalized

Morrey space LM x0p,ϕ,P (R

n). We find conditions on the pair (ϕ1,ϕ2) for the boundedness

I PΩ,α from the space LM x0

p,ϕ1,P (Rn) to another one LM x0

p,ϕ2,P (Rn), 1< p < q <∞, 1

p −1q =

αγ , and from the space LM x0

p,ϕ1,P (Rn) to the weak space W LM x0

p,ϕ2,P (Rn), 1 ≤ q <∞,

1− 1q =

αγ .

References

[1] Balakishiyev A.S., Muradova Sh.A., Orucov N.Z., Parabolic fractional integral oper-ators with rough kernels in parabolic local generalized Morrey spaces, Caspian Journal ofApplied Mathematics, Ecology and Economics, 4(1), 2016, 59-68.

62



Simplicial Algebr(oid)s and Internal CategoriesOzgun GURMEN ALANSAL1, Erdal ULUALAN1


Abstract. In this structures, we give the closed relationship among simplicial algebrasand simplicial algebroids and internal categories in the category of algebras and alge-broids.

Keywords: Crossed modules, Simplicial algebras, Simplicial algebroids.

References

[1] Arvasi Z., Porter T., Freeness conditions for 2-crossed module of commutative algebras,Applied Categorical Structures, 6, 1989, 455-477.[2] Mosa G. H., Higher dimensional algebroids and crossed complexes, PhD Thesis, Uni-versity College of NorthWales, Bangor, 1986.[3] Brown R., Spencer C.B., G-groupoids, crossed modules and the fundamental groupoidof a topological group, Proc. Konn. Ned. Akad. v. Wet., 79, 1976, 296–302.[4] Whitehead J.H.C., Combinatorial homotopy II. Bull. Amer. Math. Soc., 55, 1949,453-496.

63


Fractional Integral Associated with Schrödinger Operator onGeneralized Morrey Spaces

Ali AKBULUTKirsehir Ahi Evran University, TURKEY

Abstract. Let L = −4+V be a Schrödinger operator, where the non-negative potentialV belongs to the reverse Hölder class RHn/2, and let I L

βbe the fractional integral oper-

ator associated with L. The main goal of the work is to study the boundedness of theoperator I L

βon generalized Morrey spaces associated with Schrödinger operator Mα,V

p,ϕ ,which introduced by Guliyev in [1].

For a given potential V ∈ RHq with q ≥ n/2, we define the auxiliary functionρ(x) :=

supr>0

¦

r : r−n+2∫

B(x ,r) V (y)d y ≤ 1©

.Let ϕ(x , r) be a positive measurable function on Rn × (0,∞), 1 ≤ p <∞, α ≥

0, and V ∈ RHq, q ≥ 1. The generalized Morrey space associated with Schrödingeroperator Mα,V

p,ϕ (Rn) is the set of all functions f ∈ Lloc

p (Rn) with finite quasinorm

‖ f ‖V Mα,Vp,ϕ= sup

x∈Rn,r>0

1+r

ρ(x)

α

ϕ(x , r)−1 r−n/p‖ f ‖Lp(B(x ,r)).

Let L = −4+V with V ∈ RHn/2. The fractional integral associated with L is definedby

I Lβ

f (x) = L−β/2 f (x) =

∫ ∞

0

e−t L( f )(x) tβ/2−1d t

for 0< β < n.We find the sufficient conditions on the pair (ϕ1,ϕ2)which ensures the boundedness

of the operator I Lβ

from Mα,Vp,ϕ1

to Mα,Vq,ϕ2

, 1/p− 1/q = β/n (see [2]).This contribution is based on recent joint work with R.V. Guliyev, S. Celik and M.N.Omarova.

Keywords: Fractional integral associated with Schrödinger operator, Generalized Mor-rey space associated with Schrödinger operator.

References

[1] Guliyev V.S., Function spaces and integral operators associated with Schrödinger op-erators: an overview, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 40, 2014,178-202.[2] Akbulut A., Guliyev R.V., Celik S., Omarova M.N., Fractional integral associated withSchrödinger operator on vanishing generalized Morrey spaces, J. Math. Inequal., 12(3),2018, 789-805.

64



Fractional Integral Associated with Schrödinger Operator onVanishing Generalized Morrey Spaces

Mehriban N. OMAROVABaku State University, AZERBAIJAN

Abstract. Let L = −4+ V be a Schrödinger operator, where the non-negative poten-tial V belongs to the reverse Hölder class RHn/2, and let I L

βbe the fractional integral

operator associated with L. The main goal of the work is to study the boundedness ofthe operator I L

βon vanishing generalized Morrey spaces associated with Schrödinger

operator V Mα,Vp,ϕ .

For a given potential V ∈ RHq with q ≥ n/2, we define the auxiliary functionρ(x) :=

supr>0

¦

r : r−n+2∫

B(x ,r) V (y)d y ≤ 1©

.Let ϕ(x , r) be a positive measurable function on Rn × (0,∞), 1 ≤ p <∞, α ≥

0, and V ∈ RHq, q ≥ 1. The vanishing generalized Morrey space associated withSchrödinger operator V Mα,V

p,ϕ (Rn) is the set of all functions f ∈ Lloc

p (Rn) with finite

quasinorm

‖ f ‖V Mα,Vp,ϕ= sup

x∈Rn,r>0

1+r

ρ(x)

α

ϕ(x , r)−1 r−n/p‖ f ‖Lp(B(x ,r))

and

limr→0

supx∈Rn

1+r

ρ(x)

α

ϕ(x , r)−1 r−n/p‖ f ‖Lp(B(x ,r)) = 0. (10)

Let L = −4+V with V ∈ RHn/2. The fractional integral associated with L is definedby

I Lβ

f (x) = L−β/2 f (x) =

∫ ∞

0

e−t L( f )(x) tβ/2−1d t

for 0< β < n.We find the sufficient conditions on the pair (ϕ1,ϕ2)which ensures the boundedness

of the operator I Lβ

from V Mα,Vp,ϕ1

to V Mα,Vq,ϕ2

, 1/p− 1/q = β/n (see [1]).This contribution is based on recent joint work with A. Akbulut, R.V. Guliyev and S.Celik.

Keywords: Fractional integral associated with Schrödinger operator, Generalized Mor-rey space associated with Schrödinger operator.

65


References

[1] Akbulut A., Guliyev R.V., Celik S., Omarova M.N., Fractional integral associated withSchrödinger operator on vanishing generalized Morrey spaces, J. Math. Inequal., 12(3),2018, 789-805.

66



New Characterizations of Lipschitz Space via Commutators onOrlicz SpacesFatih DERINGOZ

Kirsehir Ahi Evran University, TURKEY

Abstract. In this talk, we give necessary and sufficient conditions for the boundednessof fractional maximal operator Mα and Riesz potential Iα on Orlicz spaces. As an appli-cation of these results, we consider the boundedness of fractional maximal commutatorMb,α, commutator of fractional maximal operator [b, Mα] and commutator of Riesz po-tential [b, Iα] on Orlicz spaces, when b belongs to the Lipschitz space, by which somenew characterizations of the Lipschitz spaces are given.This work is supported by Ahi Evran University Scientific Research Projects Coordina-tion Unit (Project Number : FEF.A4.18.019).This contribution is based on recent joint works with V.S. Guliyev and S.H. Hasanov.

Keywords: Commutator, Fractional maximal operator, Lipschitz space, Riesz potential,Orlicz space.

References

[1] Guliyev V.S., Deringoz F., Hasanov S.G., Riesz potential and its commutators on Orliczspaces, J. Inequal. Appl., 75, 2017, 18.[2] Guliyev V.S., Deringoz, F., Hasanov S.G., Fractional maximal function and its com-mutators on Orlicz spaces, Anal. Math. Phys., 9(1), 2019, 165-179.[3] Guliyev V.S., Deringoz F., Some characterizations of Lipschitz spaces via commutatorson generalized Orlicz-Morrey spaces, Mediterr. J. Math., 15(4), 2018, 19.

67


Boundedness of Variable-order Fractional Operators in VariableExponent Lebesgue Spaces Lp(·) [0, l]

Rabil AYAZOGLU(MASHIYEV)Bayburt University, TURKEY

Abstract. In the present, the Riemann-Liouville integral of variable fractional orderα(·)are shown to be bounded in the variable exponent Lebesgue spaces Lp(·) [0, l] (see [2]).One of the key tools is the inequality for the Riemann-Liouville operator and the Hardy–Littlewood maximal operator, which is of interest on its own. The p(·) and variablefractional order α (·) exponents is assumed to satisfy the usual log-Hölder continuitycondition (see [1]) .

Keywords: Bounded, Hardy–Littlewood maximal operator, Integration of variable or-der, Riemann-Liouville operator, Variable exponent Lebesgue spaces.

