Existence of nonoscillatory solutions of higher-order neutral delay difference equations with variable coefficients
type: Item
instance of: scholarly article
title: Existence of nonoscillatory solutions of higher-order neutral delay difference equations with variable coefficients
zbMATH Open document ID: 1052.39015
DOI: 10.1016/S0898-1221(03)00074-9
author: Yong Zhou
author: B. G. Zhang
publication date: 2004-08-06T00:00:00Z
review text: The authors study the higher order neutral delay difference equation of the form \[ \Delta^m(x_n+cx_{n-k})+p_nx_{n-r}-q_nx_{n-l}=0, \] where \(c\in{\mathbb R}, m\geq 1, k\geq 1, r,l\geq 0, n\geq n_0\geq 0\) are integers, \(\{p_n\}_{n=n_0}^\infty\) and \(\{q_n\}_{n=n_0}^\infty\) are sequences of nonnegative real numbers, and \(\Delta x_n=x_{n+1}-x_n\). A solution \(\{x_n\}\) of the above equation is said to be oscillatory if the numbers \(x_n\) are neither eventually all positive nor eventually all negative. Otherwise, the solution is called nonoscillatory. The authors prove the following Theorem. Suppose \(m\geq 1\) is an odd (resp. \(m\geq 2\) is an even) integer, \[ \sum_{i=n_0}^\infty i^{m-1}p_i<\infty, \qquad \sum_{i=n_0}^\infty i^{m-1}q_i<\infty, \] and one of the following two conditions is fulfilled: (i) \(c\geq 0\) and there exist a constant \(\alpha>\max\{1,c\}/(1-c)\) and a number \(N\geq n_0\) such that \(p_n\geq\alpha q_n\) (resp. \(q_n\geq\alpha p_n\)) for all \(n\geq N\); (ii) \(c<0\) and there exist a constant \(\alpha>1\) and a number \(N\geq n_0\) such that \(p_n\geq\alpha q_n\) (resp. \(q_n\geq\alpha p_n\)) for all \(n\geq N\). Then the above equation has a bounded nonoscillatory solution.
reviewed by: Oleksiy Karlovych
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 2082209
zbMATH Keywords: nonoscillatory solutions
zbMATH Keywords: neutral delay difference equation
zbMATH Keywords: bounded solution
MaRDI profile type: Publication
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published in: Computers & Mathematics with Applications
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description: scientific article; zbMATH DE number 2082209