Functional Differential and Difference Equations with ...downloads.hindawi.com/journals/specialissues/353195.pdf · Functional Differential and Difference Equations with Applications

Guest Editors: J. Diblk, E. Braverman, I. Gyri, Yu. Rogovchenko, M. Ru ic kov, and A. Zafer

Functional Differential and Difference Equations with Applications 2013

Abstract and Applied Analysis

Functional Differential and DifferenceEquations with Applications 2013

Abstract and Applied Analysis

Functional Differential and DifferenceEquations with Applications 2013

Guest Editors: J. Diblk, E. Braverman, I. Gyori, Yu. Rogovchenko,M. Ruzickova, and A. Zafer

Copyright 2013 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in Abstract and Applied Analysis. All articles are open access articles distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly cited.

Editorial Board

Dirk Aeyels, BelgiumRavi P. Agarwal, Saudi ArabiaM. O. Ahmedou, GermanyNicholas D. Alikakos, GreeceDebora Amadori, ItalyPablo Amster, ArgentinaDouglas R. Anderson, USAJan Andres, Czech RepublicGiovanni Anello, ItalyStanislav Antontsev, PortugalMohamed Kamal Aouf, EgyptNarcisa C. Apreutesei, RomaniaNatig M. Atakishiyev, MexicoFerhan M. Atici, USAIvan G. Avramidi, USASoohyun Bae, KoreaZhanbing Bai, ChinaChuanzhi Bai, ChinaDumitru Baleanu, TurkeyJzef Bana, PolandGerassimos Barbatis, GreeceMartino Bardi, ItalyRoberto Barrio, SpainFeyzi Basar, TurkeyAbdelghani Bellouquid, MoroccoDaniele Bertaccini, ItalyMichiel Bertsch, ItalyLucio Boccardo, ItalyIgor Boglaev, New ZealandMartin J. Bohner, USAJulian F. Bonder, ArgentinaGeraldo Botelho, BrazilElena Braverman, CanadaRomeo Brunetti, ItalyJanusz Brzdek, PolandDetlev Buchholz, GermanySun-Sig Byun, KoreaFabio M. Camilli, ItalyAntonio Canada, SpainJinde Cao, ChinaAnna Capietto, ItalyKwang-chih Chang, ChinaJianqing Chen, ChinaWing-Sum Cheung, Hong KongMichel Chipot, Switzerland

Changbum Chun, KoreaSoon Y. Chung, KoreaJaeyoung Chung, KoreaSilvia Cingolani, ItalyJean M. Combes, FranceMonica Conti, ItalyDiego Cordoba, SpainJuan Carlos Cortes Lopez, SpainGraziano Crasta, ItalyGuillermo P. Curbera, SpainBernard Dacorogna, SwitzerlandVladimir Danilov, RussiaMohammad T. Darvishi, IranLuis F. Pinheiro de Castro, PortugalToka Diagana, USAJesus I. Daz, SpainJosef Diblk, Czech RepublicFasma Diele, ItalyTomas Dominguez, SpainAlexander I. Domoshnitsky, IsraelMarco Donatelli, ItalyBo-Qing Dong, ChinaOndrej Dosly, Czech RepublicWei-Shih Du, TaiwanLuiz Duarte, BrazilRoman Dwilewicz, USAPaul W. Eloe, USAAhmed El-Sayed, EgyptLuca Esposito, ItalyJose A. Ezquerro, SpainKhalil Ezzinbi, MoroccoDashan Fan, USAAngelo Favini, ItalyMarcia Federson, BrazilStathis Filippas, Equatorial GuineaAlberto Fiorenza, ItalyTore Flatten, NorwayIlaria Fragala, ItalyBruno Franchi, ItalyXianlong Fu, ChinaMassimo Furi, ItalyGiovanni P. Galdi, USAIsaac Garcia, SpainJesus Garca Falset, SpainJose A. Garca-Rodrguez, Spain

Leszek Gasinski, PolandGyorgy Gat, HungaryVladimir Georgiev, ItalyLorenzo Giacomelli, ItalyJaume Gine, SpainValery Y. Glizer, IsraelLaurent Gosse, ItalyJean P. Gossez, BelgiumDimitris Goussis, GreeceJose L. Gracia, SpainMaurizio Grasselli, ItalyQian Guo, ChinaYuxia Guo, ChinaChaitan P. Gupta, USAUno Hamarik, EstoniaFerenc Hartung, HungaryBehnam Hashemi, IranNorimichi Hirano, JapanJiaxin Hu, ChinaChengming Huang, ChinaZhongyi Huang, ChinaGennaro Infante, ItalyIvan Ivanov, BulgariaHossein Jafari, IranJaan Janno, EstoniaAref Jeribi, TunisiaUn C. Ji, KoreaZhongxiao Jia, ChinaLucas Jodar, SpainJong Soo Jung, Republic of KoreaHenrik Kalisch, NorwayHamid Reza Karimi, NorwaySatyanad Kichenassamy, FranceTero Kilpelainen, FinlandSung Guen Kim, Republic of KoreaLjubisa Kocinac, SerbiaAndrei Korobeinikov, SpainPekka Koskela, FinlandVictor Kovtunenko, AustriaRen-Jieh Kuo, TaiwanPavel Kurasov, SwedenMiroslaw Lachowicz, PolandKunquan Lan, CanadaRuediger Landes, USAIrena Lasiecka, USA

Matti Lassas, FinlandChun-Kong Law, TaiwanMing-Yi Lee, TaiwanGongbao Li, ChinaPedro M. Lima, PortugalElena Litsyn, IsraelYansheng Liu, ChinaShengqiang Liu, ChinaCarlos Lizama, ChileMilton C. Lopes Filho, BrazilJulian Lopez-Gomez, SpainJinhu Lu, ChinaGrzegorz Lukaszewicz, PolandShiwang Ma, ChinaWanbiao Ma, ChinaEberhard Malkowsky, TurkeySalvatore A. Marano, ItalyCristina Marcelli, ItalyPaolo Marcellini, ItalyJesus Marn-Solano, SpainJose M. Martell, SpainMieczysaw Mastyo, PolandMing Mei, CanadaTaras Melnyk, UkraineAnna Mercaldo, ItalyChangxing Miao, ChinaStanislaw Migorski, PolandMihai Mihailescu, RomaniaFeliz Minhos, PortugalDumitru Motreanu, FranceRoberta Musina, ItalyMaria Grazia Naso, ItalyGaston M. NGuerekata, USASylvia Novo, SpainMicah Osilike, NigeriaMitsuharu Otani, JapanTurgut Ozis, TurkeyFilomena Pacella, ItalyNikolaos S. Papageorgiou, GreeceSehie Park, KoreaAlberto Parmeggiani, ItalyKailash C. Patidar, South AfricaKevin R. Payne, ItalyJosip E. Pecaric, CroatiaShuangjie Peng, ChinaSergei V. Pereverzyev, AustriaMaria Eugenia Perez, SpainDavid Perez-Garcia, SpainJosefina Perles, SpainAllan Peterson, USA

Andrew Pickering, SpainCristina Pignotti, ItalySomyot Plubtieng, ThailandMilan Pokorny, Czech RepublicSergio Polidoro, ItalyZiemowit Popowicz, PolandMaria M. Porzio, ItalyEnrico Priola, ItalyVladimir S. Rabinovich, MexicoIrena Rachunkova, Czech RepublicMaria Alessandra Ragusa, ItalySimeon Reich, IsraelWeiqing Ren, USAAbdelaziz Rhandi, ItalyHassan Riahi, MalaysiaJuan P. Rincon-Zapatero, SpainLuigi Rodino, ItalyYuriy Rogovchenko, NorwayJulio D. Rossi, ArgentinaWolfgang Ruess, GermanyBernhard Ruf, ItalyMarco Sabatini, ItalySatit Saejung, ThailandStefan G. Samko, PortugalMartin Schechter, USAJavier Segura, SpainSigmund Selberg, NorwayValery Serov, FinlandNaseer Shahzad, Saudi ArabiaAndrey Shishkov, UkraineStefan Siegmund, GermanyAbdel-Maksoud A. Soliman, EgyptPierpaolo Soravia, ItalyMarco Squassina, ItalySvatoslav Stanek, Czech RepublicStevo Stevic, SerbiaAntonio Suarez, SpainWenchang Sun, ChinaRobert Szalai, UKSanyi Tang, ChinaChun-Lei Tang, ChinaYoushan Tao, ChinaGabriella Tarantello, ItalyNasser-eddine Tatar, Saudi ArabiaSusanna Terracini, ItalyGerd Teschke, GermanyAlberto Tesei, ItalyBevan Thompson, AustraliaSergey Tikhonov, SpainClaudia Timofte, Romania

