Difference Equations in Normed Spaces: Stability and Oscillations

Difference equations appear as natural descriptions of observed evolution phenomena because most measurements of time evolving variables are discrete. They also appear in the applications of discretization methods for differential, integral and integro-differential equations.

The application of the theory of difference equations is rapidly increasing to various fields, such as numerical analysis, control theory, finite mathematics, and computer sciences. This book is devoted to linear and nonlinear difference equations in a normed space. The main methodology presented in this book is based on a combined use of recent norm estimates for operator-valued functions with the following methods and results:

The freezing methodThe Liapunov type equationThe method of majorantsThe multiplicative representation of 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 equationsDevelops the freezing method and presents recent results on Volterra discrete equationsContains an approach based on the estimates for 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
Notes
References
List of Main Symbols
Index