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

Key Features

  • 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


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.

Table of 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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© 2007
Elsevier Science
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