Oscillation of a class of partial difference equations with unbounded delay
type: Item
instance of: scholarly article
title: Oscillation of a class of partial difference equations with unbounded delay
zbMATH Open document ID: 0999.39008
DOI: 10.1016/S0898-1221(01)00175-4
author: Zhendong Xie
author: Shengli Xie
author: Chuanjun Tian
publication date: 2002-06-13T00:00:00Z
review text: Consider real double sequences \((a_{mn})_{mn}\), \((b_{mn})_{mn}\), \((p_{mn})_{mn}\), the set \(\mathbb{N}\) of natural numbers, the set \(\mathbb{Z}\) of integers, nondecreasing functions \(\sigma\) and \(\tau\) defined on \(\mathbb{N}\) and with values in \(\mathbb{Z}\) such that \(\sigma(n)<n\), \(\tau(n)<n\), \(n\in \mathbb{N}\), \(\lim_{n\to+\infty}\sigma(n)=\lim_{n\to+\infty}\tau(n)=+\infty\). Assume that there exist positive constants \(a\) and \(b\) such that for all large \(m\) and \(n\) we have \(a_{mn}\geq a\), \(b_{mn}\leq b\), \(p_{mn}\geq 0\). Consider the partial difference equation \[ A_{m+1,n}+a_{mn}A_{m,n+1} -b_{mn}A_{mn}+ p_{mn}A_{\sigma(m),\tau(n)}=0. \] A nontrivial double sequence \((A_{mn})_{mn}\) satisfying (1) for \(m\geq 0\), \(n\geq 0\), is called a solution of (1). A solution \((A_{mn})_{mn}\) of (1) is said to be eventually positive (or negative) if \(A_{mn}>0\) (or \(A_{mn}<0)\) for large \(m\) and \(n\). It is said to be oscillatory if it is neither eventually positive nor eventually negative, and nonoscillatory otherwise. The authors prove some oscillation criteria for the solutions of the equation (1). For example, if \[ \limsup_{m,n\to+\infty} p_{mn}b^{\sigma(m)-m-1}\left(\frac ab\right)^{n-\tau(n)} c^{n-\tau(n)}_{m-\sigma(m)+n-\tau(n)}>1 \] then every solution of (1) is oscillatory. The authors give some examples to illustrate the results obtained.
Mathematics Subject Classification ID: 39A11
zbMATH DE Number: 1755272
zbMATH Keywords: unbounded delay
zbMATH Keywords: partial difference equation
zbMATH Keywords: oscillation
reviewed by: Dan-Mircea Borş
MaRDI profile type: Publication
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full work available at URL: s0898-1221(01)00175-4
OpenAlex ID: W2077409766
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 1755272