Description 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.
Audience
The book is intended not only for specialists in stability theory, but for everyone interested in various applications who has had at least a first year graduate level course in analysis.
Contents Preface
1. Definitions and Preliminaries
2. Classes of Operators
3. Functions of Finite Matrices
4. Norm Estimates for Operator Functions
5. Spectrum Perturbations
6. Linear Equations with Constant Operators
7. Liapunov's Type Equations
8. Bounds for Spectral Radiuses
9. Linear Equations with Variable Operators
10. Linear Equations with Slowly Varying Coefficients
11. Nonlinear Equations with Autonomous
Linear Parts
12. Nonlinear Equations with Time-Variant Linear Parts
13. Higher Order Linear Difference Equations
14. Nonlinear Higher
Order Difference Equations
15. Input-to-State Stability
16. Periodic Solutions of Difference Equations and Orbital Stability
17. Discrete
Volterra Equations in Banach Spaces
18. Convolution type Volterra Difference Equations in Euclidean Spaces and their Perturbations
19
Stieltjes Differential Equations
20 Volterra-Stieltjes Equations
21. Difference Equations with Continuous Time
22. Steady States of Difference
Equations
Appendix A
Notes
References
List of Main Symbols
Index
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