Asymptotic behavior of solutions of nonautonomous difference equations
type: Item
instance of: scholarly article
title: Asymptotic behavior of solutions of nonautonomous difference equations
zbMATH Open document ID: 0821.39002
author: Takeshi Taniguchi
published in: Journal of Mathematical Analysis and Applications
publication date: 1995-09-27T00:00:00Z
review text: Let \(B_ r\) denote the ball of radius \(r\) in \(\mathbb{R}^ d\). The author proves that if there exist functions \(V(n,x) : \mathbb{N}_ 0 \times B_ r \to \mathbb{R}\), \(W(n,x) : \mathbb{N}_ 0 \times B_ r \to \mathbb{R}_ +\), \(\alpha : \mathbb{R}_ + \to \mathbb{R}_ +\), \(p,q : \mathbb{N}_ 0 \to \mathbb{R}_ +\) \((W(n,x), V(n,x), \alpha (x)\) are continuous in \(x, \alpha\)-nondecreasing, \(V(n,x)\) bounded from below by some real constant, \(\sum^ \infty_{k = 1} q(k) < \infty\), \(\sum^ \infty_{k = 1} p(m_ k) = \infty\) for any sequence \(\{m_ k\}\) of positive integers \(m_ k \to \infty)\) and such that \(V(n + 1, f(n,x)) - V(n,x) \leq - p(n) \alpha (W(n,x)) + q(n)\), \((n,x) \in \mathbb{N}_ 0 \times B_ r\), then for every solution \(\{x(n)\}\) of the difference equation \(x(n + 1) = f(n,x(n))\) there is \(W(n,x(n)) \to 0\) as \(n \to \infty\). The author gives examples of difference equations for which applying his result he deduces that the zero solution is asymptotically stable, and for which earlier results like discrete LaSalle's invariance principle or discrete Salvadori's theorem cannot be applied.
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 635928
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: nonautonomous difference equations
zbMATH Keywords: asymptotic stability
zbMATH Keywords: discrete LaSalle's invariance principle
zbMATH Keywords: discrete Salvadori's theorem
reviewed by: Jerzy Ryszard Popenda
MaRDI profile type: Publication
full work available at URL: jmaa.1994.1204
OpenAlex ID: W1989728062
DOI: 10.1006/JMAA.1994.1204
description: scientific article; zbMATH DE number 635928