Thanh Dien Tran, AustraliaJuan J. Trujillo, SpainCiprian A. Tudor, FranceGabriel Turinici, FranceMilan Tvrdy, Czech RepublicMehmet Unal, TurkeyStephan A. van Gils, The NetherlandsCsaba Varga, RomaniaCarlos Vazquez, SpainGianmaria Verzini, ItalyJesus Vigo-Aguiar, SpainQing-Wen Wang, ChinaYushun Wang, ChinaShawn X. Wang, CanadaYouyu Wang, ChinaJing Ping Wang, UKPeixuan Weng, ChinaNoemi Wolanski, ArgentinaNgai-Ching Wong, TaiwanPatricia J. Y. Wong, SingaporeRoderick Wong, Hong KongZili Wu, ChinaYong Hong Wu, AustraliaShanhe Wu, ChinaTie-cheng Xia, ChinaXu Xian, ChinaYanni Xiao, ChinaFuding Xie, ChinaNaihua Xiu, ChinaDaoyi Xu, ChinaZhenya Yan, ChinaXiaodong Yan, USANorio Yoshida, JapanBeong In Yun, KoreaVjacheslav Yurko, RussiaAgack Zafer, TurkeySergey V. Zelik, UKJianming Zhan, ChinaMeirong Zhang, ChinaWeinian Zhang, ChinaChengjian Zhang, ChinaZengqin Zhao, ChinaSining Zheng, ChinaTianshou Zhou, ChinaYong Zhou, ChinaQiji J. Zhu, USAChun-Gang Zhu, ChinaMalisa R. Zizovic, SerbiaWenming Zou, China

Contents

Functional Differential and Difference Equations with Applications 2013, J. Diblk, E. Braverman,I. Gyori, Yu. Rogovchenko, M. Ruzickova, and A. ZaferVolume 2014, Article ID 543797, 2 pages

General Explicit Solution of Planar Weakly Delayed Linear Discrete Systems and Pasting Its Solutions,Josef Diblk and Hana HalfarovaVolume 2014, Article ID 627295, 37 pages

Representation of the Solutions of Linear Discrete Systems with Constant Coefficients and Two Delays,Josef Diblk and Blanka MoravkovaVolume 2014, Article ID 320476, 19 pages

Spatial Approximation of Nondivergent Type Parabolic PDEs with Unbounded Coefficients Related toFinance, Fernando F. Goncalves and Maria Rosario GrossinhoVolume 2014, Article ID 801059, 11 pages

Stability Analysis of Impulsive Stochastic Functional Differential Equations with Delayed Impulses viaComparison Principle and Impulsive Delay Differential Inequality, Pei Cheng, Fengqi Yao,and Mingang HuaVolume 2014, Article ID 710150, 9 pages

Automorphisms of Ordinary Differential Equations, Vaclav Tryhuk and Veronika ChrastinovaVolume 2014, Article ID 482963, 32 pages

On the Periodicity of Some Classes of Systems of Nonlinear Difference Equations, Stevo Stevic,Mohammed A. Alghamdi, Dalal A. Maturi, and Naseer ShahzadVolume 2014, Article ID 982378, 6 pages

On Asymptotic Behavior of Solutions of Generalized Emden-Fowler Differential Equations with DelayArgument, Alexander Domoshnitsky and Roman KoplatadzeVolume 2014, Article ID 168425, 13 pages

Asymptotic Behavior of Higher-Order Quasilinear Neutral Differential Equations,Tongxing Li and Yuriy V. RogovchenkoVolume 2014, Article ID 395368, 11 pages

Existence and Global Exponential Stability of Equilibrium for Impulsive Cellular Neural NetworkModels with Piecewise Alternately Advanced and Retarded Argument, Kuo-Shou ChiuVolume 2013, Article ID 196139, 13 pages

Newton-Kantorovich and Smale Uniform Type Convergence Theorem for a Deformed Newton Methodin Banach Spaces, Rongfei Lin, Yueqing Zhao, Zdenek Smarda, Yasir Khan, and Qingbiao WuVolume 2013, Article ID 923898, 8 pages

Asymptotic Behavior of Solutions to a Vector Integral Equation with Deviating Arguments,Cristobal Gonzalez and Antonio Jimenez-MeladoVolume 2013, Article ID 957696, 7 pages

Stability and Global Hopf Bifurcation Analysis on a Ratio-Dependent Predator-Prey Model with TwoTime Delays, Huitao Zhao, Yiping Lin, and Yunxian DaiVolume 2013, Article ID 321930, 15 pages

Boundedness of Solutions for a Class of Second-Order Periodic Systems, Shunjun Jiang and Yan DingVolume 2013, Article ID 267572, 9 pages

Moment Equations in Modeling a Stable Foreign Currency Exchange Market in Conditions ofUncertainty, Josef Diblk, Irada Dzhalladova, Maria Michalkova, and Miroslava RuzickovaVolume 2013, Article ID 172847, 11 pages

Singular Initial Value Problem for Certain Classes of Systems of Ordinary Differential Equations,Josef Diblk, Josef Rebenda, and Zdenek SmardaVolume 2013, Article ID 207352, 12 pages

Stability of Impulsive Differential Systems, Andrejs Reinfelds and Lelde SermoneVolume 2013, Article ID 253647, 11 pages

Infinitely Many Periodic Solutions to Delay Differential Equations via Critical Point Theory,Xiaosheng Zhang and Duo WangVolume 2013, Article ID 526350, 7 pages

On Linear Difference Equations for Which the Global Periodicity Implies the Existence of anEquilibrium, Istvan Gyori and Laszlo HorvathVolume 2013, Article ID 971394, 5 pages

Oscillation Theorems for Even Order Damped Equations with Distributed Deviating Arguments,Chunxia Gao and Peiguang WangVolume 2013, Article ID 393892, 10 pages

Impulsive Boundary Value Problems for Planar Hamiltonian Systems, Zeynep Kayar and Agack ZaferVolume 2013, Article ID 892475, 6 pages

Oscillation of Half-Linear Differential Equations with Delay, Simona Fisnarova and Robert MarkVolume 2013, Article ID 583147, 6 pages

Some New Nonlinear Weakly Singular Inequalities and Applications to Volterra-Type DifferenceEquation, Kelong Zheng, Wenqiang Feng, and Chunxiang GuoVolume 2013, Article ID 912874, 6 pages

Heat Transfer Analysis on the Hiemenz Flow of a Non-Newtonian Fluid: A Homotopy Method Solution,Yasir Khan and Zdenek SmardaVolume 2013, Article ID 342690, 5 pages

Asymptotic Behavior of Solutions to a Linear Volterra Integrodifferential System,Yue-Wen Cheng and Hui-Sheng DingVolume 2013, Article ID 245905, 5 pages

Regularity of a Stochastic Fractional Delayed Reaction-Diffusion Equation Driven by Levy Noise,Tianlong Shen, Jianhua Huang, and Jin LiVolume 2013, Article ID 807459, 11 pages

A Half-Inverse Problem for Impulsive Dirac Operator with Discontinuous Coefficient, Yalcn GulduVolume 2013, Article ID 181809, 4 pages

Duffing-Type Oscillator with a Bounded from above Potential in the Presence of Saddle-CenterBifurcation and Singular Perturbation: Frequency Control, Robert Vrabel, Pavol Tanuska, Pavel Vazan,Peter Schreiber, and Vladimir LiskaVolume 2013, Article ID 848613, 7 pages

EditorialFunctional Differential and Difference Equations withApplications 2013

J. Diblk,1 E. Braverman,2 I. Gyri,3 Yu. Rogovchenko,4 M. RRDiIkov,5 and A. Zafer6

1 Brno University of Technology, Brno, Czech Republic2 University of Calgary, Calgary, Canada3University of Pannonia, Veszprem, Hungary4University of Agder, Kristiansand, Norway5University of Zilina, Zilina, Slovakia6Middle East Technical University, Ankara, Turkey

Correspondence should be addressed to J. Diblk; [email protected]

Received 27 January 2014; Accepted 27 January 2014; Published 12 May 2014

Copyright 2014 J. Diblk et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This annual issue comes as a sequel to two special issues,Recent Progress in Differential andDifference Equations editedby the four members of the present team and FunctionalDifferential and Difference Equations with Applications withthe same editorial team, both published by the Abstract andApplied Analysis in 2011 and 2012, respectively.

In the call for papers prepared by the Guest Editors andposted on the journals web page, we encouraged submissionof state-of-the-art contributions on a wide spectrum of topicssuch as asymptotic behavior of solutions, boundedness andperiodicity of solutions, nonoscillation and oscillation ofsolutions, representation of solutions, stability, numericalalgorithms, and computational aspects, as well as applica-tions to real-world phenomena. This invitation was warmlywelcomed by the mathematical community; more than sixtymanuscripts addressing important problems in the above-mentioned fields of qualitative theory of functional differen-tial and difference equations were submitted to the EditorialOffice and went through a thorough peer refereeing process.Twenty-seven carefully selected research articles collectedin this special issue reflect modern trends and advances infunctional differential and difference equations. Sixty-sevenauthors from fourteen countries (China, Czech Republic,Georgia, Hungary, Israel, Latvia, Norway, Portugal, SaudiArabia, Serbia, Slovak Republic, Spain, Turkey, and Ukraine)have contributed to the success of this thematic collection ofpapers.

Questions related to the existence of periodic solutionsand their stability properties traditionally attract attentionof researchers working in the qualitative theory of differen-tial, functional differential, and difference equations. In thisissue, the reader will find results that relate periodicity oflinear autonomous nonhomogeneous difference equations tothe existence of equilibria. Systems of nonlinear differenceequations whose all well-defined solutions are periodic areconsidered. For some classes of nonlinear systems with delay,it is shown that the presence of the time delay results in theexistence of periodic solutions.

Existence of oscillating solutions for half-linear differen-tial equations with delay and for even order damped equa-tions with distributed deviating arguments is studied. Non-linear oscillations in the context of saddle-center bifurcationin the dynamical system describing a singularly perturbedforced oscillator of Duffings type with a nonlinear restoringand a nonperiodic external driving force are examined.Boundedness of solutions to a class of second-order periodicsystems with singularities is considered as well.