References

[1] Samko S.G., B. Ross B., Integration and differentiation to a variable fractional order,Integral Transforms and Special Functions, 1, 1993, 277-300.[2] Sharapudinov I.I., On the topology of the space Lp(t)([0, 1]), Math. Notes 26 (3-4),1979, 796-806, Translation of Mat. Zametki, 26 (4), 1979, 613-632.

68



Uniform Boundedness of Kantorovich Operators in LebesgueSpaces with Variable Exponent

Rabil AYAZOGLU(MASHIYEV)Bayburt University, TURKEY

Abstract. In this paper we give a boundedness theorem for theKantorovich-type op-erators in the variable exponent Lebesgue spaces Lp(·) ([0, l]) (see [4]). It is also provedthat convergence of a sequence of Kantorovich-type operators to thefunction f in Lebesguespaces with variable exponent (see [1-3]).

Keywords: Boundedness and convergence properties, Kantorovich-type operators, Lebesguespaces with variable exponent.

References

[1] Altomare F., Campiti M., Korovkin-type Approximation Theory and its Applications,de Gruyter Studies in Mathematics, 17, Walter de Gruyter Co., Berlin, 1994.[2] Burenkov V., Ghorbanalizadeh A., Sawano Y., Uniform boundedness of Kantorovichoperators in Morrey spaces, Positivity, 22(4), 2018, 1097-1107.[3]Maier V., Lp-approximation by Kantorovic operators, Analysis Mathematica, 4, 1978,289-295.[4] Sharapudinov I.I., On the topology of the space Lp(t)([0, 1]), Math. Notes 26 (3-4),796-806, Translation of Mat. Zametki, 26(4), 1979, 613-632.

69


The Welland Inequality on Hypergroups

Mubariz G. HAJIBAYOVNational Aviation Academy, AZERBAIJAN

Institute of Mathematics and Mechanics, AZERBAIJAN

Abstract. Let (K ,∗) be a hypergroup and δx a point measure on K . The measure λ onK is called Haar measure if for every Borel measurable function f on K ,

∫

K

f (δx ∗δy)dλ(y) =

∫

K

f (y)dλ(y) (x ∈ K).

Define generalized translation operator T x , x ∈ K , by

T x f (y) =

∫

K

f d(δx ∗δy)

for all y ∈ K .Let K be hypergroup with Haar measure λ. The convolution of two functions is definedby

f ∗ g(x) =

∫

K

T x f (ys)g(y)dλ(y).

Let (K ,∗) be a hypergroup, with quasi-metric ρ, Haar measure λ and all balls B(x , r) =y ∈ K : ρ(x , y) < r be λ-measurable. We will say Haar measure λ is a doubling onan identity, if there exists a constant Cλ > 0, not depending r > 0, such that

λB(e, 2r)≤ CλλB(e, r).

If this condition holds the triple (K ,∗,λ) we will call a space of homogeneous type onan identity.To avoid trivial measures we will always assume that 0< λB(e, r)< +∞, for all r > 0.Given a space of homogeneous type (K ,∗,λ) on an identity, we will call that it is areverse doubling space on an identity if there exists a constant 0 < γ < 1 such that forevery r > 0 such that B(e, r) 6= K ,

λB(e,r2)≤ γλB(e, r).

Lemma. Let (K ,∗) be a hypergroup, with quasi-metric ρ and Haar measure λ,(K ,ρ,λ) reverse doubling space on an identity. Suppose thatλ(K) = +∞ or diam (K)<+∞. Then λe= 0.

70



Let (K ,∗) be a hypergroup, with quasi-metric ρ and Haar measure λ and 0< β < 1.Forλ-locally integrable function f on hypergroup K , define fractional maximal operator

Mβ f (x) = supr>0

1λB(e, r)1−β

(| f | ∗χB(e,r))(x)

and fractional integral

Iβ f (x) = ( f ∗λB(e,ρ(e, ·))β−1)(x)

on hypergroup K .Theorem. Let (K ,∗) be a hypergroup, with quasi-metric ρ and Haar measure λ,

(K ,ρ,λ) reverse doubling space on an identity, ε is a positive number satisfying ε <minβ , 1 − β. Assume also λ(K) = +∞ or diam (K) < +∞. Then there exists apositive constant C such that for every f ∈ L1

loc(K) and for every x ∈ K ,

|Iβ f (x)| ≤ Cq

Mβ−ε f (x)Mβ+ε f (x)

71


Weighted Inequality for (p, q)-admissible B-potential OperatorsFatai A. ISAYEV

Institute of Mathematics and Mechanics of ANAS, AZERBAIJAN

Abstract. The main goal of the work is to prove the boundedness of (p, q)-admissiblepotential operators, associated with the Laplace-Bessel differential operator Bk,n =

n∑

i=1

∂ 2

∂ x2i+

k∑

j=1

γ j

x j

∂∂ x j

((p, q)-admissible Bk,n–potential operators) on a weighted Lebesgue

spaces Lp,ω,γ(Rnk,+) including their weak versions. These conditions are satisfied by

most of the operators in harmonic analysis, such as the Bk,n–fractional maximal opera-tor, Bk,n–potential integral operators and so on. Sufficient conditions on weighted func-tionsω andω1 are given so that (p, q)-admissible Bk,n–potential operators are boundedfrom Lp,ω,γ(Rn

k,+) to Lq,ω1,γ(Rnk,+) for 1 < p < q <∞ and weak (p, q)-admissible Bk,n–

potential operators are bounded from Lp,ω,γ(Rnk,+) to Lq,ω1,γ(Rn

k,+) for 1 ≤ p < q <∞(see [1]).

Keywords: Weighted Lebesgue space, (p, q)-admissible Bk,n-potential operators, Bk,n-fractional maximal operator, Bk,n-potential integral operators.

References

[1] E.A. Gadjieva, F.A. Isayev, A. Kucukaslan, Two-weighted inequality for (p, q)-admissibleBk,n-potential operators in weighted Lebesgue spaces, Trans. Natl. Acad. Sci. Azerb. Ser.Phys.-Tech. Math. Sci., 36(1), 2016, 36-53.

72



On q-Meyer König-Zeller-Durrmeyer OperatorsDilek SOYLEMEZ

Ankara University, TURKEY

Abstract. In this talk, we consider q-integrated Meyer-König-Zeller-Durrmeyer oper-ators defined by Govil and Gupta. We mention about some approximation propertiesof these operators. Moreover, taking into account the Abel convergence, we give someapproximation results. Finally, we compute rate of the Abel convergence for these op-erators.

Keywords: Abel convergence, q-Beta function, q-Meyer-König-Zeller-Durrmeyer oper-ators, Rate of convergence.

References

[1] Govil N. K., Gupta V., Convergence of q-Meyer-König-Zeller-Durrmeyer operators, Ad-vanced Studies in Contemporary Mathematics, 19(1), 2009, 97-108.[2] Powell R. E., Shah S.M., Summability Theory and Its Applications, Prentice-Hall ofIndia, New Delhi, 1988.[3] Söylemez D., Ünver, M., Korovkin type theorems for Cheney Sharma Operators viasummability methods, Res. Math., 73(3), 2017, 1601-1612.[4] Ünver M., Abel transforms of positive linear operators, In: ICNAAM 2013. AIP Con-ference Proceedings, 1558, 2013, 1148-1151.

73


Oscillatory Integrals with Variable Calderon-Zygmund Kernel onVanishing Generalized Morrey Spaces

Aysel A. AHMADLIKutahya Dumlupinar University, TURKEY

Abstract. The classical Morrey spaces were introduced by Morrey (1938) to studythe local behavior of solutions to second-order elliptic partial differential equations.Moreover, various Morrey spaces are defined in the process of study. Guliyev, Mizuharaand Nakai introduced generalized Morrey spaces Mp,ϕ(Rn).

Let ϕ(x , r) be a positive measurable function on Rn× (0,∞) and 1≤ p <∞. Thegeneralized Morrey space Mp,ϕ(Rn) is the set of all functions f ∈ Lloc

p (Rn) with finite

quasinorm

‖ f ‖V Mp,ϕ= sup

x∈Rn,r>0ϕ(x , r)−1 r−n/p ‖ f ‖Lp(B(x ,r)).

Suppose that k is the standard Calderón-Zygmund kernel. That is, k ∈ C∞(Rn\0)is homogeneous of degree −n, and

∫

Σk(x)dσx = 0, where Σ= x ∈ Rn : |x |= 1. The

oscillatory integral operator Tλ is defined by

Tλ f (x) = p.v.

