Inequalities for finite difference equations. (English) Zbl 0987.39001

Pachpatte, B. G.

Pure and Applied Mathematics, Marcel Dekker. 247. New York, NY: Marcel Dekker. viii, 514 p. (2002).

The book offers a detailed account of basic difference inequalities, which provide explicit bounds on unknown solutions of difference and sum-difference equations, respectively. As a rule, these inequalities can be considered as (partly rather complicated) generalizations of the discrete version of Gronwall’s Lemma, but also comparison theorems are presented. The inequalities concern linear and nonlinear equations, equations of first and of higher order, one- and multidimensional equations.
Difference inequalities are useful for stability investigations in connection with Lyapunov’s second method, for the proof of continuous dependence of the solution on the equation and the initial data, and for estimates in numerical analysis. Applications are given to perturbed equations, to stochastic difference equations, and hints refer to possible applications concerning physical systems, control systems, biological models and economics.
The book contains material from 220 references, among them 83 papers of the author, and several results unpublished before.

Reviewer: Lothar Berg (Rostock)

MSC:

39A05	General theory of difference equations
39-02	Research exposition (monographs, survey articles) pertaining to difference and functional equations
39A11	Stability of difference equations (MSC2000)
39A12	Discrete version of topics in analysis
26D15	Inequalities for sums, series and integrals

Keywords:

finite difference equations; difference inequalities; textbook; stability; sum-difference equations; Gronwall’s Lemma; comparison theorems; linear; nonlinear; Lyapunov’s second method; perturbed equations; stochastic difference equations