Difference equations in normed spaces: Stability and oscillations

M.I. Gil’ (Eds.)0444527133, 9780444527134, 9780080469355

Many problems for partial difference and integro-difference equations can be written as difference equations in a normed space. This book is devoted to linear and nonlinear difference equations in a normed space. Our aim in this monograph is to initiate systematic investigations of the global behavior of solutions of difference equations in a normed space. Our primary concern is to study the asymptotic stability of the equilibrium solution. We are also interested in the existence of periodic and positive solutions. There are many books dealing with the theory of ordinary difference equations. However there are no books dealing systematically with difference equations in a normed space. It is our hope that this book will stimulate interest among mathematicians to develop the stability theory of abstract difference equations. Note that even for ordinary difference equations, the problem of stability analysis continues to attract the attention of many specialists despite its long history. It is still one of the most burning problems, because of the absence of its complete solution, but many general results available for ordinary difference equations (for example, stability by linear approximation) may be easily proved for abstract difference equations. The main methodology presented in this publication is based on a combined use of recent norm estimates for operator-valued functions with the following methods and results: a) the freezing method; b) the Liapunov type equation; c) the method of majorants; d) the multiplicative representation of solutions. In addition, we present stability results for abstract Volterra discrete equations. The book consists of 22 chapters and an appendix. In Chapter 1, some definitions and preliminary results are collected. They are systematically used in the next chapters. In, particular, we recall very briefly some basic notions and results of the theory of operators in Banach and ordered spaces. In addition, stability concepts are presented and Liapunov’s functions are introduced. In Chapter 2 we review various classes of linear operators and their spectral properties. As examples, infinite matrices are considered. In Chapters 3 and 4, estimates for the norms of operator-valued and matrix-valued functions are suggested. In particular, we consider Hilbert-Schmidt, Neumann-Schatten, quasi-Hermitian and quasiunitary operators. These classes contain numerous infinite matrices arising in applications. In Chapter 5, some perturbation results for linear operators in a Hilbert space are presented. These results are then used in the next chapters to derive bounds for the spectral radiuses. Chapters 6-14 are devoted to asymptotic and exponential stabilities, as well as boundedness of solutions of linear and nonlinear difference equations. In Chapter 6 we investigate the linear equation with a bounded constant operator acting in a Banach space. Chapter 7 is concerned with the Liapunov type operator equation. Chapter 8 deals with estimates for the spectral radiuses of concrete operators, in particular, for infinite matrices. These bounds enable the formulation of explicit stability conditions. In Chapters 9 and 10 we consider nonautonomous (time-variant) linear equations. An essential role in this chapter is played by the evolution operator. In addition, we use the “freezing” method and multiplicative representations of solutions to construct the majorants for linear equations. Chapters 11 and 12 are devoted to semilinear autonomous and nonautonomous equations. Chapters 13 and 14 are concerned with linear and nonlinear higher order difference equations. Chapter 15 is devoted to the input-to-state stability. In Chapter 16 we study periodic solutions of linear and nonlinear difference equations in a Banach space, as well as the global orbital stability of solutions of vector difference equations. Chapters 17 and 18 deal with linear and nonlinear Volterra discrete equations in a Banach space. An important role in these chapter is played by operator pencils. Chapter 19 deals with a class of the Stieltjes differential equations. These equations generalize difference and differential equations. We apply estimates for norms of operator valued functions and properties of the multiplicative integral to certain classes of linear and nonlinear Stieltjes differential equations to obtain solution estimates that allow us to study the stability and boundedness of solutions. We also show the existence and uniqueness of solutions as well as the continuous dependence of the solutions on the time integrator. Chapter 20 provides some results regarding the Volterra–Stieltjes equations. The Volterra–Stieltjes equations include Volterra difference and Volterra integral equations. We obtain estimates for the norms of solutions of the Volterra–Stieltjes equation. Chapter 21 is devoted to difference equations with continuous time. In Chapter 22, we suggest some conditions for the existence of nontrivial and positive steady states of difference equations, as well as bounds for the stationary solutions. – Deals systematically with difference equations in normed spaces – Considers new classes of equations that could not be studied in the frameworks of ordinary and partial difference equations – Develops the freezing method and presents recent results on Volterra discrete equations – Contains an approach based on the estimates for norms of operator functions

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Table of contents :
Content:
Preface
Pages v-x

Chapter 1 Definitions and preliminaries Original Research Article
Pages 1-19

Chapter 2 Classes of operators Original Research Article
Pages 21-32

Chapter 3 Functions of finite matrices Original Research Article
Pages 33-55

Chapter 4 Norm estimates for operator functions Original Research Article
Pages 57-74

Chapter 5 Spectrum perturbations Original Research Article
Pages 75-83

Chapter 6 Linear equations with constant operators Original Research Article
Pages 85-103

Chapter 7 Liapunov’s type equations Original Research Article
Pages 105-111

Chapter 8 Bounds for spectral radiuses Original Research Article
Pages 113-142

Chapter 9 Linear equations with variable operators Original Research Article
Pages 143-152

Chapter 10 Linear equations with slowly varying coefficients Original Research Article
Pages 153-161

Chapter 11 Nonlinear equations with autonomous linear parts Original Research Article
Pages 163-172

Chapter 12 Nonlinear equations with time-variant linear parts Original Research Article
Pages 173-186

Chapter 13 Higher order linear difference equations Original Research Article
Pages 187-199

Chapter 14 Nonlinear higher order difference equations Original Research Article
Pages 201-214

Chapter 15 Input-to-state stability Original Research Article
Pages 215-222

Chapter 16 Periodic solutions of difference equations and orbital stability Original Research Article
Pages 223-237

Chapter 17 Discrete volterra equations in banach spaces Original Research Article
Pages 239-260

Chapter 18 Convolution type volterra difference equations in euclidean spaces and their perturbations Original Research Article
Pages 261-273

Chapter 19 Stieltjes differential equations Original Research Article
Pages 275-289

Chapter 20 Volterra-stieltjes equations Original Research Article
Pages 291-297

Chapter 21 Difference equations with continuous time Original Research Article
Pages 299-306

Chapter 22 Steady states of difference equations Original Research Article
Pages 307-323

Appendix A. Functions of non-compact operators
Pages 325-339

Notes
Pages 341-345

References
Pages 347-358

List of main symbols
Page 359

Index
Pages 361-362