Oscillation of fourth-order delay dynamic equations
type: Item
instance of: scholarly article
title: Oscillation of fourth-order delay dynamic equations
zbMATH Open document ID: 1321.34121
author: Chenghui Zhang
author: Ravi P. Agarwal
author: Tongxing Li
author: Martin Bohner
published in: Science China. Mathematics
publication date: 2015-03-26T00:00:00Z
review text: The authors consider the fourth order delay dynamic equation \[ \bigg(c(t)\,\big(b(t)\,(a(t)\,x^\Delta(t))^\Delta\big)^\Delta\bigg)^\Delta+p(t)\,x(\tau(t))=0 \tag{*} \] on a time scale \(\mathbb{T}\), which is unbounded above. The coefficients \(a(t)\), \(b(t)\), \(c(t)\) are positive and rd-continuous and the delay \(\tau(t)\) is rd-continuous from \(\mathbb{T}\) to \(\mathbb{T}\) with \(\tau(t)\leq t\) and \(\tau(t)\to\infty\) as \(t\to\infty\). The authors derive several new oscillation criteria for equation (\(*\)), which improve the existing results on this subject. The paper contains interesting comparison of the current oscillation results for various nonlinear differential, difference, and dynamic equations. Four particular examples show the application of the main results in the situations, where the known criteria cannot be applied. Reviewer's remark: The reviewer believes that the assumption on the positivity of the rd-continuous coefficients \(a(t)\), \(b(t)\), \(c(t)\) should be strengthened to the condition \[ \inf_{t\in[t_0,t_1]_{\mathbb{T}}} r(t)>0 \quad\text{for all }\;t_1\in[t_0,\infty)_{\mathbb{T}}, \quad r(\cdot)\in\{a(\cdot),\,b(\cdot),\,c(\cdot)\}. \tag{**} \] Condition (\(**\)) guarantees the rd-continuity of the functions \(1/r(t)\), hence it guarantees the existence of the integrals involving \(1/r(t)\), see also [\textit{R. Hilscher} and \textit{C. C. Tisdell}, Electron. J. Differ. Equ. 2008, Paper No. 68, 21 p. (2008; Zbl 1176.34059)]. The paper will be useful for researchers in the oscillation theory of differential, difference, and dynamic equations.
reviewed by: Roman Šimon Hilscher
Mathematics Subject Classification ID: 34N05
Mathematics Subject Classification ID: 34K11
zbMATH DE Number: 6419694
zbMATH Keywords: Swift-Hohenberg equation
zbMATH Keywords: oscillation
zbMATH Keywords: delay dynamic equation
zbMATH Keywords: fourth-order
zbMATH Keywords: time scale
MaRDI profile type: Publication
full work available at URL: s11425-014-4917-9
OpenAlex ID: W2259008109
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DOI: 10.1007/S11425-014-4917-9
description: scientific article; zbMATH DE number 6419694