Existence of bounded and unbounded nonoscillatory solutions of nonlinear partial difference equations
type: Item
instance of: scholarly article
title: Existence of bounded and unbounded nonoscillatory solutions of nonlinear partial difference equations
zbMATH Open document ID: 1117.39011
author: Yong Zhou
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-06-06T00:00:00Z
review text: The author considers the higher order nonlinear partial difference equation of neutral type \[ \triangle_n^h\triangle_m^r(y(m,n)+cy(m-k,n-l))+F(m,n,y(m-\tau,n-\sigma))=0 \] where \(h,r\in \mathbb{N}(1)\), \(k,l,\tau,\sigma\in \mathbb{N}(0)\), \(c\in \mathbb{R}\) and \(F:\mathbb{N}\times \mathbb{N}\times \mathbb{R}\mapsto \mathbb{R}\) is continuous as a map between topological spaces with the discrete topology on \(R\). Existence of a nonoscillatory solution satisfying the inequalities \[ a\leq {{y(m,n)}\over{m^{r_1}n^{h_1}}}\leq b \] for some integers \(r_1\leq r\), \(h_1\leq h\) and some interval \(0<a<b\) is considered in two cases \smallskip \(1^o\) \(c\neq -1\) : the above \(r_1\), \(h_1\), \(a\), \(b\) are such that \[ \displaystyle{\sum_{m_0}^{\infty}\sum_{n_0}^{\infty}m^{r-1}n^{h-1}\sup_{w\in[a,b]} | F(m,n,w(m-r)^{r_1}(n-\sigma)^{h_1})| <\infty} \] \(2^o\) \(c=-1\) : the above \(r_1\), \(h_1\), \(a\), \(b\) are such that \[ \sum_{m_0}^{\infty}\sum_{n_0}^{\infty}m^rn^h\sup_{w\in[a,b]} | F(m,n,w(m-r)^{r_1}(n-\sigma)^{h_1})| <\infty \]
Mathematics Subject Classification ID: 39A11
zbMATH DE Number: 5161772
zbMATH Keywords: nonoscillatory solutions
zbMATH Keywords: bounded solutions
zbMATH Keywords: unbounded solutions
zbMATH Keywords: higher order nonlinear partial difference equation of neutral type
reviewed by: Vladimir Răsvan
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.10.059
OpenAlex ID: W1986551402
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DOI: 10.1016/J.JMAA.2006.10.059
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description: scientific article; zbMATH DE number 5161772