Difference Equations in Normed Spaces

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 Volte

Book by Gil Michael