Stability problems always attract interest of researchers,and this special issue is not an exception. The reader willfind papers on the stability of impulsive stochastic functionaldifferential equations with delayed impulses and stabilityof differential systems under permanently arising impulses,an application of moment equations in a model of thestable foreign currency exchange market in conditions of

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 543797, 2 pageshttp://dx.doi.org/10.1155/2014/543797

http://dx.doi.org/10.1155/2014/543797

2 Abstract and Applied Analysis

uncertainty, a study on the global exponential stability ofequilibria for impulsive cellular neural network models withpiecewise alternately advanced and retarded arguments, andanalysis of stability and global Hopf bifurcation phenomenain a ratio-dependent predator-prey model with two timedelays.

New results for boundary value problems are alsoreported, including the existence and uniqueness theoremfor solutions of nonhomogeneous impulsive boundary valueproblem for planar Hamiltonian systems and general resultson the solvability of singular initial value problems.

Several articles deal with the behavior of solutions atinfinity. Namely, explicit formulas for planar weakly delayedlinear discrete systems are derived; results on asymptoticbehavior of solutions to generalized Emden-Fowler differen-tial equations with delayed argument, higher-order quasilin-ear neutral differential equations, vector integral equationswith deviating arguments, and linear Volterra integrodif-ferential systems are reported along with the applicationsof nonlinear weakly singular inequalities to Volterra-typedifference equations.

Other important problems discussed in this special issueare related to the representation of the solutions of lineardiscrete systems with constant coefficients and two delays,automorphisms of ordinary differential equations, spatialdiscretization of the Cauchy problem for a multidimensionallinear parabolic partial differential equation of the secondorder with nondivergent operator, and unbounded time-and space-dependent coefficients. Newton-Kantorovich andSmale type convergence theorems are used in a deformedNewton method with the third-order convergence for solv-ing nonlinear equations. Regularity of a mild solution fora stochastic fractional delayed reaction-diffusion equationdriven by Levy space-time white noise and an inverseproblem for Dirac differential operators with discontinu-ity conditions and discontinuous coefficient are studied.Finally, a mathematical model for the incompressible two-dimensional/axisymmetric non-Newtonian fluid flows andheat transfer analysis in the region of stagnation point over astretching/shrinking sheet and axisymmetric shrinking sheetis presented.

It is not possible to provide in this short editorial note adetailed description for all papers included in this volume.However, it is clear that they reflect contemporary trendsin the development of the qualitative theory of functionaldifferential equations and feature important applications. Webelieve that this special issue challenges researchers withnew unsolved problems and introduces many new ideas anduseful techniques.

J. DiblkE. Braverman

I. GyoriYu. Rogovchenko

M. RuzickovaA. Zafer

Research ArticleGeneral Explicit Solution of Planar Weakly Delayed LinearDiscrete Systems and Pasting Its Solutions

Josef Diblk1,2 and Hana Halfarov1,2

1 Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology,602 00 Brno, Czech Republic

2 Department of Mathematics, Faculty of Electrical Engineering, Brno University of Technology, 616 00 Brno, Czech Republic

Correspondence should be addressed to Josef Diblk; [email protected]

Received 3 September 2013; Accepted 21 October 2013; Published 29 April 2014

Academic Editor: Miroslava Ruzickova

Copyright 2014 J. Diblk and H. Halfarova. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Planar linear discrete systems with constant coefficients and delays ( + 1) = () + =1

(

) are considered where

Z0:= {0, 1, . . . ,},

1, 2, . . . ,

are constant integer delays, 0 <

1< 2< <

, , 1, . . . , are constant 2 2

matrices, and : Z

R2. It is assumed that the considered system is weakly delayed. The characteristic equations of suchsystems are identical with those for the same systems but without delayed terms. In this case, after several steps, the space ofsolutions with a given starting dimension 2(

+ 1) is pasted into a space with a dimension less than the starting one. In a sense,

this situation is analogous to one known in the theory of linear differential systems with constant coefficients and special delayswhen the initially infinite dimensional space of solutions on the initial interval turns (after several steps) into a finite dimensionalset of solutions. For every possible case, explicit general solutions are constructed and, finally, results on the dimensionality of thespace of solutions are obtained.

1. Introduction

1.1. Preliminary Notions and Properties. We use the followingnotation: for integers , , , we define Z

:= {, +

1, . . . , }, where = or = is admitted, too.Throughout this paper, using notation Z

, we always assume

. In the paper, we deal with the discrete planar system

( + 1) = () +

=1

(

) , (1)

where 1, 2, . . . ,

are constant integer delays, 0 <

1 .

Proof. We start with computing determinant defined by(5). We get

=

12

34

, (17)

where1= 11+ 1

111

+ 2

112

+ +

11

,

2= 12+ 1

121

+ 2

122

+ +

12

,

3= 21+ 1

211

+ 2

212

+ +

21

,

4= 22+ 1

221

+ 2

222

+ +

22

.

(18)

Expanding the determinant on the right-hand side alongsummands of the first column, we get

=

11

12+ 1

121+ 2

122+ +

12

2122+ 1

221+ 2

222+ +

22

+ 1

1

1112+ 1

121+ 2

122+ +

12

1

2122+ 1

221+ 2

222+ +

22

+ 2

2

1112+ 1

121+ 2

122+ +

12

2

2122+ 1

221+ 2

222+ +

22

+

+

1112+ 1

121+ 2

122+ +

12

2122+ 1

221+ 2

222+ +

22

+

1 12+ 1

121+ 2

122+ +

12

0 22+ 1

221+ 2

222+ +

22

.

(19)

Expanding each of the above determinants along sum-mands of the second column, we have

=

1112

2122

+ 1

111

12

211

22

+ 2

112

12

212

22

+ +

11

12

21

22

+

11

0

211

+ 1

[

1

1112

1

2122

+ 1

1

111

12

1

211

22

+ 2

1

112

12

1

212

22

+ +

1

11

12

1

21

22

+

1

110

1

211

]

+ 2

[

2

1112

2

2122

+ 1

2

111

12

2

211

22

+ 2

2

112

12

2

212

22

+ +

2

11

12

2

21

22

+

2

110

2

211

]

+

+

[

1112

2122

+ 1

111

12

211

22

+ 2

112

12

212

22

+ +

11

12

21

22

+

110

211

]

+ [

1 12

0 22

+ 1

1 1

12

0 1

22

+ 2

1 2

12

0 2

22

+ +

1

12

0

22

+

1 0

0 1

] .

(20)After simplification, we get

=

11

12

21

22

1+1

(1

11+ 1

22)

2+1

(2

11+ 2

22) +

+1

(

11+

22)

+ 1

[

1112

1

211

22

+

1

111

12

2122

]

+ 2

[

1112

2

212

22

+

2

112

12

2122

]

+ +

[

1112

21

22

+

11

12

2122

]

+ 12

[

1

111

12

2

212

22

+

2

112

12

1

211

22

]

+ + 1

[

1

111

12

21

22

+

11

12

1

211

22

]

+ 23

[

2

112

12

3

213

22

+

3

113

12

2

212

22

]

+ + 2

[

2

112

12

21

22

+

11

12

2

212

22

]

+ + 1

[

1

111

12

21

22

+

11

12

1

211

22

]

+ 21

1

111

12

1

211

22

+ 22

2

112

12

2

212

22

+ + 2

11

12

21

22

.

(21)


Now we see that for (8) to hold; that is,

= det( +

=1

)

= det ( )

=

11

12

21

22

,

(22)

conditions (13)(16) are both necessary and sufficient.

Lemma 4. Conditions (13)(16) are equivalent to

tr = det = 0, (23)

det ( + ) = det, (24)

det ( + V) = 0, (25)

where , V = 1, 2, . . . , and V > .

Proof. (I) We show that assumptions (13)(16) imply (23)(25). It is obvious that condition (23) is equivalent to (13), (14).Now we consider

det ( + ) =

11+

1112+

12

21+

2122+

22

. (26)

Expanding the determinant on the right-hand side alongsummands of the first column and then expanding each ofthe determinants along summands of the second column, wehave

det ( + ) =

1112+

12

2122+

22

+

1112+

12

2122+

22

=

1112

2122

+

11

12

21

22

+

1112

2122

+

11

12

21

22

= [due to (15) and (16)] = det.

(27)

Now we consider

det ( + V) =

11+

V11

12+

V12

21+

V21

22+

V22

. (28)

Expanding the determinant on the right-hand side alongsummands of the first column and then expanding each of

the determinants along summands of the second column, wehave

det ( + V) =

11

12+

V12

21

22+

V22

+

V11

12+

V12

V21

22+

V22

=

11

12

21

22

+

11V12

21V22

+

V11

12

V21

22

+

V11V12

V21V22

= [due to (15) , (17)] = 0.

(29)

(II) Now we prove that assumptions (23)(25) imply (13)and (16). Due to equivalence of (13) and (14) with (23), itremains to be shown that (23)(25) imply (15) and (16).

If (24) holds, then, from computations in (27), we see that

11

12

21

22

+

1112

2122

+

11

12

21

22

= 0, (30)

and because of (23) we get (15).Finally, we show that (23) and (25) imply (16). From (29)

(using (23)) we get

det ( + V) =

11V12

21V22

+

V11

12

V21

22

= 0, (31)

that is, (16) holds.

