Oscillation and nonoscillation for second-order linear difference equations
type: Item
instance of: scholarly article
title: Oscillation and nonoscillation for second-order linear difference equations
zbMATH Open document ID: 0973.39007
DOI: 10.1016/S0898-1221(99)00308-9
author: Yong Zhou
author: B. G. Zhang
publication date: 2001-11-25T00:00:00Z
review text: The authors study the oscillatory properties of the equation \[ \Delta^{2}x_{n-1}+p_{n}x_{n}=0, \quad n=1,2,\ldots, \tag{1} \] where \(\Delta\) is the forward difference, \(\Delta x_{n}=x_{n+1}-x_{n}\), \(\Delta^{2}x_{n}=\Delta(\Delta x_{n})\), \(\{p_{n}\}^{\infty}_{n=1}\) is a real nonnegative sequence. Let \(\alpha_{0}=3-2\sqrt{2}\) and \(\mathbb N\) be the set of natural numbers. The following sufficient oscillatory conditions are given: if there exists \(n_{0}\in\mathbb{N}\) such that for every \(m\in\mathbb{N}\) \[ \sum_{i=2^{m}n_{0}}^{2^{m+1}n_{0}-1}p_{i}\leq \frac{\alpha_{0}}{2^{m+1}n_{0}}, \] then equation (1) is nonoscillatory; if there exists \(n_{0}\in\mathbb{N}\) and \(\alpha>\alpha_{0}\) such that for every \(m\in\mathbb{N}\) \[ \sum_{i=2^{m}n_{0}}^{2^{m+1}n_{0}-1}p_{i}\geq \frac{\alpha}{2^{m}n_{0}}, \] then equation (1) is oscillatory.
Mathematics Subject Classification ID: 39A11
zbMATH DE Number: 1428542
zbMATH Keywords: second-order linear difference equations
zbMATH Keywords: oscillation
zbMATH Keywords: nonoscillation
reviewed by: Oleg Anashkin
MaRDI profile type: Publication
cites work: Q4033029
cites work: Oscillations of delay difference equations in a critical state
cites work: Q4849090
cites work: Oscillation and nonoscillation of second order linear differential equations
cites work: On the existence of positive solutions of higher order difference equations
Wikidata QID: Q127473215
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 1428542