∫

Rn

eiλΦ(x ,y)k(x − y)ϕ(x , y) f (y)d y, (11)

where λ ∈ R,ϕ ∈ C∞0 (Rn × Rn), the space of infinitely differentiable functions on

Rn×Rn with compact supports, and Φ is a real-analytic function or a real-C∞(Rn×Rn)function satisfying that for any (x0, y0) ∈ suppϕ, there exists ( j0, k0), 1 ≤ j0, k0 ≤ n,such that ∂ 2Φ(x0, y0)/∂ x j0∂ yk0

does not vanish up to infinite order. These operatorshave arisen in the study of singular integrals supported on lower dimensional varieties,and the singular Radon transform.

The main goal of the work is study the boundedness of the oscillatory singularintegrals with variable Calderón-Zygmund kernel on the generalized Morrey spacesM p,ϕ(Rn) and the vanishing generalized Morrey spaces V M p,ϕ(Rn).


Keywords: Generalized Morrey space, Oscillatory integral, Variable Calderón-Zygmundkernels.

74



Lipschitz Estimates for Rough Fractional Multilinear IntegralOperators on Vanishing Generalized Morrey Spaces

Ramin V. GULIYEVKutahya Dumlupinar University, TURKEY

Abstract. Let γ = (γ1,γ2, . . . ,γn), and γi (i = 1, 2, . . . , n) be nonnegative integers.Denote |γ|=

∑ni=1 γi, γ!= γ1!γ2! . . .γn!, xγ = xγ1

1 xγ22 . . . xγn

n and

Dγ =∂ |γ|

∂ γ1 x1 ∂ γ2 x2 . . . ∂ γn xn.

Suppose that Ω ∈ Ls(Sn−1) (s > 1) is homogeneous of degree zero on Rn withzero means value on Sn−1, A is a function defined on Rn. Rough fractional multilinearintegral operator IA,m

Ω,α , is defined by

IA,mΩ,α f (x) =

∫

Rn

Rm(A; x , y)|x − y|n−α+m−1

Ω(x − y) f (y)d y,

where 0< α < n, and Rm(A; x , y) is the m-th remainder of Taylor series of A at x abouty . More precisely,

Rm(A; x , y) = A(x)−∑

|γ|<m

1γ!

DγA(y)(x − y)γ.

The main goal of the work is to prove the Lipschitz boundedness for a class offractional multilinear operators IA,m

Ω,α with rough kernels Ω ∈ Ls(Sn−1), s > n/(n − α)on the vanishing generalized Morrey spaces V M x0

p,ϕ , where the functions A belong tohomogeneous Lipschitz space Λβ , 0 < β < 1. We find the sufficient conditions onthe pair (ϕ1,ϕ2) which ensures the boundedness of the operators IA,m

Ω,α from V M x0p,ϕ1

toV M x0

q,ϕ2for 1< p < q <∞ and 1/p−1/q = (α+β)/n. In all cases the conditions for the

boundedness of the operator IA,mΩ,α is given in terms of Zygmund-type integral inequalities

on (ϕ1,ϕ2), which do not assume any assumption on monotonicity of ϕ1(x , r), ϕ2(x , r)in r.


Keywords: Fractional multilinear integral, Lipshitz function, Rough kernel, Vanishinggeneralized Morrey space.

75


Approximation by Nörlund Means in Weighted Generalized GrandSmirnov Classes

Ahmet TESTICIBalikesir University, TURKEY

Abstract. Let G ⊂ C be a domain bounded by a Carleson curve Γ . The set of allmeasurable functions f such that

‖ f ‖Lp),θ (Γ ,ω) := sup0<ε<p−1

εθ1|Γ |

∫

Γ

| f (x)|p−εω (x) d x

1/(p−ε)

<∞,

constitute weighted generalized grand Lebesgue space Lp),θ (Γ ,ω) . We define general-ized grand Smirnov class Ep),θ (G,ω) of analytic functions by

Ep),θ (G,ω) :=

f ∈ E1 (G,ω) : f ∈ Lp),θ (Γ ,ω)

.

For f ∈ Ep),θ (G,ω) norm is defined by ‖ f ‖Ep),θ (G,ω) := ‖ f ‖Lp),θ (Γ ,ω). We denote byE p),θ (G,ω) the closure of Smirnov class Ep (G,ω) of analytic function with respect tonorm Ep),θ (G,ω). The generalized grand Smirnov class are generalization of classicalSmirnov class.

Let Ep),θ (G,ω) , θ > 0 be weighted generalized grand Smirnov class of analyticfunctions where ω belongs to Muckenhoupt class Ap (Γ ) and 1 < p <∞. In this casethe constructive characterization of functions belongs to E p),θ (G,ω) are obtained in[1]. In E p),θ (G,ω) the generalized grand Lipschitz class in [1].

Let (pn)∞n=0 be sequence of positive real numbers. The Nörlund means of the p − ε

Faber series of f ∈ Ep),θ (G,ω) are defined as

N Gn ( f , z) :=

1Pn

n∑

k=0

pn−kSGn ( f , z)

where Pn =n∑

k=0pk and SG

n ( f , z) is the n− th partial sum of p−ε Faber series of f . In this

talk we investigate the approximation properties of N Gn ( f , z) in the generalized grand

Lipschitz class under the some restriction on the sequence (pn) .

Keywords: Faber series, Nörlund means, Weighted generalized grand Smirnov classes.

76



References

[1] Israfilov D.M., Testici A., Approximation In Weighted Generalized Grand SmirnovClasses, Studia Scientiarum Mathematicarum Hungarica, 54(4), 2017, 471-488.

77


Characterization of Commutators of Parabolic Fractional IntegralOperator in Parabolic Generalized Orlicz-Morrey Spaces

Gulnara A. ABASOVAAzerbaijan State University of Economics, AZERBAIJAN

Abstract. Let P be a real n× n matrix, all of its eigenvalues having positive real part.Let At = t P , t > 0, and set γ = t rP. Then, there exists a quasi-distance ρ associatedwith P such that ρ(At x) = tρ(x), t > 0 for every x ∈ Rn; ρ(0) = 0, ρ(x − y) =ρ(y− x)≥ 0 and ρ(x− y)≤ k(ρ(x−z)+ρ(y−z)); d x = ργ−1dσ(w)dρ, where ρ =ρ(x), w = Aρ−1 x and dσ(w) is a C∞ measure on the ellipsoid w : ρ(w) = 1. Then,Rn,ρ, d x becomes a space of homogeneous type in the sense of Coifman-Weiss. Theballs with respect to ρ, centered at x of radius r, are just the ellipsoids E (x , r) =y ∈ Rn : ρ(x − y)< r, with the Lebesgue measure |E (x , r)| = vρrγ, where vρ is thevolume of the unit ellipsoid in Rn.

The parabolic fractional integral I Pα

f , 0< α < γ, of a function f ∈ Lloc1 (R

n) is definedby

I Pα

f (x) =

∫

Rn

f (y)ρ(x − y)γ−α

d y.

In this talk, we give necessary and sufficient condition for the Adams-Guliyev typeboundedness (see [1]) of parabolic fractional integral and its commutators in parabolicgeneralized Orlicz-Morrey spaces (see [2]).

Keywords: BMO, Commutator, Parabolic fractional integral, Parabolic generalizedOrlicz-Morrey space.

References

[1] Guliyev Vagif S., Boundedness of the maximal, potential and singular operators in thegeneralized Morrey spaces, J. Inequal. Appl., 2009, Art. ID 503948, 20 pp.[2] Eroglu A., Abasova G. A., Guliyev V. S., Characterization of parabolic fractional inte-gral and its commutators in parabolic generalized Orlicz-Morrey spaces, Azerb. J. Math.,9(1), 2019, 92-107.

78



On the Equivalence of the K-functional and the Modulus ofContinuity on the Variable Exponent Morrey Spaces

Arash GHORBANALIZADEHInstitute for Advanced studies in Basic Sciences (IASBS) Zanjan, IRAN

Abstract. In this work, we show the modulus of smoothness in the variable exponentsMorrey spaces and K-functional with respect the variable exponents Morrey spaces–thevariable exponents Sobolev–Morrey spaces are equivalent. As an application, we char-acterize the Nikol’skii–Besov–variable exponents Morrey spaces via real interpolation.

Keywords: Modulus of continuity, Nikol’skii–Besov–variable exponents Morrey spaces,Peetre’s K-functional.

