The asymptotic stability of difference equations
type: Item
instance of: scholarly article
title: The asymptotic stability of difference equations
zbMATH Open document ID: 0964.39013
DOI: 10.1016/S0893-9659(99)00136-6
published in: Applied Mathematics Letters
publication date: 2001-07-11T00:00:00Z
review text: The paper discusses the stability properties of the linear delay difference equation \[ x_{n+1}-x_n=-a_nx_{n-k} , \tag{1} \] where \(a_n\geq 0\) and \(k\geq 1\). As the main result it is proved that if \[ \sum_{n=0}^{\infty}a_n=\infty \tag{"(C1)"} \] and \[ \underset{n\to\infty}\limsup \sum_{j=n-2k}^{n-1}a_j<\frac{7}{4} , \tag{"(C2)"} \] then every solution \(\{x_n\}\) of (1) tends to zero as \(n\to\infty\). Moreover, the presented example demonstrates that the constant in (C2) is sharp enough. The proof technique is also generalized to the case of nonlinear difference equation \[ x_{n+1}-x_n=f_n(x_{n-k}), \tag{2} \] where \(f_n(x)/x\leq 0\) for \(x\neq 0\) and \(f_n(0)=0\). The sufficient conditions for the asymptotic stability of (2) are simple modifications of (C1), (C2).
reviewed by: Jan Čermák
Mathematics Subject Classification ID: 39A11
zbMATH DE Number: 1429481
zbMATH Keywords: linear delay difference equation
zbMATH Keywords: nonlinear difference equation
zbMATH Keywords: asymptotic stability
author: István Kovácsvölgyi
MaRDI profile type: Publication
cites work: Stability of solutions of linear nonautonomous difference equations
cites work: Global stability of a linear nonautonomous delay difference equation1
cites work: Asymptotic formulae for the solutions of a linear delay difference equation
cites work: A stability criterion for a neutral difference equation with delay
cites work: Asymptotic behavior of solutions of first order nonlinear delay difference equations
cites work: Q4003305
cites work: Q3137926
description: scientific article; zbMATH DE number 1429481