1.4. Problem under Consideration. The aim of this paper is togive explicit formulas for solutions of weakly delayed systemsand to show that, after several steps, the dimension of thespace of all solutions, being initially equal to the dimension2(+ 1) of the space of initial data (3) generated by discrete

functions , is reduced to a dimension less than the initialone on an interval of the form Z

with an > 0. In other

words, we will show that the 2(+ 1)-dimensional space

of all solutions of (1) is pasted to a less-dimensional space ofsolutions on Z

. This problem is solved directly by explic-

itly computing the corresponding solutions of the Cauchyproblems with each of the cases arising being considered.Theunderlying idea for such investigation is simple. If (1) is aweakly delayed system, then the corresponding characteristicequation has only two eigenvalues instead of 2(

+ 1)

eigenvalues in the case of systems with nonweak delays. Thisexplains why the dimension of the space of solutions becomesless than the initial one. The final results (Theorems 1013)provide the dimension of the space of solutions. Our resultsgeneralize the results in [1, 2], where system (1) with = 1and = 2 was analyzed.

1.5. Auxiliary Formula. For the readers convenience, werecall one explicit formula (see, e.g., [3]) for the solutions oflinear scalar discrete nondelayed equations used in this paper.We consider initial-value problem for the first order lineardiscrete nonhomogeneous equation

( + 1) = () + () , (0) = 0, Z

0

, (32)

Abstract and Applied Analysis 5

with C and : Z0

C. Then, it is easy to verify thatunique solution of this problem is

() = 0

0+

1

=0

1

() , Z

0+1. (33)

Throughout the paper, we adopt the customary notation forthe sum:

=+F() = 0, where is an integer, is a positive

integer, and F denotes the function considered indepen-dently of whether it is defined for indicated arguments or not.

Note that the formula (33) is used many times in recentliterature to analyze asymptotic properties of solutions ofvarious classes of difference equations, including nonlinearequations. We refer, for example, to [48] and to relevantreferences therein.

2. General Solution of Weakly Delayed System

If (8) holds, then (5) and (7) have only two (and the same)roots simultaneously. In order to prove the properties of thefamily of solutions of (1) formulated in the introduction, wewill discuss each combination of roots, that is, the cases of tworeal and distinct roots, a pair of complex conjugate roots, and,finally, a double real root.

Although computations in Sections 1.2 and 1.3 wereperformed under assumption that = 0, results of this partremain valid also if one or both roots of characteristicequation (7) are zero.

2.1. Jordan Forms of theMatrix and Corresponding Solutionsof Problem (1) and (3). It is known that, for every matrix, there exists a nonsingular matrix transforming it to thecorresponding Jordan matrix form . This means that

= 1

, (34)

where has the following four possible forms (denotedbelow as

1, 2, 3, 4), depending on the roots of the

characteristic equation (7), that is, on the roots of

2

(11+ 22) + (

1122 1221) = 0. (35)

If (35) has two real distinct roots 1, 2, then

1= (

10

0 2

) , (36)

if the roots are complex conjugate, that is, 1,2= with

= 0, then

2= (

) (37)

and, finally, in the case of one double real root 1,2= , we

have either

3= (

0

0 ) (38)

or

4= (

1

0 ) . (39)

The transformation () = 1() transforms (1) into asystem

( + 1) = () +

=1

( ) , Z

0(40)

with = 1, = (), = 1, . . . , , and , = 1, 2.

Together with (40), we consider an initial problem

() =

() , (41)

Z0

with : Z0

R2 where () = 1() isthe initial function corresponding to the initial function in(3).

Next, we consider all four possible cases (36)(39) sepa-rately.

We define

1() := (0,

1())

, 2() := (

2() , 0)

,

Z0

.

(42)

Assuming that (1) is a weakly delayed system, by Lemma 2,the system (40) is weakly delayed system again.

2.1.1. Case (36) of Two Real Distinct Roots. In this case, wehave =

1and

1= diag(

1,

2). The necessary and

sufficient conditions (13)(16) for (40) turn into

11+

22= 0, (43)

11

12

21

22

=

11

22

12

21= 0, (44)

10

21

22

+

11

12

0 2

= 1

22+ 2

11= 0, (45)

11

12

V21

V22

+

V11

V12

21

22

= 0. (46)

Since 1= 2, (43) and (45) yield

11=

22= 0, then, from

(44), we get 12

21= 0, so that either

21= 0 or

12= 0. In

view of assumptions =, = 1, 2, . . . , , we conclude thatonly the following cases I, II are possible:

(I) 11=

22=

21= 0, 12

= 0, = 1, 2, . . . , ,

(II) 11=

22=

12= 0, 21

= 0, = 1, 2, . . . , .

In Theorem 5 both cases I, II are analyzed.

Theorem 5. Let (1) be a weakly delayed system and (35) hastwo real distinct roots

1, 2. If case (I) holds, then the solution


of the initial problem (1), (3) is () = (), Z

, where() has the form

() =

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{

() Z0

,

1

(0) +

1

=0

1

1[

=1

122(

)]

Z1+1

1,

...

1

(0) +

1

=0

1

1[

=+1

122(

)]

+

=1

12[

=0

1

12(

)

+2(0)

1

=+1

1

1

2]

Z+1+1

+2,

= 1, 2, . . . , 1,

...

1

(0) +

=1

12[

=0

1

12(

)

+2(0)

1

=+1

1

1

2]

Z+2.

(47)

If case (II) is true, then the solution of initial problem (1), (3) is() = (), Z

, where () has the form

() =

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{

() Z0

,

1

(0) +

1

=0

1

2[

=1

211(

)]

Z1+1

1,

...

1

(0) +

1

=0

1

2[

=+1

211(

)]

+

=1

21[

=0

1

21(

)

+1(0)

1

=+1

11

2]

Z+1+1

+2,

= 1, 2, . . . , 1,

...

1

(0) +

=1

21[

=0

1

21(

)

+1(0)

1

=+1

11

2]

Z+2.

(48)

Proof. If case (I) is true, then the transformed system (40)takes the form

1( + 1) =

11() +

=1

122(

) , (49)

2( + 1) =

22() ,

Z

0,

(50)

and if case (II) holds, then (40) takes the form

1( + 1) =

11() , (51)

2( + 1) =

22() +

=1

211(

) ,

Z

0.

(52)

We investigate only the initial problem (49), (50), (41) sincethe initial problem (51), (52), (41) can be examined in a similarway.

From (50), (41), we get

2() = {

2() if Z0

,

2

2(0) if Z

1,

(53)

then (49) becomes

1( + 1)

=

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{

11() +

=1

12

2(

) if Z1

0,

11() +

1

121

2

2(0)

+

=2

12

2(

) if Z2

1+1,

11() +

2

=1

12

2

2(0)

+

=3

12

2(

) if Z3

2+1,

...

11() +

=1

12

2

2(0)

+

=+1

12

2(

) if Z+1

+1,

= 3, 4, . . . , 1,

...

11() +

=1

12

2

2(0) if Z

+1.

(54)


First, we solve this equation for Z10. This means that we

consider the problem

1( + 1) =

11()

+

=1

12

2(

) , Z

1

0,

1(0) =

1(0) .

(55)

With the aid of formula (33), we get

1() =

1

1(0) +

1

=0

1

1[

=1

12

2(

)] ,

Z1+1

1.

(56)

Now we solve (54) for Z21+1

with initial data deducedfrom (56); that is, we consider the problem

1( + 1) =

11() +

1

121

2

2(0)

+

=2

12

2(

) , Z

2

1+1,

1(1+ 1) =

1+1

1

1(0) +

1

=0

1

1[

=1

12

2(

)] .

(57)

Applying formula (33) we get (for Z2+11+2)

1() =

(1+1)

11(1+ 1)

+

1

=1+1

1

1[1

121

2

2(0)

+

=2

12

2(

)]

= 11

1[1+1

1

1(0)

+

1

=0

1

1[

=1

12

2(

)]]

+

1

=1+1

1

1[1

121

2

2(0)

+

=2

12

2(

)]

=

1

1(0) +

1

=0

1

1[

=1

12

2(

)]

+

1

=1+1

1

1[1

121

2

2(0)

+

=2

12

2(

)]

=

1

1(0) +

1

=0

1

1[

=2

12

2(

)]

+ 1

12[

1

=0

1

1

2(

1)

+

2(0)

1

=1+1

1

11

2] .

(58)

Now we solve (54) for Z32+1


1( + 1) =

11() +

2

=1

12

2

2(0)

+

=3

12

2(

) , Z

3

2+1,

1(2+ 1) =

2+1

1

1(0) +

2

=0

2

1[

=2

12

2(

)]

+ 1

12[

1

=0

2

1

2(

1)

+

2(0)

2

=1+1

2

11

2] .

(59)

Applying formula (33) yields (for Z3+12+2)

1()

= (2+1)

11(2+ 1)

+

1

=2+1

1

1[

2

=1

12

2

2(0) +

=3

12

2(

)]

= 21

1[2+1

1

1(0) +

2

=0

2

1[

=2

12

2(

)]


+ 1

12[

1

=0

2

1

2(

1)

+

2(0)

2

=1+1

2

11

2]]

+

1

=2+1

1

1[

2

=1

12

2

2(0) +

=3

12

2(

)]

=

1

1(0) +

2

=0

1

1[

=2

12

2(

)]

+ 1

12[

1

=0

1

1

2(

1)

+

2(0)

2

=1+1

1

11

2]

+

1

=2+1

1

1[

2

=1

12

2

2(0)

+

=3

12

2(

)]

=

1

1(0) +

1

=0

1

1[

=3

12

2(

)]

+ 1

12[

1

=0

1

1

2(

1)

+

2(0)

1

=1+1

1

11

2]

+ 2

12[

2

=0

1

1

2(

2)

+

2(0)

1

=2+1

1

12

2] .

