Difference Equations in Normed Spaces

Stability and Oscillations


  • Michael Gil

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

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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.


Book information

  • Published: January 2007
  • Imprint: ELSEVIER
  • ISBN: 978-0-444-52713-4

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 Operators10. Linear Equations with Slowly Varying Coefficients11. Nonlinear Equations with Autonomous Linear Parts12. Nonlinear Equations with Time-Variant Linear Parts13. Higher Order Linear Difference Equations14. Nonlinear Higher Order Difference Equations15. Input-to-State Stability16. Periodic Solutions of Difference Equations and Orbital Stability17. Discrete Volterra Equations in Banach Spaces18. Convolution type Volterra Difference Equations in Euclidean Spaces and their Perturbations19 Stieltjes Differential Equations20 Volterra-Stieltjes Equations21. Difference Equations with Continuous Time22. Steady States of Difference EquationsAppendix ANotesReferencesList of Main SymbolsIndex