Difference Equations in Normed Spaces

Stability and Oscillations

Michael Gil

• Publisher: Elsevier Science
• Year: 2007
• eBook ISBN: 9780080469355
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

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 method- The Liapunov type equation- The method of majorants- The 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 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

Table of contents

• Cover
• Copyright Page
• Preface
• Table of Contents
• Chapter 1 Definitions and Preliminaries
• Chapter 2 Classes of Operators
• Chapter 3 Functions of Finite Matrices
• Chapter 4 Norm Estimates for Operator Functions
• Chapter 5 Spectrum Perturbations
• Chapter 6 Linear Equations with Constant Operators
• Chapter 7 Liapunov’s Type Equations
• Chapter 8 Bounds for Spectral Radiuses
• Chapter 9 Linear Equations with Variable Operators
• Chapter 10 Linear Equations with Slowly Varying Coefficients
• Chapter 11 Nonlinear Equations with Autonomous Linear Parts
• Chapter 12 Nonlinear Equations with Time-Variant Linear Parts
• Chapter 13 Higher Order Linear Difference Equations
• Chapter 14 Nonlinear Higher Order Difference Equations
• Chapter 15 Input-to-State Stability
• Chapter 16 Periodic Solutions of Difference Equations and Orbital Stability
• Chapter 17 Discrete Volterra Equations in Banach Spaces
• Chapter 18 Convolution type Volterra Difference Equations in Euclidean Spaces and their Perturbations
• Chapter 19 Stieltjes Differential Equations
• Chapter 20 Volterra–Stieltjes Equations
• Chapter 21 Difference Equations with Continuous Time
• Chapter 22 Steady States of Difference Equations
• Appendix A. Functions of Non-Compact Operators
• Notes
• References
• List of Main Symbols
• Index