(60)

From (56), (58), and (60) we deduce that expected formof the solution of the initial problem for Z

1+1

withinitial data derived from the solution of previous equation for Z

1

2+1

is

1() =

1

1(0) +

1

=0

1

1[

=

12

2(

)]

+

1

=1

12[

=0

1

1

2(

)

+

2(0)

1

=+1

1

1

2]

if Z+11+2.

(61)

We solve (54) for Z+1+1


1( + 1) =

11() +

=1

12

2

2(0)

+

=+1

12

2(

) , Z

+1

+1,

1(+ 1) =

+1

1

1(0) +

=0

1[

=

12

2(

)]

+

1

=1

12[

=0

1

2(

)

+

2(0)

=+1

1

2] .

(62)

Applying formula (33) yields (for Z+1+1+2

)

1()

= (+1)

11(+ 1)

+

1

=+1

1

1[

=1

12

2

2(0) +

=+1

12

2(

)]

= (+1)

1[+1

1

1(0) +

=0

1[

=

12

2(

)]

+

1

=1

12[

=0

1

2(

)

+

2(0)

=+1

1

2]]

+

1

=+1

1

1[

=1

12

2

2(0)

+

=+1

12

2(

)]

=

1

1(0) +

=0

1

1[

=

12

2(

)]


+

1

=1

12[

=0

1

1

2(

)

+

2(0)

=+1

1

1

2]

+

1

=+1

1

1[

=1

12

2

2(0)

+

=+1

12

2(

)]

=

1

1(0)+

=0

1

1[

12

2(

)

+

=+1

12

2(

)]

+ 1

12[

1

=0

1

1

2(

1)

+

2(0)

=1+1

1

11

2]

+ 2

12[

2

=0

1

1

2(

2)

+

2(0)

=2+1

1

12

2]

+

+ 1

12[

1

=0

1

1

2(

1)

+

2(0)

=1+1

1

11

2]

+ 1

12[

2(0)

1

=+1

1

11

2]

+ 2

12[

2(0)

1

=+1

1

12

2]

+ . . .

+

12[

2(0)

1

=+1

1

1

2]

+

1

=+1

1

1[

=+1

12

2(

)]

=

1

1(0) +

1

=0

1

1[

=+1

12

2(

)]

+

=1

12[

=0

1

1

2(

)

+

2(0)

1

=+1

1

1

2] .

(63)

In the end we solve (54) for Z+1

with initial datadeduced from (63); that is, we consider the problem

1( + 1) =

11() +

=1

12

2

2(0) , Z

+1,

1(+ 1) =

+1

1

1(0) +

=0

1

12

2(

)

+

1

=1

12[

=0

1

2(

)

+

2(0)

=+1

1

2] .

(64)

Applying formula (33) yields (for Z+2)

1() =

(+1)

11(+ 1)

+

1

=+1

1

1[

=1

12

2

2(0)]

= (+1)

1[+1

1

1(0) +

=0

1

12

2(

)

+

1

=1

12[

=0

1

2(

)

+

2(0)

=+1

1

2]]

+

1

=+1

1

1[

=1

12

2

2(0)]

=

1

1(0) +

=0

1

1

12

2(

)


+

1

=1

12[

=0

1

1

2(

)

+

2(0)

=+1

1

1

2]

+

1

=+1

1

1[

=1

12

2

2(0)]

=

1

1(0) +

=0

1

1

12

2(

)

+ 1

12[

1

=0

1

1

2(

1)

+

2(0)

=1+1

1

11

2]

+ 2

12[

2

=0

1

1

2(

2)

+

2(0)

=2+1

1

12

2]

+

+ 1

12[

1

=0

1

1

2(

1)

+

2(0)

=1+1

1

11

2]

+ 1

12[

2(0)

1

=+1

1

11

2]

+ 2

12[

2(0)

1

=+1

1

12

2]

+

+

12[

2(0)

1

=+1

1

1

2]

=

1

1(0) +

=1

12[

=0

1

1

2(

)

+

2(0)

1

=+1

1

1

2] .

(65)

Summing up all particular cases (56)(65) we have

1() =

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{

1() if Z0

,

1

1(0) +

1

=0

1

1[

=1

12

2(

)]

if Z1+11

,

1

1(0) +

1

=0

1

1[

=2

12

2(

)]

+ 1

12[

1

=0

1

1

2(

1)

+

2(0)

1

=1+1

1

11

2]

if Z2+11+2,

1

1(0) +

1

=0

1

1[

=3

12

2(

)]

+1

12[

1

=0

1

1

2(

1)

+

2(0)

1

=1+1

1

11

2]

+2

12[

2

=0

1

1

2(

2)

+

2(0)

1

=2+1

1

12

2]

if Z3+12+2,

...

1

1(0) +

1

=0

1

1[

=+1

12

2(

)]

+

=1

12[

=0

1

1

2(

)

+

2(0)

1

=+1

1

1

2]

if Z+1+1+2,

...

1

1(0)

+

=1

12[

=0

1

1

2(

)

+

2(0)

1

=+1

1

1

2]

if Z+2.

(66)

Now, taking into account (42), formula (47) is a consequenceof (53) and (66). Formula (48) can be proved in a similar way.

Finally, we note that both formulas (47), (48) remain validfor 12=

21= 0. In this case, the transformed system


(1) reduces to a system without delays. This possibility isexcluded by condition (2).

2.1.2. Case (37) of Two Complex Conjugate Roots. The neces-sary and sufficient conditions (13)(16) take the forms (43),(44), (46), and

21

22

+

11

12

= (

11+

22) + (

12

21) = 0,

(67)

where , V = 1, 2, . . . , and V > .The system of conditions (43), (44), and (67) gives

12=

21, (11)2

= (

12)2 and admits only one possibility; namely,

11=

22=

12=

21= 0. (68)

Consequently, = , = .The initial problem (1), (3) reduces to a problem without

delay

( + 1) = () ,

() = () , Z0

(69)

and, obviously,

() = {

() if Z0

,

(0) if Z1.

(70)

From this discussion, the next theorem follows.

Theorem 6. There exists no weakly delayed system (1) if hasthe form (37).

Finally, we note that the assumption (2) alone excludesthis case.

2.1.3. Case (38) of Double Real Root. In this case we have =

3and

3= diag(, ). For (40), the necessary and

sufficient conditions (13)(16) are reduced to (43), (44), (46),and

0

21

22

+

11

12

0

= (

11+

22) = 0, (71)

where = 1, 2, . . . , .From (43), (44), and (71), we get

12

21= (

11)

2

. Fromthe condition (46) we get

11V22

12V21+

22V11

21V12= 0, (72)

where , V = 1, 2, . . . , and V > . Multiplying (72) by 12V12,

we have

11V22

12V12 (

12)

2

V21V12

+

22V11

12V12

21

12(V12)2

= 0.

(73)

Substituting 12

21= (

11)

2

, V12V21= (

V11)2 into (73) and

using (43) we obtain

11V11

12V12+ (

12)

2

(V11)2

11V11

12V12+ (

11)

2

(V12)2

= 0.

(74)

The equation (74) can be written as

(

12V11 V12

11)

2

= 0,

12V11= V12

11.

(75)

Nowwewill analyse the two possible cases: 12

21= 0 and

12

21= 0.

For the case 12

21= 0, we have from (43), (44) that

11=

22= 0 and

12= 0 or

21= 0. For

12= 0 and

21= 0,

condition (46) gives V12= 0, where , V = 1, 2, . . . , and V > .

Then, from (43), (44) for = V, we get V11= V22= 0 and

V21

= 0.For 21= 0 and

12= 0, condition (46) gives V

21= 0,

where , V = 1, 2, . . . , and V > , then, from (43), (44) for = V, we get V

11= V22= 0 and V

12= 0.

Now we discuss the case 12

21= 0. From conditions (43),

(44), we have 12

21= (

11)

2

and 11

22= 0. This yields

11= 0, 22

= 0 and, from (75), we have V11

= 0, V12

= 0. Byconditions (43), (44) for V = , we get V

22= 0, V21

= 0.From the assumptions =, we conclude that only the

following cases ((I), (II), (III)) are possible:

(I) 11=

22=

21= 0, 12

= 0,

(II) 11=

22=

12= 0, 21

= 0,

(III) 12

21= 0,

where = 1, 2, . . . , .

2.1.4. Case 12

21= 0

Theorem 7. Let (1) be a weakly delayed system, (35) has atwofold root

1,2= ,

12

21= 0 and the matrix has the

form (38). Then the solution of the initial problem (1), (3) is() = (), Z

, where in case 21= 0, () has the

form


() =

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{

() Z0

,

3

(0) +

1

=0

1

[

=1

122(

)]

Z1+1

1,

...

3

(0) +

1

=0

1

[

=+1

122(

)]

+

=1

12[

=0

1

2(

)

+ ( 1 ) 1

2(0) ]

Z+1+1

+2,

= 1, 2, . . . , 1,

...

3

(0)

+

=1

12[

=0

1

2(

)

+ ( 1 ) 1

2(0) ]

Z+2.

(76)

If 12= 0 is true then the solution of initial problem (1), (3) is

() = (), Z

, where () has the form

() =

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{

() Z0

,

3

(0) +

1

=0

1

[

=1

211(

)]

Z1+1

1,

...

