Linearized oscillation theorems for certain nonlinear delay partial difference equations
type: Item
instance of: scholarly article
title: Linearized oscillation theorems for certain nonlinear delay partial difference equations
zbMATH Open document ID: 0907.39017
DOI: 10.1016/S0898-1221(97)00294-0
publication date: 1999-03-04T00:00:00Z
review text: Consider the difference equation \[ A_{m+1,n} +A_{m,n+1} -A_{m,n} +p_{m,n} f(A_{m-k, n-1})= 0,m,n=0,1, \dots \tag{1} \] where \(f\in C(\mathbb{R}, \mathbb{R})\), \(p_{m,n}>0\), and corresponding with it the linear equation \[ A_{m+1,n} +A_{m,n+1} -A_{m,n} +pA_{m-k,n-1} =0,\;p>0. \tag{2} \] A solution \(\{A_{i,j}\}\) to (2) is said to be proper if there exist positive constants \(M, \alpha\) and \(\beta\) such that \(| A_{m,n} |\leq M\alpha^m \beta^n\) for all large \(m\) and \(n\). Let \(\lim\inf_{m,n \to\infty} p_{m,n} =p>0\), \((f(x))/x>0\) for \(x\neq 0\) and \(\lim_{x\to 0} (f(x))/x=1\), then if every proper solution to (2) is oscillatory the same holds for (1). A similar criterion is obtained for positivity of solutions.
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 1192707
zbMATH Keywords: nonlinear delay partial difference equations
zbMATH Keywords: linearization
zbMATH Keywords: oscillatory solution
zbMATH Keywords: positive solution
author: B. G. Zhang
author: Jianshe Yu
reviewed by: Dobiesław Bobrowski
MaRDI profile type: Publication
cites work: On the Partial Difference Equations of Mathematical Physics
cites work: Q4000373
cites work: Q3732204
cites work: Q4314904
cites work: Oscillation of a class of delay partial difference equations
cites work: Q4882601
cites work: Q4862646
full work available at URL: s0898-1221(97)00294-0
OpenAlex ID: W2029798417
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 1192707