References

[1] Israfilov D.M, Testici A., Approximation problems in the Lebesgue spaces with variableexponent, Journal of Mathematical Analysis and Applications, 459(1), 2018, 112-123.[2] Almeida A., Hasanov J., Samko S., Maximal and potential operators in variable ex-ponent Morrey spaces, Georgian Mathematical Journal, 15(2), 2008, 195-208.[3] Cruz-Uribe D., Fiorenza A., Variable Lebesgue spaces: Foundations and harmonicanalysis, Springer Science & Business Media, 2013.[4] Diening L., Harjulehto P., Hästö P., Ruzicka M., Lebesgue and Sobolev spaces withvariable exponents, Springer, 2011.[5] Guliyev V.S., Ghorbanalizadeh A., Sawano Y., Approximation by trigonometric poly-nomials in variable exponent Morrey spaces, Analysis and Mathematical Physics, 2018,1-21.[6] Ghorbanalizadeh A., Roohi seraji R., On the equivalence of the K-functional and themodulus of continuity on the periodic variable exponent Morrey spaces, Periodica Mathe-matica Hungarica.[7] Burenkov V., Ghorbanalizadeh A., Sawano Y., On the equivalence of the K-functionaland the modulus of continuity on the Morrey spaces, submitted.

79


Commutators of Maximal Operator Associated with the DunklOperators on Orlicz Spaces

Yagub Y. MAMMADOV1, Fatma A. MUSLIMOVA1

1Nakhchivan Teacher-Training Institute, AZERBAIJAN

Abstract. On the real line, the Dunkl operators

Dν( f )(x) :=d f (x)

d x+ (2ν+ 1)

f (x)− f (−x)2x

, ∀ x ∈ R, ∀ν≥ −1/2

are differential-difference operators associated with the reflection group Z2 on R.Let ν > −1/2 be a fixed number and mν be the weighted Lebesgue measure on R

given by dmν(x) :=

2ν+1Γ (ν+ 1)−1|x |2ν+1 d x , ∀ x ∈ R. For any x ∈ R and r > 0,

let B(x , r) := y ∈ R : |y| ∈ ]max0, |x | − r, |x |+ r[ .The maximal commutator Mb,ν associated with Dunkl operator on the real line and

with a locally integrable function b ∈ Lloc1 (R, dmν) is defined by

Mb,ν f (x) := supr>0(mνB(x , r))−1

∫

B(x ,r)

|b(x)− b(y)| | f (y)| dmν(y), ∀ x ∈ R.

It is well known that maximal operator and its commutators play an important rolein harmonic analysis. In this work, in the framework of this analysis in the setting R,we study the boundedness of the maximal commutator Mb,ν and the commutator of themaximal operator, [b, Mν], on Orlicz spaces LΦ(R, dmν), when b belongs to the spaceBMO(R, dmν), by which some new characterizations of the space BMO(R, dmν) aregiven (see [1]).

Theorem 1. Let Φ be a Young function with Φ ∈ ∇2. Then the condition b ∈BMO(R, dmν) is necessary and sufficient for the boundedness of Mb,ν on LΦ(R, dmν).Let b−(x) = 0, if b(x) ≥ 0 and b−(x) = |b(x)|, if b(x) < 0, b+(x) := |b(x)| − b−(x).From Theorem 1 we deduce the following conclusion.

Corollary. Let Φ be a Young function with Φ ∈ ∇2. Then the conditions b+ ∈BMO(R, dmν) and b− ∈ L∞(R, dmν) are sufficient for the boundedness of [b, Mν] onLΦ(R, dmν).

Keywords: BMO, Commutator of maximal operator, Dunkl operator, Orlicz space.

80



References

[1]Mammadov Y.Y., Muslumova F.A., Commutators of maximal operator associated withthe Dunkl operators on Orlicz spaces, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech.Math. Sci., 38(4), 2018, 106-114.

81


Fractional Maximal Operator Associated with the Dunkl Operatorson Orlicz Spaces

Fatma A. MUSLIMOVANakhchivan State University, AZERBAIJAN

Abstract. On the real line, the Dunkl operators

Dν( f )(x) :=d f (x)

d x+ (2ν+ 1)

f (x)− f (−x)2x

, ∀ x ∈ R, ∀ν≥ −1/2

are differential-difference operators associated with the reflection group Z2 on R. Themain goal of the work is to find necessary and sufficient conditions for the boundednessof the fractional maximal operator Mα,ν on Orlicz spaces LΦ(R, dmν).

For any x ∈ R and r > 0, let dmν(x) :=

2ν+1Γ (ν+ 1)−1|x |2ν+1 d x and B(x , r) :=

y ∈ R : |y| ∈ ]max0, |x | − r, |x |+ r[ .The fractional maximal operator Mα,ν, 0≤ α < 2ν+2 associated by Dunkl operator

on the real line is given by

Mα,ν f (x) := supr>0(mνB(x , r))−1+ α

2ν+2

∫

B(x ,r)

| f (y)| dmν(y), ∀ x ∈ R.

In this work, in the framework of the Dunkl analysis in the setting R, we shall give anecessary and sufficient condition for the boundedness of fractional maximal operatorMα,ν on Orlicz spaces LΦ(R, dmν) and weak Orlicz spaces W LΦ(R, dmν).

A function Φ : [0,∞) → [0,∞] is called a Young function if Φ is convex, left-continuous, lim

r→+0Φ(r) = Φ(0) = 0 and lim

r→∞Φ(r) =∞. The set of Young functions such

that 0< Φ(r)<∞ for 0< r <∞ is denoted by Y .The following result completely characterizes the boundedness of Mα,ν on Orlicz

spaces LΦ(R, dmν).Theorem. Let 0< α < 2ν+ 2, Φ,Ψ be Young functions and Φ ∈ Y . The condition

r−α

2ν+2 Φ−1(r)≤ C Ψ−1(r) (12)

for all r > 0, where C > 0 does not depend on r, is necessary and sufficient for theboundedness of Mα,ν from LΦ(R, dmν) to W LΨ(R, dmν). Moreover, if Φ ∈ ∇2, the con-dition (12) is necessary and sufficient for the boundedness of Mα,ν from LΦ(R, dmν) toLΨ(R, dmν).

This contribution is based on recent joint work with Y.Y. Mammadov.

Keywords: BMO, Commutator of maximal operator, Dunkl operator, Orlicz space.

82



Approximation Properties of Faber SeriesDaniyal ISRAFILOV1, Emine KIRHAN1

1Balikesir University, TURKEY

Abstract. In this talk we discuss the approximation problems in the variable exponentLebesgue spaces defined on the simply connected domains of the complex plane. Un-der some restrictive conditions on the boundary of domains we will estimate the rate ofconvergence by the partial sums of Faber series, constructed via Faber polynomials andconsidered functions. The error we will estimate in the norms of different generaliza-tions of the classical Smirnov classes of analytic functions, such as variable exponent,grand variable exponent Smirnov classes.

This work was supported by Balikesir University Grant No: 2018/071[D25] : "In-equalities in variable exponent spaces".

Keywords: Maximal Convergence, Smirnov classes, Variable exponent.

83


Sublinear Operators with Rough Kernel Generated byCalderon-Zygmund Operators on Generalized Weighted Morrey

Spaces

Vugar H. HAMZAYEV

Nakhchivan Teacher-Training Institute, AZERBAIJAN

Abstract. The classical Morrey spaces were introduced by Morrey (1938) to studythe local behavior of solutions to second-order elliptic partial differential equations.Moreover, various Morrey spaces are defined in the process of study. Guliyev, Mizuharaand Nakai (1994) introduced generalized Morrey spaces Mp,ϕ(Rn); Komori and Shirai(2009) defined weighted Morrey spaces Lp,κ(w); Guliyev (2012) gave a concept of thegeneralized weighted Morrey spaces Mp,ϕ(w) which could be viewed as extension ofboth Mp,ϕ and Lp,κ(w). In [1], the boundedness of the classical operators and theircommutators in spaces Mp,ϕ(w) was also studied.

Let Ω ∈ Ls(Sn−1) with 1 < s ≤ ß be homogeneous of degree zero. Suppose thatTΩ represents a linear or a sublinear operator, such that that for any f ∈ L1(Rn) withcompact support and x /∈ supp f

|TΩ f (x)| ≤ c0

∫

Rn

|Ω(x − y)||x − y|n

| f (y)|d y, (13)

where c0 is independent of f and x .Let 1≤ p <∞, ϕ be a positive measurable function on Rn× (0,∞) and w be non-

negative measurable function on Rn. The generalized weighted Morrey space Mp,ϕ(w)is the set of all functions f ∈ Lloc

p,w(Rn) with finite norm

‖ f ‖Mp,ϕ(w) = supx∈Rn,r>0

ϕ(x , r)−1 w(B(x , r))−1p ‖ f ‖Lp,w(B(x ,r)).

The main goal of this work to study the boundedness of a large class of sublinearoperators with rough kernel TΩ on the generalized weighted Morrey spaces Mp,ϕ(w) forwith q′ ≤ p < ß, p 6= 1 and w ∈ Ap/q′ or 1< p < q and w1−p′ ∈ Ap′/q′ , whereΩ ∈ Lq(Sn−1)with q > 1 be homogeneous of degree zero (see [2]).