3

(0) +

1

=0

1

[

=+1

211(

)]

+

=1

21[

=0

1

1(

)

+ ( 1 ) 1

1(0) ]

Z+1+1

+2,

= 1, 2, . . . , 1,

...

3

(0)

+

=1

21[

=0

1

1(

)

+ ( 1 ) 1

1(0) ]

Z+2.

(77)

Proof. Case (I) means that 12

= 0. Then (40) turns into thesystem

1( + 1) =

1() +

=1

122(

) , Z

0

2( + 1) =

2() ,

(78)

and, if 21

= 0, (40) turns into the system

1( + 1) =

1() ,

2( + 1) =

2() +

=1

211(

) , Z

0.

(79)

System (78) can be solved in much the same way as thesystems (49), (50) if we put

1= 2= , and the discussion

of the system (79) goes along the same lines as that of thesystems (51), (52) with

1= 2= . Formulas (76) and (77)

are consequences of (47), (48).

2.1.5. Case 12

21= 0. For Z0

, we define

() := (

11[

1() +

1

12

1

11

2()] ,

(

11)

2

12

[

1() +

1

12

1

11

2()])

.

(80)

Theorem 8. Let system (1) be a weakly delayed system, (35)admits two repeated roots

1,2= ,

12

21= 0 and the matrix

3has the form (38). Then the solution of the initial problem


(1), (3) is given by () = (), Z

, where () has theform

() =

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{

() Z0

,

3

(0) +

1

=0

1

[

=1

(

)]

Z1+1

1,

...

3

(0) +

1

=0

1

[

=+1

(

)]

+

=1

[

=0

1

(

)

+ ( 1 ) 1

(0) ]

Z+1+1

+2,

= 1, 2, . . . , 1,

...

3

(0)

+

=1

[

=0

1

(

)

+ ( 1 ) 1

(0) ]

Z+2.

(81)

Proof. In this case, all the entries of are nonzero and, from(43), (44), and (71), we get

= (

11

12

(

11)

2

12

11

), (82)

where = 1, 2, . . . , , then, the system (40) reduces to

1( + 1) =

1() +

=1

[

111(

) +

122(

)] ,

(83)

2( + 1)

= 2()

=1

[

[

(

11)

2

12

1(

) +

112(

)]

]

,

(84)

where Z0. It is easy to see (multiplying (84) by 1

12/1

11

and summing both equations) that

1( + 1) +

1

12

1

11

2( + 1) = [

1() +

1

12

1

11

2()] ,

Z

0.

(85)

Equation (85) is a homogeneous equation with respect to theunknown expression

1() + (

1

12

1

11

)2() , (86)

then, using (33), we obtain

1() +

1

12

1

11

2()

=

{{{{

{{{{

{

1() +

1

12

1

11

2() if Z0

,

[

1(0) +

1

12

1

11

2(0)] if Z

1.

(87)

With the aid of (87), we rewrite the systems (83), (84) asfollows:

1( + 1) =

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{

1() +

=1

11[

1(

)

+

1

12

1

11

2(

)]

if Z10,

1() +

1

111[

1(0) +

1

12

1

11

2(0)]

+

=2

11[

1(

)+

1

12

1

11

2(

)]

if Z21+1,

1() +

2

=1

11

[

1(0)+

1

12

1

11

2(0)]

+

=3

11[

1(

)+

1

12

1

11

2(

)]

if Z32+1,

...

1() +

=1

11

[

1(0) +

1

12

1

11

2(0)]

+

=+1

11[

1(

)+

1

12

1

11

2(

)]

if Z+1+1,

= 3, 4, . . . , 1,

...

1() +

=1

11

[

1(0) +

1

12

1

11

2(0)]

if Z+1,


2( + 1) =

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

{

2()

=1

(

11)

2

12

[

1(

)

+

1

12

1

11

2(

)]

if Z10,

2()

(1

11)

2

1

12

1

[

1(0) +

1

12

1

11

2(0)]

=2

(

11)

2

12

[

1(

1)

+

1

12

1

11

2(

1)]

if Z21+1,

2()

2

=1

(

11)

2

12

[

1(0) +

1

12

1

11

2(0)]

=3

(

11)

2

12

[

1(

)

+

1

12

1

11

2(

)]

if Z32+1,

...

2()

=1

(

11)

2

12

[

1(0) +

1

12

1

11

2(0)]

=+1

(

11)

2

12

[

1(

)

+

1

12

1

11

2(

)]

if Z+1+1,

= 3, 4, . . . , 1,

...

2()

=1

(

11)

2

12

[

1(0) +

1

12

1

11

2(0)]

if Z+1.

(88)

First, we solve this system for Z10

and consider theproblems

1( + 1) =

1()

+

=1

11[

1(

) +

1

12

1

11

2(

)]

if Z10,

1(0) =

1(0) ,

2( + 1) =

2()

=1

(

11)

2

12

[

1(

) +

1

12

1

11

2(

)]

if Z10,

2(0) =

2(0) .

(89)

With the aid of formula (33), we get

1()

=

1(0) +

1

=0

1

(

=1

11[

1(

) +

1

12

1

11

2(

)]) ,

Z1+1

1,

(90)

2()

=

2(0)

1

=0

1

(

=1

(

11)

2

12

[

1(

) +

1

12

1

11

2(

)]) ,

Z1+1

1.

(91)

Now we solve system (88) for Z21+1; that is, we

consider the problem (with initial data deduced from (90),(91))

1( + 1)

= 1() +

1

111

[

1(0) +

1

12

1

11

2(0)]

+

=2

11[

1(

) +

1

12

1

11

2(

)]

if Z21+1,

1(1+ 1)

= 1+1

1(0) +

1

=0

1

(

=1

11[

1(

) +

1

12

1

11

2(

)]) ,


2( + 1)

= 2()

(1

11)

2

1

12

1

[

1(0) +

1

12

1

11

2(0)]

=2

(

11)

2

12

[

1(

1) +

1

12

1

11

2(

1)]

if Z21+1,

2(1+ 1)

= 1+1

2(0)

1

=0

1

(

=1

(

11)

2

12

[

1(

) +

1

12

1

11

2(

)]) .

(92)

Formula (33) yields (for Z2+11+2)

1()

= (1+1)

1(1+ 1) +

1

=1+1

1

(1

111

[

1(0) +

1

12

1

11

2(0)]

+

=2

11[

1(

)

+

1

12

1

11

2(

)])

= 11

[1+1

1(0) +

1

=0

1

(

=1

11[

1(

)

+

1

12

1

11

2(

)])]

+

1

=1+1

1

(1

111

[

1(0) +

1

12

1

11

2(0)]

+

=2

11[

1(

)

+

1

12

1

11

2(

)])

=

1(0) +

1

=0

1

(

=1

11[

1(

)

+

1

12

1

11

2(

)])

+

1

=1+1

1

(1

111

[

1(0) +

1

12

1

11

2(0)]

+

=2

11[

1(

)

+

1

12

1

11

2(

)])

=

1(0) +

1

=0

1

(

=2

11[

1(

)

+

1

12

1

11

2(

)])

+

1

=0

1

1

11[

1(

1) +

1

12

1

11

2(

1)]

+ ( 1 1) 1

111

1

[

1(0) +

1

12

1

11

2(0)]

=

1(0) +

1

=0

1

(

=2

11[

1(

)

+

1

12

1

11

2(

)])

+ 1

11(

1

=0

1

[

1(

1) +

1

12

1

11

2(

1)]

+ ( 1 1) 1

1

[

1(0)

+

1

12

1

11

2(0)]) ,

(93)

2()

= (1+1)

2(1+ 1)

1

=1+1

1

(

(1

11)

2

1

12

1

[

1(0) +

1

12

1

11

2(0)]

+

=2

(

11)

2

12

[

1(

)


+

1

12

1

11

2(

)])

= 11[

[

1+1

2(0)

1

=0

1

(

=1

(

11)

2

12

[

1(

)

+

1

12

1

11

2(

)])

]

]

1

=1+1

1

(

(1

11)

2

1

12

1

[

1(0) +

1

12

1

11

2(0)]

+

=2

(

11)

2

12

[

1(

)

+

1

12

1

11

2(

)])

=

2(0)

1

=0

1

(

=1

(

11)

2

12

[

1(

)

+

1

12

1

11

2(

)])

1

=1+1

1

(

(1

11)

2

1

12

1

[

1(0) +

1

12

1

11

2(0)]

+

=2

(

11)

2

12

[

1(

)

+

1

12

1

11

2(

)])

=

2(0)

1

=0

1

(

=2

(

11)

2

12

[

1(

)

+

1

12

1

11

2(

)])

1

=0

1

(1

11)

2

1

12

[

1(

1) +

1

12

1

11

2(

1)]

( 1 1) 1

1

(1

11)

2

1

12

[

1(0) +

1

12

1

11

2(0)]

=

2(0)

1

=0

1

(

=2

(

11)

2

12

[

1(

)

+

1

12

1

11

12

2(

)])

(1

11)

2

1

12

(

1

=0

1

[

1(

1) +

1

12

1

11

2(

1)]

+ ( 1 1) 1

1

[

1(0) +

1

12

1

11

2(0)]) .

