Difference Equations in Normed Spaces

Many problems for partial difference and integro-difference equations. This book is devoted to linear and nonlinear difference equations in a normed space. Our aim is to initiate systematic investigations in the field of global equations. 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 with systematically with difference equations in a normed space. . It is also possible to develop the stability of the difference in equations.
Note that even for the long term, despite the fact that it has long history. It is still one of the most burning problems,
but many general results are available for ordinary difference equations
(for example, stability by linear approximation) may have been proved for abstract difference equations.

The main methodology presented in this publication is based on a recent standard
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 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 have very briefly notices. Banach and ordered spaces. In addition, stability concepts are presented and Liapunov's functions are introduced. In Chapter 2 in the review of 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 appearing in applications. In Chapter 5, some perturbation results for linear operators. Hilbert space are presented. Chapters to derive bounds for the spectral radiuses. Chapters 6-14 are devoted to asymptotic and exponential stabilities, as well as the boundaries of linear and nonlinear differences equations. In Chapter 6 in investigative linear equation with a bounded constant operator in a Banach space. Chapter 7 is concerned with the Liapunov type operator equation. Chapter 8 deals with estimates for the spectral radios 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 operator. In addition, we use the "freezing" method and multiplicative representations of solutions 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 in the study of non-linear equations in a banal space, as well as the global orbital stability of solutions. Chapters 17 and 18 deal with linear and nonlinear Volterra discrete equations in a Banach space. An important role in this chapter is played by operator pencils. Chapter 19 deals with the class of the Stieltjes differential equations.
These equations generalize difference and differential equations. We apply for estimates of values and features of the multiplicative classes. We also show the existence and unification of solutions. Chapter 20 provides some results regarding the Volterra - Stieltjes equations. The Volterra - Stieltjes equations include Volterra difference and Volterra integral equations. Volterra - Stieltjes equation. Chapter 21 is devoted to difference equations with continuous time. In Chapter 22, we recommend the conditions for difference in the field 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 framework of ordinary and partial difference equations
- Voltra discrete equations
- Contains an approach based on the norms of operator functions

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
Notebook
References
List of Main Symbols
index