Keywords: Ap weights, Calderón-Zygmund operator, Commutator, Generalized weightedMorrey spaces, Rough kernel, Sublinear operator.

84



References

[1] Guliyev V.S., Generalized weighted Morrey spaces and higher order commutators ofsublinear operators, Eurasian Math. J., 3(3), 2012, 33-61.[2] Hamzayev V., Sublinear operators with rough kernel generated by Calderón-Zygmundoperators and their commutators on generalized weighted Morrey spaces, Trans. Natl.Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci., 38(1), 2018, 79-94.

85


Fractional Multilinear Integrals with Rough Kernels on GeneralizedWeighted Morrey Spaces

Amil A. HASANOVGandja State University, AZERBAIJAN

Abstract. The classical Morrey spaces were introduced by Morrey (1938) to studythe local behavior of solutions to second-order elliptic partial differential equations.Moreover, various Morrey spaces are defined in the process of study. Guliyev, Mizuharaand Nakai (1994) introduced generalized Morrey spaces Mp,ϕ(Rn); Komori and Shirai(2009) defined weighted Morrey spaces Lp,κ(w); Guliyev (2012) gave a concept of thegeneralized weighted Morrey spaces Mp,ϕ(w) which could be viewed as extension ofboth Mp,ϕ and Lp,κ(w). In [1], the boundedness of the classical operators and theircommutators in spaces Mp,ϕ(w) was also studied.

The generalized weighted Morrey space Mp,ϕ(w) is the set of all functions f ∈Lloc

p,w(Rn) with finite norm

‖ f ‖Mp,ϕ(w) = supx∈Rn,r>0

ϕ(x , r)−1 w(B(x , r))−1p ‖ f ‖Lp,w(B(x ,r)).

Let us give the definition of the multilinear fractional integral operator as follows:

T A1,A2,...,AkΩ,α f (x) =

∫

Rn

Ω(x − y)|x − y|n−α+N

k∏

j=1

Rm j(A j; x , y) f (y)d y,

where 0< α < n, N =k∑

j=1

(m j − 1), min1≤ j≤k m j ≥ 2, Ω is homogeneous of degree zero

and Ω ∈ Ls(Sn−1), s > 1, Rm j(A j; x , y) = A(x)−

∑

|γ|<m j

1γ! D

γA j(y)(x − y)γ.In this talk, we study the boundedness of fractional multilinear integral operators

with rough kernels T A1,A2,...,AkΩ,α , which is generalization of the higher-order commutator

of the rough fractional integral on the generalized weighted Morrey spaces Mp,ϕ(w).We find the sufficient conditions on the pair (ϕ1,ϕ2) with w ∈ Ap,q which ensures theboundedness of the operators T A1,A2,...,Ak

Ω,α from Mp,ϕ1(wp) to Mp,ϕ2

(wq) for 1< p < q <∞(see [2]).

Keywords: BMO, Fractional multilinear integral, Generalized weighted Morrey space,Rough kernel.

86



References

[1] Guliyev V.S., Generalized weighted Morrey spaces and higher order commutators ofsublinear operators, Eurasian Math. J., 3(3), 2012, 33-61.[2] Akbulut A., Hasanov A., Fractional multilinear integrals with rough kernels on gen-eralized weighted Morrey spaces, Open Math., 14(1), 2016, 1023-1038.

87


New Characterizations of BMO Space via Commutators on OrliczSpaces

Sabir G. HASANOVGanja State University, AZERBAIJAN

Abstract. The main goal of this talk to give necessary and sufficient conditions for theboundedness of fractional maximal operator Mα and Riesz potential Iα on Orlicz spaces.As an application of these results, we consider the boundedness of fractional maximalcommutator Mb,α, commutator of fractional maximal operator [b, Mα] and commutatorof Riesz potential [b, Iα] on Orlicz spaces, when b belongs to the BMO space, by whichsome new characterizations of the BMO spaces are given.

This contribution is based on recent joint works with V.S. Guliyev and F. Deringoz.

Keywords: BMO space, commutator, Fractional maximal operator, Orlicz space, Rieszpotential.

References

[1] Guliyev V.S., Deringoz F., Hasanov S.G., Riesz potential and its commutators on Orliczspaces, J. Inequal. Appl., 2017, 75, 18 p.[2] Guliyev V.S., Deringoz F., Hasanov S.G., Fractional maximal function and its commu-tators on Orlicz spaces, Anal. Math. Phys., 9(1), 2019, 165-179.[3] Guliev V.S., Deringoz F., Hasanov S.G., Commutators of a fractional maximal oper-ator on Orlicz spaces, (Russian) Mat. Zametki, 104(4), 2018, 516-526; translation inMath. Notes, 104(3-4), 2018, 498-507.

88



On Embeddings of Grand Sobolev-Morrey Spaces with DominantMixed Derivatives

Alik M. NAJAFOVAzerbaijan University of Architecture and Construction, AZERBAIJAN

Institute of Mathematics and Mechanics of ANAS, AZERBAIJAN

Abstract. In this abstract we introduce and investigate, from the point of view of em-bbedding theory, certain properties of functions from the grand grand Sobolev-Morreyspaces type with dominant mixed derivatives S l

p),c),a,αW (G). The norm in spacesS l

p),c),a,αW (G) defined as

‖ f ‖S lp),c),a,αW (G) =

∑

e⊆en

‖Dlef ‖p),c),a,α;G,

‖ f ‖p),c),a,α;G = f ‖Lp),c),a,α(G) =

= supx∈G,

0<ε<sm

1∏

j∈en

tc j a−εα j

j

ε

|Gtc(x)|

∫

Gtc (x)

| f (y)|p−εd y

1p−ε

where G ⊂ Rn is bounded domain,en = 1,2, . . . , n;l ∈ N n;l e = (l e1, l e

2, . . . , l en), l e

j = l j

( j ∈ e), l ej = 0( j ∈ en\e) 1 < p <∞, a ∈ [0,1], c ∈ (0,∞)n, sm = min(sm

1 , sm2 , . . . , sm

n ),sm

j =minp− 1,c j aα j, α j ≥ 0,( j ∈ en), d j ( j ∈ en) is diaqonal of Itc(x).

In case a = 0, α j( j ∈ en) = 0 the S lp),c),a,αW (G) coinsided a spaces S l

p),a,cW (G)introduced and studied in [1].

By method integral representations are proved embedding theorems of the type

1. Dν : S lp),c),a,αW (G) ,→ Lq−ε(G) (p < q ≤∞);

2. Dν : S lp),c),a,αW (G) ,→ S l ′

q−εW (G) (p < q ≤∞), l1 ∈ N n, l1j < l j( j ∈ en) is holds;

3. It is also proved that for the function from space f ∈ S lp),c),a,αW (G) the generalized

derivatives Dν f satisfy the Hölder condition in the metric Lq−ε(G)or C(G).

89


References

[1] Najafov A.M., Alekberli S.A., The embedding theorems of spaces type grand Sobolev-MOrrey with dominant mixed derivatives, MADEA-8, June 2018, 90p.

90



Some Results on Hausdoff OperatorsRadouan DAHER

Faculty of Sciences Aïn Chock Univesity Hassan II, MOROCCO

Abstract. Liflyand and Móricz studied the Hausdorff operator on the real Hardy spaceand proved the boundedness property and the commuting relations of this operator andHilbert transformations. The main objective of this paper is extending these results tothe two contexts of Dunkl theory and the Jacobi hypergroup.

This talk is based on the joint work with T. Kawazoe and F. Saadi.

91


Continuous State Space Branching Process as Limit of Sequence ofBellman-Harris Processes

Soltan A. ALIYEV1, Sahib A. ALIYEV2

1Institute of Control Systems of ANAS, AZERBAIJAN2Nakhchivan Teacher-Training Institute, AZERBAIJAN

Abstract. Let η(t), t ≥ 0 be the number of particles of Bellman-Harris branchingprocess at the moment of time t (η(0) = 1).

Distribution function of the lifetime of one particle l we denote G(t) = Pl ≤ t.At the end of life the particle moves to k newbons of the particles (k = 0,1, 2, . . .) withprobability Pk respectively. The number of particles in 0,1, 2, . . .-th generation of theprocess η(t) is denoted by ξ0,ξ1, . . .. It is known, that the variants ξ0,ξ1, . . . form theGalton-Watson branching process with the generating function

F(S) =∞∑

k=0

PkSk = M

Sξ | ξ0 = 1

, |s| ≤ 1.

Let F(t; s) be the generating function of the number of particle in the Bellman-Harrisbranching process at the moment of time t.