(94)

Now we solve (88) for Z32+1; that is, we consider the

problem (with initial data deduced from (93), (94))

1( + 1)

= 1() +

2

=1

11

[

1(0) +

1

12

1

11

2(0)]

+

=3

11[

1(

) +

1

12

1

11

2(

)]

if Z32+1,

1(2+ 1)

= 2+1

1(0) +

2

=0

2

(

=2

11[

1(

) +

1

12

1

11

2(

)])

+ 1

11(

1

=0

2

[

1(

1) +

1

12

1

11

2(

1)]

+ (2 1) 21

[

1(0) +

1

12

1

11

2(0)]) ,

2( + 1)

= 2()

2

=1

(

11)

2

12

[

1(0) +

1

12

1

11

2(0)]

=3

(

11)

2

12

[

1(

) +

1

12

1

11

2(

)]

if Z32+1,


2(2+ 1)

= 2+1

2(0)

2

=0

2

(

=2

(

11)

2

12

[

1(

) +

1

12

1

11

12

2(

)])

(1

11)

2

1

12

(

1

=0

2

[

1(

1) +

1

12

1

11

2(

1)]

+ (2 1) 21

[

1(0) +

1

12

1

11

2(0)]) .

(95)

Applying formula (33) yields (for Z3+12+2)

1()

= (2+1)

1(2+ 1) +

1

=2+1

1

(

2

=1

11

[

1(0) +

1

12

1

11

2(0)]

+

=3

11[

1(

) +

1

12

1

11

2(

)])

= 21

[2+1

1(0) +

2

=0

2

(

=2

11[

1(

)

+

1

12

1

11

2(

)])

+ 1

11(

1

=0

2

[

1(

1)

+

1

12

1

11

2(

1)]

+ (2 1) 21

[

1(0) +

1

12

1

11

2(0)])]

+

1

=2+1

1

(

2

=1

11

[

1(0) +

1

12

1

11

2(0)]

+

=3

11[

1(

)

+

1

12

1

11

2(

)])

=

1(0) +

2

=0

1

(

=2

11[

1(

)

+

1

12

1

11

2(

)])

+ 1

11(

1

=0

1

[

1(

1) +

1

12

1

11

2(

1)]

+ (2 1) 1

1

[

1(0) +

1

12

1

11

2(0)])

+

1

=2+1

1

(

2

=1

11

[

1(0) +

1

12

1

11

2(0)]

+

=3

11[

1(

)

+

1

12

1

11

2(

)])

=

1(0) +

1

=0

1

(

=3

11[

1(

)

+

1

12

1

11

2(

)])

+

2

=0

1

2

11[

1(

2) +

1

12

1

11

2(

2)]

+ 1

11(

1

=0

1

[

1(

1) +

1

12

1

11

2(

1)]

+ (2 1) 1

1

[

1(0) +

1

12

1

11

2(0)])

+ ( 1 2) (1

111

1

[

1(0) +

1

12

1

11

2(0)]

+ 2

111

2

[

1(0) +

1

12

1

11

2(0)])

=

1(0) +

1

=0

1

(

=3

11[

1(

)

+

1

12

1

11

2(

)])


+ 1

11(

1

=0

1

[

1(

1) +

1

12

1

11

2(

1)]

+ ( 1 1) 1

1

[

1(0) +

1

12

1

11

2(0)])

+ 2

11(

2

=0

1

[

1(

2) +

1

12

1

11

2(

2)]

+ ( 1 2) 1

2

[

1(0) +

1

12

1

11

2(0)]) ,

(96)2()

= (2+1)

2(2+ 1)

1

=2+1

1

(

2

=1

(

11)

2

12

[

1(0) +

1

12

1

11

2(0)]

+

=3

(

11)

2

12

[

1(

) +

1

12

1

11

2(

)])

= 21

[2+1

2(0)

2

=0

2

(

=2

(

11)

2

12

[

1(

)

+

1

12

1

11

2(

)])

(

11)

2

1

12

(

1

=0

2

[

1(

1)

+

1

12

1

11

2(

1)]

+ (2 1) 21

[

1(0) +

1

12

1

11

2(0)])]

1

=2+1

1

(

2

=1

(

11)

2

12

[

1(0) +

1

12

1

11

2(0)]

+

=3

(

11)

2

12

[

1(

)

+

1

12

1

11

2(

)])

=

2(0)

2

=0

1

(

=2

(

11)

2

12

[

1(

)

+

1

12

1

11

2(

)])

(1

11)

2

1

12

(

1

=0

1

[

1(

1) +

1

12

1

11

2(

1)]

+ (2 1) 1

1

[

1(0) +

1

12

1

11

2(0)])

1

=2+1

1

(

2

=1

(

11)

2

12

[

1(0) +

1

12

1

11

2(0)]

+

=3

(

11)

2

12

[

1(

)

+

1

12

1

11

2(

)])

=

2(0)

1

=0

1

(

=3

(

11)

2

12

[

1(

)

+

1

12

1

11

2(

)])

2

=0

1

(2

11)

2

2

12

[

1(

2)+

1

12

1

11

2(

2)]

(1

11)

2

1

12

(

1

=0

1

[

1(

1)

+

1

12

1

11

2(

1)]

+ (2 1) 1

1

[

1(0) +

1

12

1

11

2(0)])

( 1 2)

(

(1

11)

2

1

12

1

1

[

1(0) +

1

12

1

11

2(0)]

+

(2

11)

2

2

12

1

2

[

1(0) +

1

12

1

11

2(0)])


=

2(0)

1

=0

1

(

=3

(

11)

2

12

[

1(

)

+

1

12

1

11

2(

)])

(1

11)

2

1

12

(

1

=0

1

[

1(

1)+

1

12

1

11

2(

1)]

+ ( 1 1) 1

1

[

1(0) +

1

12

1

11

2(0)])

(2

11)

2

2

12

(

2

=0

1

[

1(

2) +

1

12

1

11

2(

2)]

+ ( 1 2) 1

2

[

1(0) +

1

12

1

11

2(0)]) .

(97)

From (93)(97) we deduce that expected form of thesolution of the initial problem for Z

1+1

with initialdata derived from the solution of previous equation for Z1

2+1

is

1()

=

1(0) +

1

=0

1

(

=

11[

1(

) +

1

12

1

11

2(

)])

+

1

=1

[

11(

=0

1

[

1(

) +

1

12

1

11

2(

)]

+ ( 1 ) 1

[

1(0) +

1

12

1

11

2(0)])]

if Z+11+2,

1()

=

2(0)

1

=0

1

(

=

(

11)

2

12

[

1(

) +

1

12

1

11

2(

)])

1

=1

[

[

(

11)

2

12

(

=0

1

[

1(

) +

1

12

1

11

2(

)]

+ ( 1 ) 1

[

1(0) +

1

12

1

11

2(0)])

]

]

if Z+11+2.

(98)

We solve (88) for Z+1+1


1( + 1)

= 1()

+

=1

11

[

1(0) +

1

12

1

11

2(0)]

+

=+1

11[

1(

) +

1

12

1

11

2(

)]

if Z+1+1,

1(+ 1)

= +1

1(0)

+

=0

(

=

11[

1(

) +

1

12

1

11

2(

)])

+

1

=1

11(

=0

[

1(

) +

1

12

1

11

2(

)]

+ ( )

[

1(0) +

1

12

1

11

2(0)]) ,

2( + 1)

= 2()

=1

(

11)

2

12

[

1(0) +

1

12

1

11

2(0)]

=+1

(

11)

2

12

[

1(

) +

1

12

1

11

2(

)]

if Z+1+1,


2(+ 1)

= +1

2(0)

=0

(

=

(

11)

2

12

[

1(

) +

1

12

1

11

2(

)])

1

=1

(

11)

2

12

(

=0

[

1(

) +

1

12

1

11

2(

)]

+ ( )

[

1(0) +

1

12

1

11

2(0)]) .

(99)

Applying formula (33) yields (for Z+1+1+2

)

1()

= (+1)

1(+ 1) +

1

=+1

1

(

=1

11

[

1(0) +

1

12

1

11

2(0)]

+

=+1

11[

1(

) +

1

12

1

11

2(

)])

= 1

[+1

1(0) +

=0

(

=

11[

1(

) +

1

12

1

11

2(

)])

+

1

=1

11(

=0

[

1(

)

+

1

12

1

11

2(

)]

+ ( )

[

1(0) +

1

12

1

11

2(0)])]

+

1

=+1

1

(

=1

11

[

1(0) +

1

12

1

11

2(0)]

+

=+1

11[

1(

)

+

1

12

1

11

2(

)])

=

1(0) +

=0

1

(

=

11[

1(

) +

1

12

1

11

2(

)])

+

1

=1

11(

=0

1

[

1(

) +

1

12

1

11

2(

)]

+ ( ) 1

[

1(0) +

1

12

1

11

2(0)])

+

1

=+1

1

(

=1

11

[

1(0) +

1

12

1

11

2(0)]

+

=+1

11[

1(

)

+

1

12

1

11

2(

) ])

=

1(0) +

1

=0

1

(

=+1

11[

1(

) +

1

12

1

11

2(

)])

+

11

=0

1

[

1(

) +

1

12

1

11

2(

)]

+ 1

11(

1

=0

1

[

1(

1) +

1

12

1

11

2(

1)]

+ ( 1) 1

1

[

1(0) +

1

12

1

11

2(0)])

+ 2

11(

2

=0

1

[

1(

2) +

1

12

1

11

2(

2)]

+ ( 2) 1

2

[

1(0) +

1

12

1

11

2(0)])

+

+ 1

11(

1

=0

1

[

1(

1)

