Necessary and sufficient conditions for oscillations of linear delay partial difference equations
type: Item
instance of: scholarly article
title: Necessary and sufficient conditions for oscillations of linear delay partial difference equations
zbMATH Open document ID: 0965.39003
DOI: 10.1155/S1026022697000265
author: B. G. Zhang
author: Shutang Liu
published in: Discrete Dynamics in Nature and Society
publication date: 2001-07-23T00:00:00Z
full work available at URL: https://eudml.org/doc/121115
review text: The authors present oscillation results for the linear partial difference equation \[ A_{m,n}=\sum_{i=1}^up_iA_{m-k_i,n-l_i}+ \sum_{j=1}^vq_jA_{m+\tau_j,n+\sigma_j}, \tag{1} \] where \(p_i, q_j\) are \(r\times r\) matrices, \(A_{m,n}=(a_{m,n}^1,a_{m,n}^2,\dots ,a_{m,n}^r)^T\), \(k_i,l_i,\tau_j,\sigma_j\in\mathbb N_{0}\), \(i=1,2,\dots u\), \(j=1,2,\dots v\) and \(u,v\) are positive integers. The main result yields a sufficient and necessary condition for all proper solutions of (1) to be componentwise oscillatory. This condition is that the characteristic equation \[ \text{ det} \Big (\sum_{i=1}^up_i\lambda^{-k_i}\mu^{-l_i}-I+ \sum_{j=1}^vq_j\lambda^{\tau_j}\mu^{\sigma_j}\Big)=0 \] admits no positive roots. Considering the scalar linear difference equation \[ a_{m,n}=\sum_{i=1}^up_ia_{m-k_i,n-l_i}+ \sum_{j=1}^vq_ja_{m+\tau_j, n+\sigma_j} , \tag{2} \] the corresponding characteristic equation becomes \[ 1=\sum_{i=1}^up_i\lambda^{-k_i}\mu^{-l_i}+ \sum_{j=1}^vq_j\lambda^{\tau_j}\mu^{\sigma_j} . \tag{3} \] Then the main result applied to some special equations (2) enables to formulate oscillation criteria in terms of the coefficients of (3) and thus generalizes some previous results of the authors.
reviewed by: Jan Čermák
Mathematics Subject Classification ID: 39A11
zbMATH DE Number: 1337766
zbMATH Keywords: linear delay partial difference equations
zbMATH Keywords: oscillation
zbMATH Keywords: characteristic equation
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1337766