F(t; s) =∞∑

k=0

Pη(t) = kSk = M

Sη(t) | η(0) = 1

.

Now we consider the sequence of Bellman-Harris processη(m)(t), m= 1,2, . . ., t ≥ 0with generating function

F m(t; s) = M

Sη(m)(t) | η(m)(0) = 1

and with lifetime of particle lm.Denote

G(t)m = Plm ≤ t, am =

∫ ∞

0

tdGm(t)<∞, Am =dF m(s)

ds|lS=i , lim

m→∞am = a,

and suppose that Am→ 1 as m→∞ and there exists a finite limitb(m)(r t/a)

bm(t/a)

, as n = [t/a],

t →∞, m→∞, η(1− Am)→ 0, where b(m)t are positive constants.On this conditions the theorem about convergence of sequence of the Bellman-

Harris normalized processes

µ(m)t (τ) =η(m)(τt)

Bm, µ(m)t (0) =

[xBm(t)]Bm(t)

, Bm(t) = b(m)[t/a]

to the critical continuous state space branching process is proved.

92



The Determination of the Right-hand Side in a Semilinear HeatConduction Problem with the Nonlocal Dirichlet Boundary

ConditionArzu ERDEM COSKUN

Kocaeli University, TURKEY

Abstract. In this study we consider the inverse problem of determining the nonlinearright-hand side of a quasi-linear parabolic equation and prove a uniqueness theorem.Stability estimate for the solution are obtained. A method of representing the nonlinearright-hand side explicitly is proposed for the special case.

Keywords: Inverse problem, Nonlocal Dirichlet boundary condition, Semilinear parabolicproblem, Stability, Uniqueness.

References

[1] Bushuyev I., Global uniqueness for inverse parabolic problems with final observation,Inverse Problems, 11, 1995, 11-16.[2] Choulli M., An inverse problem for a semilinear parabolic equation, Inverse Problems,10(5), 1994, 1123-1132.[3]Dehghan M., An inverse problem of findinf a source parameter in a semilinear parabolicequation, Applied Mathematical Modelling, 25(9), 2001, 743-754.[4] Isakov V., On uniqueness in inverse problems for semilinear parabolic equations, Archivefor Rational Mechanics and Analysis, 124(1), 1993, 1-12.[5] Lorenzi A., An inverse problem for a semilinear parabolic equation, Annali Di Math-ematica Pura Ed Applicata, 131(1), 1982, 145-166.[6] Polyanin A.D., Handbook of Nonlinear Partial Differential Equations, Chapman andHall/CRC, 203.[7] Slodicka M., Semilinear parabolic problems with nonlocal Dirichlet boundary condi-tions, Inverse Problems in Science and Engineering, 19(5), 2011, 706-716.

93


Relativistic Derivation Formalism of Electromagnetic FieldEquations in Quaternion Algebra

Mustafa Emre KANSU1, Murat TANISLI2, Suleyman DEMIR2

1Kutahya Dumlupinar University, TURKEY2Eskisehir Technical University, TURKEY

Abstract. Quaternion algebra is one of the member of higher dimensional algebra,which satisfies non-commutative but associative mathematical tools [1]. They are of-ten used in many subfields of physics such as electromagnetism. The most fundamentalproperty of the electromagnetic field equations is to be invariant under Lorentz’s trans-formations [2]. In this study, the more basic and useful expressions of the electromag-netic field equations including monopole terms are obtained via complex quaternions.Besides, by using the well-known relativistic approaches for electromagnetism [3], thenew adapted quaternionic forms are seperately derived from the moving media in adetailed manner.

Keywords: Electromagnetic fields, Lorentz transformations, Monopole, Quaternion.

References

[1] Hamilton W.R., Elements of Quaternions, Chelsea Publishing, New York, 1969.[2] Jackson J.D., Classical Electrodynamics, Wiley - Sons, USA, 1999.[3] Tolan T., Tanisli M., Demir S., Octonic Form of Proca-Maxwell’s Equations and Rela-tivistic Derivation of Electromagnetism, Int. J. Theor. Phys., 52(12), 2013, 4488-4506.

94



Formulation of Maxwell-type Gravity Equations with DualQuaternions

Ismail AYMAZ1, Mustafa Emre KANSU1


Abstract. It is known that there are some analogies between electromagnetism and lin-ear gravity. As the electric charge is valid for electromagnetism on account of Coulomb’slaw, the mass is the effective term in linear gravity associated with Newton’s laws [1].Quaternion algebra is one of the different and alternative mathematical tools in orderto formulate and represent some physical systems. Quaternions have non-commutativebut associative algebraic structures in four components [2], and they have used in manysubfields of physics. In this study, the Maxwell-type equations are rewritten by usingdual quaternion form, which is one kind of quaternion algebra including dual unit [3].The equations of gravitation for static fields have been rearranged in a compact, basicand alternative way in dual quaternion notation.

Keywords: Dual Quaternion, Dual unit, Linear gravity, Static fields.

References

[1] Demir S., Tanisli M., Biquaternionic Proca-type generalization of gravity, Eur. Phys.J. Plus, 126(51), 2011, 1-8.[2] Hamilton W.R., Elements of Quaternions, Chelsea Publishing, New York, 1969.[3]Demir S., Ozdas K., Dual quaternion reformulation of Electromagnetism, Acta PhysicaSlovaca, 53(6), 2003, 429-436.

95


A Necessary Condition and Sufficient Condition for theSummability of the Discrete Hilbert Transform

Aynur F. HUSEYNLIBaku State University, AZERBAIJAN

Abstract. We find the necessary condition and sufficient condition for summability ofthe discrete Hilbert transform of the sequence from the class l1.

Let b = bnn∈Z ∈ l1. The sequence H(b) = (H b)nn∈Z is called the Hilbert trans-form of the sequence b = bnn∈Z , where

(H b)n =∑

m 6=n

bm

n−m, n ∈ Z .

M.Riesz (see [4]) proved that, if b ∈ lp, p > 1, then H(b) ∈ lp and the inequality

‖H(b)‖lp≤ Cp‖b‖lp

(14)

holds. Weighted analogues of (14) are investigated in the works [1-4].If b ∈ l1, then the sequence Hb belong to the class

⋂

p>1lp, but doesn’t belong to the

class l1. In this case R. Hunt, B. Muckenhoupt and R. Wheeden (see [3]) proved that

the distribution function (H b)(λ) =

n ∈ Z : |(H b)n| > λ

≡∑

n∈Z:|(H b)n|>λ1 of the

Hilbert transform of the sequence b satisfies the condition

∀λ > 0 : |(H b)(λ)| ≤C0

λ‖b‖l1 ,

where C0 is an absolute constant.Theorem 1. Let b ∈ l1. Then to include H b ∈ l1, it is necessary that the equation

∑

n∈Z

bn = 0

holds.Theorem 2. If the sequence b ∈ l1 satisfies the conditionsi)∑

n∈Zbn = 0;

ii)∑

n∈Z|bm| ln(e+ |m|)<∞, then H b ∈ l1 and the inequality

‖H b‖l1 ≤ 6∑

m∈Z

|bm| ln(e+ |m|)

holds.

96



Theorem 3. Under the conditions of Theorem 2 the equation∑

n∈Z

(H b)n = 0

holds.

References

[1] Andersen K.F., Inequalities with weights for Discrete Hilbert Transforms, Canad. Math.Bul., 20, 1977, 9-16.[2] De Carli L., Samad S., One-parameter groups and discrete Hilbert transform, Canad.Math. Bull., 59, 2016, 497-507.[3] Hunt R., Muckenhoupt B., Wheeden R., Weighted norm inequalities for the conjugatefunction and Hilbert Transform, Trans. of Amer. Math. Sos., 176(2), 1973, 227-251.[4] Riesz M., Sur les fonctions conjuguees, Mathematische Zeitschrift, 27, 1928, 218-244.

97


Approximation of Hypersingular Integral Operators with HilbertKernel on Hölders Spaces

Chinara A. GADJIYEVABaku State University, AZERBAIJAN

Abstract. Let the function ϕ(t) is Lebesgue integrable on [0, 2π] and 2π- periodic.Definition. If a finite limit,

limε→0+

∫ t−ε

t−π

cscτ− t

2

ϕ(τ)dτ+

∫ t+π

t+ε

cscτ− t

2

ϕ(τ)dτ− 4ϕ(t) ln1ε

exist, then the value of this limit referred to as the hypersingulyar integral of the func-

tion

cscτ−t2

ϕ(τ) on T0 and is denoted by∫ 2π

0

cscτ−t2

ϕ(τ)dτ.