+

1

12

1

11

2(

1)]


+ ( 1) 1

1

[

1(0) +

1

12

1

11

2(0)])

+ ( 1 )

(1

1

1

11[

1(0) +

1

12

1

11

2(0)]

+ 1

2

2

11[

1(0) +

1

12

1

11

2(0)] +

+ 1

1

1

11[

1(0) +

1

12

1

11

2(0)]

+1

11[

1(0) +

1

12

1

11

2(0)])

=

1(0) +

1

=0

1

(

=+1

11[

1(

) +

1

12

1

11

2(

)])

+ 1

11(

1

=0

1

[

1(

1) +

1

12

1

11

2(

1)]

+ ( 1 1) 1

1

[

1(0) +

1

12

1

11

2(0)])

+ 2

11(

2

=0

1

[

1(

2) +

1

12

1

11

2(

2)]

+ ( 1 2) 1

2

[

1(0) +

1

12

1

11

2(0)])

+

+ 1

11(

1

=0

1

[

1(

1)

+

1

12

1

11

2(

1)]

+ ( 1 1) 1

1

[

1(0) +

1

12

1

11

2(0)])

+

11(

=0

1

[

1(

) +

1

12

1

11

2(

)]

+ ( 1 ) 1

[

1(0) +

1

12

1

11

2(0)])

=

1(0) +

1

=0

1

(

=+1

11[

1(

) +

1

12

1

11

2(

)])

+

=1

11(

=0

1

[

1(

)

+

1

12

1

11

2(

)]

+ ( 1 ) 1

[

1(0) +

1

12

1

11

2(0)]) ,

(100)2()

= (+1)

2(+ 1)

1

=+1

1

(

=1

(

11)

2

12

[

1(0) +

1

12

1

11

2(0)]

+

=+1

(

11)

2

12

[

1(

)

+

1

12

1

11

2(

)])

= 1

[+1

2(0)

=0

(

=

(

11)

2

12

[

1(

)

+

1

12

1

11

2(

)])

1

=1

(

11)

2

12

(

=0

[

1(

)

+

1

12

1

11

2(

)]


+ ( )

[

1(0) +

1

12

1

11

2(0)])]

1

=+1

1

(

=1

(

11)

2

12

[

1(0) +

1

12

1

11

2(0)]

+

=+1

(

11)

2

12

[

1(

)

+

1

12

1

11

2(

)])

=

2(0)

=0

1

(

=

(

11)

2

12

[

1(

) +

1

12

1

11

2(

)])

1

=1

(

11)

2

12

(

=0

1

[

1(

)

+

1

12

1

11

2(

)]

+ ( ) 1

[

1(0) +

1

12

1

11

2(0)])

1

=+1

1

(

=1

(

11)

2

12

[

1(0) +

1

12

1

11

2(0)]

+

=+1

(

11)

2

12

[

1(

)

+

1

12

1

11

2(

)])

=

2(0)

1

=0

1

(

=+1

(

11)

2

12

[

1(

) +

1

12

1

11

2(

)])

(

11)2

12

=0

1

[

1(

) +

1

12

1

11

2(

)]

(1

11)

2

1

12

(

1

=0

1

[

1(

1) +

1

12

1

11

2(

1)]

+ ( 1) 1

1

[

1(0) +

1

12

1

11

2(0)])

(2

11)

2

2

12

(

2

=0

1

[

1(

2) +

1

12

1

11

2(

2)]

+ ( 2) 1

2

[

1(0) +

1

12

1

11

2(0)])

+

(1

11)

2

1

12

(

1

=0

1

[

1(

1) +

1

12

1

11

2(

1)]

+ ( 1)1

1

[

1(0) +

1

12

1

11

2(0)])

( 1 )

(1

1

(1

11)

2

1

12

[

1(0) +

1

12

1

11

2(0)]

+ 1

2

(2

11)

2

2

12

[

1(0) +

1

12

1

11

2(0)] +

+ 1

1

(1

11)

2

1

12

[

1(0) +

1

12

1

11

2(0)]

+1

(

11)2

12

[

1(0) +

1

12

1

11

2(0)])

=

2(0)

1

=0

1

(

=+1

(

11)

2

12

[

1(

) +

1

12

1

11

2(

)])

(1

11)

2

1

12

(

1

=0

1

[

1(

1) +

1

12

1

11

2(

1)]

+ ( 1 1) 1

1

[

1(0) +

1

12

1

11

2(0)])


(2

11)

2

2

12

(

2

=0

1

[

1(

2) +

1

12

1

11

2(

2)]

+ ( 1 2) 1

2

[

1(0) +

1

12

1

11

2(0)])

+

(1

11)

2

1

12

(

1

=0

1

[

1(

1)

+

1

12

1

11

2(

1)]

+ ( 1 1) 1

1

[

1(0) +

1

12

1

11

2(0)])

(

11)2

12

(

=0

1

[

1(

)

+

1

12

1

11

2(

)]

+ ( 1 ) 1

[

1(0) +

1

12

1

11

2(0)])

=

2(0)

1

=0

1

(

=+1

(

11)

2

12

[

1(

) +

1

12

1

11

2(

)])

=1

(

11)

2

12

(

=0

1

[

1(

)

+

1

12

1

11

2(

)]

+ ( 1 ) 1

[

1(0) +

1

12

1

11

2(0)]) .

(101)

In the end, we solve (88) for Z+1

with initial datadeduced from (100) and (101); that is, we consider the problem

1( + 1) =

1() +

=1

11

[

1(0) +

1

12

1

11

2(0)]

if Z+1,

1(+ 1)

= +1

1(0) +

=0

11

[

1(

) +

1

12

1

11

2(

)]

+

1

=1

11(

=0

[

1(

)

+

1

12

1

11

2(

)]

+ ( )

[

1(0) +

1

12

1

11

2(0)]) ,

2( + 1)

= 2()

=1

(

11)

2

12

[

1(0) +

1

12

1

11

2(0)] if Z

+1,

2(+ 1)

= +1

2(0)

=0

(

11)2

12

[

1(

) +

1

12

1

11

2(

)]

1

=1

(

11)

2

12

(

=0

[

1(

)

+

1

12

1

11

2(

)]

+ ( )

[

1(0) +

1

12

1

11

2(0)]) .

(102)


Applying formula (33) yields (for Z+2)

1()

= (+1)

1(+ 1) +

1

=+1

1

(

=1

11

[

1(0) +

1

12

1

11

2(0)])

= 1

[+1

1(0)

+

=0

11[

1(

)

+

1

12

1

11

2(

)]

+

1

=1

11(

=0

[

1(

)

+

1

12

1

11

2(

)]

+ ( )

[

1(0) +

1

12

1

11

2(0)])]

+

1

=+1

1

(

=1

11

[

1(0) +

1

12

1

11

2(0)])

=

1(0) +

=0

1

11[

1(

)

+

1

12

1

11

2(

)]

+

1

=1

11(

=0

1

[

1(

)

+

1

12

1

11

2(

)]

+ ( ) 1

[

1(0) +

1

12

1

11

2(0)])

+

1

=+1

1

(

=1

11

[

1(0) +

1

12

1

11

2(0)])

=

1(0) +

=0

1

11[

1(

)

+

1

12

1

11

2(

)]

+ 1

11(

1

=0

1

[

1(

1) +

1

12

1

11

2(

1)]

+ ( 1) 1

1

[

1(0) +

1

12

1

11

2(0)])

+ 2

11(

2

=0

1

[

1(

2) +

1

12

1

11

2(

2)]

+ ( 2) 1

2

[

1(0) +

1

12

1

11

2(0)])

+

+ 1

11(

1

=0

1

[

1(

1)

+

1

12

1

11

2(

1)]

+ ( 1) 1

1

[

1(0) +

1

12

1

11

2(0)])

+ ( 1 )

(1

1

1

11[

1(0) +

1

12

1

11

2(0)]

+ 1

2

2

11[

1(0) +

1

12

1

11

2(0)] +

+ 1

1

1

11[

1(0) +

1

12

1

11

2(0)]

+1

11[

1(0) +

1

12

1

11

2(0)])


=

1(0) +

1

11(

1

=0

1

[

1(

1)

+

1

12

1

11

2(

1)]

+ ( 1 1) 1

1

[

1(0) +

1

12

1

11

2(0)])

+ 2

11(

2

=0

1

[

1(

2) +

1

12

1

11

2(

2)]

+ ( 1 2) 1

2

[

1(0) +

1

12

1

11

2(0)])

+

+ 1

11(

1

=0

1

[

1(

1)

+

1

12

1

11

2(

1)]

+ ( 1 1) 1

1

[

1(0) +

1

12

1

11

2(0)])

+

11(

=0

1

[

1(

) +

1

12

1

11

2(

)]

+ ( 1 ) 1

[

1(0) +

1

12

1

11

2(0)])

=

1(0)

+

=1

11(

=0

1

[

1(

) +

1

12

1

11

2(

)]

+ ( 1 ) 1

[

1(0) +

1

12

1

11

2(0)]) ,

(103)

2()

= (+1)

2(+ 1)

1

=+1

1

(

=1

(

11)

2

12

[

1(0) +

1

12

1

11

2(0)])

= 1

[+1

2(0)

=0

(

11)2

12

[

1(

) +

1

12

1

11

2(

)]

1

=1

(

11)

2

12

(

=0

[

1(

)

+

1

12

1

11

2(

)]

+ ( )

[

1(0) +

1

12

1

11

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