Oscillations of second-order nonlinear partial difference equations
type: Item
instance of: scholarly article
title: Oscillations of second-order nonlinear partial difference equations
zbMATH Open document ID: 1058.39008
published in: Rocky Mountain Journal of Mathematics
publication date: 2004-10-01T00:00:00Z
review text: The paper deals with the partial difference equation \[ T(\Delta_1,\Delta_2)[c_{mn}T(\Delta_1,\Delta_2)(y_{mn})]+ p_{mn}(y_{m+1,n}+y_{m,n+1})^\nu=0, \] where \(T(\Delta_1,\Delta_2)=\Delta_1+\Delta_2+I\), \(\Delta_1y_{mn}=y_{m+1,n}-y_{mn}\), \(\Delta_2y_{mn}=y_{m,n+1}-y_{mn}\), \(Iy_{mn}=y_{mn}\); \(c_{mn}>0\), \(p_{mn}>0\), \(m\geq0\), \(n\geq0\); \(\nu>0\) is a quotient of odd positive integers. Denote \(r_{mn}=\sum_{i=m}^\infty\sum_{j=n}^\infty1/c_{ij}\) and suppose that \(r_{00}<\infty\). The authors establish the following result: All solutions of the above equation are oscillatory if \(\nu\neq1\) and \(\sum_{i=0}^\infty\sum_{j=0}^\infty(1/2)^{i+j}p_{ij} r_{i+1,j}^\theta=\infty\), where \(\theta=\max\{1,\nu\}\geq1\).
Mathematics Subject Classification ID: 39A11
zbMATH DE Number: 2104658
zbMATH Keywords: oscillation
zbMATH Keywords: second-order partial difference equation
author: Shutang Liu
author: Guanrong Chen
reviewed by: Oleg Anashkin
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181069875
OpenAlex ID: W2095271260
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DOI: 10.1216/RMJM/1181069875
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description: scientific article; zbMATH DE number 2104658