Let Λα(T0), 0 < α ≤ 1, be the space of 2π - periodic, Hölder continuous functionswith exponent α in real axis i.e. the space of the functions which satisfies the followingcondition

∃M > 0 ∀t1, t2 ∈ R : |ϕ(t1)−ϕ(t2)| ≤ M · |t1 − t2|α,

with the norm

‖ϕ‖α = ‖ϕ‖∞ + h(ϕ;α),

where

‖ϕ‖∞ =maxt∈γ0

|ϕ(t)|,

h(ϕ;α) = sup¦ |ϕ(t1)−ϕ(t2)|

|t1 − t2|: t1, t2 ∈ R, t1 6= t2

©

Consider the hypersingular integral operator

( eH (0)ϕ)(t) =1

4π

∫ 2π

0

cscτ− t

2

ϕ(τ)dτ,

where ϕ ∈ Λα(T0).Theorem. Hypersingular integral operator eH (0) is bounded from the space Λα(T0)

into the space Λα−ε(T0), for all 0< α≤ 1 and 0< ε < α.Now consider the sequences of operators

( eH (0)n ϕ)(t) =1

2n

n−1∑

k=0

cscπ(2k+ 1)

2n

ϕ

t +π(2k+ 1)

n

−ϕ(t)

+ln 4πϕ(t), t ∈ T0, n ∈ N .

Theorem. For any ϕ ∈ Λα(T0), λ < α≤ 1 the following estimate holds

‖ eH (0)ϕ − eH (0)n ϕ‖∞ ≤C20 ln(n+ 1)

nα· h(ϕ;α), n= 1, 2, . . . .

98



References

[1] Aliyev R.A., A new constructive method for solving singular integral equations, Math-ematical Notes, 79(2), 2006, 803-827.[2] Aliyev R.A., Amrakhova A.F., A new constructive method for the solution of integralequations with Hilbert kernel, Proceeding of the Institute of Mathematics and Mechan-ics: Ural branche of the Russian Academy of Sciences, 18(4), 2012, 14-25. (in Russia)[3] Aliev R.A, Gadjieva Ch.A., Approximation of hypersingular integral operators withCauchy kernel, Numerical Functional Analysis and Optimization, 37(9), 2016, 1055-1065.

99


On Singular Operators in Vanishing Generalized Variable ExponentMorrey Spaces and Applications to Bergman Type Spaces

Alexey KARAPETYANSSouthern Federal University, RUSSIA

Abstract. We give a proof of the boundedness of Bergman projection in generalizedvariable exponent vanishing Morrey spaces over the unit disc and the upper half-plane.To this end, we prove the boundedness of Calderón-Zygmund operators in general-ized variable exponent vanishing Morrey spaces. We give the proof of the latter in thegeneral context of real functions since it is new in such setting and is of independentinterest. We also study the approximation by mollified dilations and give the estimateof growth of functions near the boundary.

Joint work with S. Samko and H. Rafeiro.

100



Some Relations Between Partially q-Poly-Euler Polynomials of theSecond Kind

Burak KURTAkdeniz University, TURKEY

Abstract. In recent years, many researchers studied on the Euler, poly-Euler and q-poly-Euler polynomials and numbers. They introduced and investigated some properties ofthese polynomials including several identities for them.

In this paper, we define the partially q-poly-Euler polynomials of the second kind.We also prove some relations between the partially q-poly-Euler polynomials of thesecond kind and the q-Bernoulli polynomials Bn,q(x , y). By using these polynomials andnumbers. In addition, we obtain many identities relations including the Roger-Szégo

polynomials, the partially q-poly-Euler polynomials of the second kindö

En,q (x , y) andq-Hermite polynomials Hn,q(x)

Keywords: Euler polynomials and numbers, q-Euler polynomials of second kind, q-poly Bernoulli polynomials, Poly-logarithm function, q-Stirling numbers of the secondkind.

101


Grand Lebesgue Spaces on Sets of Infinite MeasureSalauddin UMARKHADZHIEV

Kh. Ibragimov Complex Institute of the Russian Academy of Sciences,Academy of Sciences of the Chechen Republic, RUSSIA

Abstract. We study operators of harmonic analysis in grand Lebesgue spaces on setsof infinite measure. To define such grand spaces, we introduce the notion of so calledgrandizer in terms of integrability of which we get a criterion for grand Lebesgue spaceto extend the classical Lebesgue space. We obtain a general statement for the bounded-ness of a class of sublinear operators, which is applied to a number of concrete operatorsof harmonic analysis such as maximal, potential, Hardy operators and others.

102



Condition for The Affilation of the Function to Genarilized HöldersClass Hγ

αβ(R+m+k,k)

Asim A. AKBAROVBaku State University, AZERBAIJAN

Abstract. Let Rn-be an Euclidean space of dimension n≥ 1,

R+m+k,k = (x1, . . . , xm, xm+1, . . . , xm+k) ∈ Rm+k : xm+1 > 0, . . . , xm+k > 0,

ex = (x ′, xm+1, . . . , xm+k, 0, . . . , 0), y = (y ′, ym+1, . . . , ym+k, eym+1, . . . , eym+k),

x ′, y ′ ∈ Rn, d y = d y1 . . . d ym+k deym+1 . . . deym+k, rex ,y = |ex − y|,

γ > 0, α > 0, β ∈ R, Γ - is a boundary R+m+k,k. Put

Γ = x ∈ Rm+k : xm+1 . . . xm+k = 0, rΓ (x) =min|x − y|; y ∈ Γ .

Lets take a weight function

ρ(x) = rαΓ(x)(1+ |x |)β−α ≡ (minxm+1, . . . , xm+k)α(1+ |x |)β−α, x ∈ Rm+k.

Introduce a Hölder weight space - Hγ

αβ(R+m+k,k). By definition [1], [2] u ∈ Hγ

αβ(R+m+k,k),

if limx→z∈Γ

u(x)ρ(x) = 0, limx→∞

u(x)ρ(x) = 0 and the finite norm

‖u‖Hγαβ=

∑

x ,y∈R+m+k,k

(|u(x)ρ(x)− u(y)ρ(y)|d−γ(x , y)),

where

d(x , y) = |x − y|((1+ |x |)(1+ |y|))−1.

Let x ∈ R+m+k,k. Denote

Ux =¦

y ∈ Rm+2k : |ex − y|<rΓ (x)

2

©

; l = 2γ+ β −α.

Theorem. If u ∈ Hγ

αβ(R+m+k,k), then

a)

u(y ′,q

y2m+1 + ey

2m+1, . . . ,

q

y2m+k + ey

2m+k)

≺ ‖u‖Hγαβ(A(y, ey))γ−α(1+ |y|)−l

‖u‖Hγαβ(A(y, ey))γ−α(1+ |y ′|+ B(y, ey))−l ,

103


where A(y, ey) = min|ym+i| + |eym+i|, i = 1, . . . , k, B(y, ey) =k∑

i=1(|ym+i| + |eym+i|); b)

∀x ∈ R+m+k,k, ∀y ∈ Ux

u

y ′,q

y2m+1 + ey

2m+1, . . . ,

q

y2m+k + ey

2m+k

− u(x)

≺ ‖u‖Hγαβ·ρ−1(x) · dγ(ex , y)

‖u‖Hγαβ·ρ−1(x) ·

|ex − y|(1+ |x |)2

γ

.

References

[1] Abdullayev S.K., Akbarov A.A., Weight Hölder estimates of singular integrals gen-erated by generalized shift operator, Vestnik Baku State Univ. Ser. Fiz-Mat., 3, 2007,24-31.[2] Abdullayev S.K., Agarzayev B.K., Hölder weight estimates of singular integrals gener-ated by generalized shift operator, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math.Sci., 24(1), 2004, 9-18.

104



The Solution of a Class of Nonlinear Integral Equations byNewton-Kantorovich Method

Fuad A. ABDULLAYEVBaku State University, AZERBAIJAN

Abstract. Consider the problem of finding a solution of the following nonlinear singularintegral equation

f (t, u(t))−1π

∫ 1

−1

u(τ)τ− t

dτ− D = 0 (15)

satisfying

u(−1) = u(1) = 0 (16)

where f (t, u) is given function.Here is required to find the function which satisfies the conditions (15)-(16) and

the constant D. This problem is equivalent to the following Riemann-Hilbert problem[1]. There is required to find the functions W (z) = u(z) + iv(z), z = x + i y whichholomorphic on the upper half plane y > 0 of the complex plane C , such that it isbounded at an infinity point and the following boundary conditions

¨

u= 0, |x | ≥ 1,

V (x) + f (x , u(x)) = 0, |x |< 1(17)

holds on the real axis y = 0.From the theory of singular integral equations it follows that the problem (15)-(16)

is equivalent to the following problem [2]:

u(t) =p

1− t2

π

∫ 1

−1

f (τ, u(τ))p

1−τ2(τ− t)dτ, (18)

D =1π

∫ 1

−1

f (τ, u(τ))p

1−τ2dτ. (19)

We will consider the equation (18) in the space Hα = ϕ ∈ Hα[−1,1] | ϕ(−1) =ϕ(1) = 0, 0 < α < 1

2 , where Hα[−1,1]- is the space of the functions which satisfiesHölder condition with exponent α.

Consider the following operator

(Pu)(t) = u(t) +p

1− t2

π

∫ 1

−1

f (τ, u(τ))p

1−τ2(τ− t)dτ.

105


Suppose that the function f (t, u) is defined in Ω= (t, u) | − 1≤ t ≤ 1,−∞< u<+∞ and satisfies the following condition:

1) f (t, 0) ∈ Hα,2) There exists ∃l > 0 such that for all∀(t1, u1), (t2, u2) ∈ Ω the following inequality

| f (t1, u1)− f (t2, u2)| ≤ l

|t1 − t2|α + |u1 − u2|

is true.Then it follows that operator P is bounded in Hα. The following theorem is trueTheorem: Suppose that there exist the derivatives f ′u(t, u), f ′′u2(t, u) in Ω and the

following conditions holds:a) f (t, 0), f ′u(t, 0), f ′′u2(t, 0) ∈ Hα,b) There exist constants ∃l0, l1, l2 > 0 such that for all ∀(t1, u1), (t2, u2) ∈ Ω, the

following inequalities

| f (t1, u1)− f (t2, u2)| ≤ l0

|t1 − t2|α + |u1 − u2|

| f iui(t1, u1)− f i

ui(t2, u2)| ≤ li

|t1 − t2|α + |u1 − u2|

i = 1, 2

are true.Then the operator P is a Frechet-differentiable at any point of the space Hα and its

Frechet derivatives have bounded inverse. Using proven theorem in the present paperthe problem (15)-(16) is solved by the Newton-Kantorovich Method.

References

[1] Junghanns P., Silbermann B., Numerical analysis of the quadrature method for solvinglinear and nonlinear singular integral equations, Karl-Marx-Stadt, 1988, 895.[2] Muskheleshvili N.I., Singular Integral equations, Moscow, "Nauka", 1968, 511.

106



Some Progress on Reversible RingsHandan KOSE

Kirsehir Ahi Evran University, TURKEY

Abstract. In ring theory the nilpotent elements of a ring are very important tool instudying the structure of the ring. In this work, we deal with some version of reversibil-ity with nilpotent elements on rings.

Keywords: Nilpotent elements, Reversible rings.

107


Constructing a System of First Kind Chebyshev PolynomialsSahib A. ALIYEV

Nakhchivan Teacher-Training Institute, AZERBAIJAN

Abstract. It is known that the system of orthogonal and orthonormal polynomials withrespect to weight function used more widely in the theory of harmonic analysis wasstudy [1, 2] and etc.

In this work, using the condition of existence and uniqueness of the system of or-thonormal polynomials, the system of first kind Chebyshev polynomials, is considered.

Lemma. The n-th degree polynomial Qn(k) by the system of polynomials F0(x), F1(x),F2(x), . . . , Fn(x) can be shown in a unique way as follows(see [4])

Qn(k) = a0F0(k) + a1F1(k) + . . .+ anFn.

Theorem. For arbitrary weight function h(x) there exists a unique system of poly-nomials Pn(x)with positive higher order term coefficient and satisfying the orthonor-mality condition.

Later we said the algebraic polynomials Tn(x) = cos(n arccosx) to be first kindChebyshev polynomials. Using the theorem, the system of first kind Chebyshev orthog-onal polynomials is constructed.

By means of the new method, using the existence and uniqueness condition, thesystem of first kind Chebyshev polynomials is constructed. As a result, the terms of theChebyshev polynomials T0(x) =

1pπ

, T1(x) =q

2π x T2(x) = (2x2−1)

q

2π . . . are found.

References

[1] Uvarenkov I.M., Maller M.Z., Mathematical Analysis course, Textbook for phys.Math. Dept. of Ped. Univer. vol. II. M. Prosveshenie, 1976. (in Russian)[2] Suetin P.K., Classic orthogonal polynomials, Edition 2., M. Nauka, 1979, 416. (inRussian)[3] Fichtenholts E.T., Course of differential and integral calculus, Vol. I,II, III. Nauka,1969. (in Russian)[4] Aliyev S., Agayev E., Existence and uniqueness conditions of the system of orthonor-mal polynomials with respect to weight function, Elm eserler, Nakhchivan, 4, NSU-2017,Azerbaijan.

108

Author Index

A.ABKAR, 45

A. FundaSAGLAMER, 35

A. SerdarNAZLIPINAR, 44, 47

AbdellahGHERBI, 58

AbdulhamitKUCUKASLAN, 53

AhmetBOZ, 49EROGLU, 59TESTICI, 76

Akbar B.ALIEV, 51

AlexeyKARAPETYANS, 100

AliAKBULUT, 64

Alik M.NAJAFOV, 15, 89

Amil A.HASANOV, 86

AmiranGOGATISHVILI, 5

AndreaSCAPELLATO, 46

ArashGHORBANALIZADEH, 79

ArzuERDEM COSKUN, 93

Asim A.AKBAROV, 103

AyhanSERBETCI, 34

Aynur F.HUSEYNLI, 96

Aynur N.MAMMADOVA, 54

Aysel A.AHMADLI, 74

AysunYURTTAS, 4

Aytekin E.ABDULLAYEVA, 55

Azizgul M.GASUMOVA, 20

BarbarosBASTURK, 44

BurakKURT, 101

CanayAYKOL, 34

CansuKESKIN, 56

Chinara A.GADJIYEVA, 98

DaniyalISRAFILOV, 83

DenizUCAR, 40

109


DilekSOYLEMEZ, 73

Dunya R.ALIYEVA, 54

EkremSAVAS, 12

ElçinYUSUFOGLU, 37

EmineKIRHAN, 83

ErdalULUALAN, 63

EsraKAYA, 26

FadilJARADAT, 29

Faig M.NAMAZOV, 18

Fatai A.ISAYEV, 72

FatihDERINGOZ, 67

Fatma A.MUSLIMOVA, 80, 82

Firida F.MUSTAFAYEVA, 31

FrancescaANCESCHI, 10

Fuad A.ABDULLAYEV, 105

Gulnara A.ABASOVA, 78

Gulshan Kh.SHAFIYEVA, 51

HandanKOSE, 107

HaticeARMUTCU, 61

Hummet K.

MUSAEV, 60

IsmailAYMAZ, 95EKINCIOGLU, 25, 26

Ismail NaciCANGUL, 4

Kamala H.SAFAROVA, 57

KhalidHILAL, 48

KonulSULYEMANOVA, 19YASINLI, 30

Lale R.ALIYEVA, 55

Leyla Sh.KADIMOVA, 22

llkemTURHAN CETINKAYA, 37

M.H.TALSHBOOL, 29

Maria AlessandraRAGUSA, 10, 11

Mehriban N.OMAROVA, 65

MiloudASSAL, 17

MojtabaRASTGOO, 45

Mubariz G.HAJIBAYOV, 70

MugeTOGAN, 4

MuratTANISLI, 94

Mustafa EmreKANSU, 94, 95

Mustafa KemalBERKTAS, 36

110



Nilufer R.RUSTAMOVA, 23

NisrineMAAROUF, 48

NurtenKILIÇ, 35

NurullahYILMAZ, 41

OzgunGURMEN ALANSAL, 63

PankajJAIN, 9

PrzemyslawGORKA, 6

RabilAYAZOGLU(MASHIYEV), 68, 69

RadouanDAHER, 91

Ramin V.GULIYEV, 75

Rovshan A.BANDALIYEV, 3

Rovshan F.BABAYEV, 27

S. ElifnurEKINCIOGLU, 47

Sabir G.HASANOV, 88

SadikDELEN, 4

Sahib A.ALIYEV, 92, 108

SahseneALTINKAYA, 39, 42

Saikat

KANJILAL, 52Sain T.

ALEKBERLI, 15Salauddin

UMARKHADZHIEV, 102Semra

DOGRUOZ, 36Sergio

POLIDORO, 10Sevda

YILDIZ, 33Shahla

GALANDAROVA, 19Shemsiyye A.

MURADOVA, 62Sibel

YALCIN TOKGOZ, 39, 42Soltan A.

ALIYEV, 92Suleyman

DEMIR, 94

Tahir S.GADJIEV, 18, 19, 30

Tugce UnverYILDIZ, 50

Vagif S.GULIYEV, 7

Veysel FuatHATIPOGLU, 40

Victor I.BURENKOV, 2

Vugar H.HAMZAYEV, 84

Yagub Y.MAMMADOV, 80

111

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