Oscillation of nonlinear dynamic equations on time scales.
type: Item
instance of: scholarly article
title: Oscillation of nonlinear dynamic equations on time scales.
zbMATH Open document ID: 1045.39012
DOI: 10.1016/S0096-3003(02)00829-9
published in: Applied Mathematics and Computation
publication date: 2003-12-03T00:00:00Z
review text: The author investigates oscillatory properties of the nonlinear second order dynamic equation on a time scale \[ (p(t)x^\Delta(t))^\Delta+q(t)f(x(\sigma(t)))=0, \tag{*} \] where \(p,q,f\) are rd-continuous positive functions, \(\sigma\) is the time scale right jump operator, and the nonlinearity \(f\) satisfies \(xf(x)>0\) and \(f(x)>Kx\) for \(x\neq 0\). Under the last assumption, the nonlinear equation (*) is a Sturmian majorant, in a certain sense, of the linear equation \[ (p(t)x^\Delta(t))^\Delta+Kq(t)x(\sigma(t))=0. \tag{**} \] The standard Riccati technique is applied to (**) and this approach then implies oscillation of the nonlinear equation (*). A typical result is the following statement. Theorem. Suppose that \(\int^\infty p^{-1}(t)\Delta t=\infty\). Further, suppose that there exist a positive function \(\delta\) and a positive constant \(M\) such that \[ \limsup_{t\to \infty}\int^t \left[ K\delta(\sigma (s)) q(s)- \frac{(\delta^\Delta (s))^2p(s)(s+M\mu(s))}{4s\delta(\sigma(s))}\right] \Delta s=\infty, \] where \(\mu(t)=\sigma(t)-t\) is the gaininess of the time scale under consideration, then every solution of (*) oscillates. Applying this criterion with \(\delta(t)\equiv t\) to the Sturm-Liouville differential equation \(x''+q(t)x=0\) considered as a perturpation of the Euler differential equation \(y''+\frac{1}{4t^2}y=0\), i.e. the former equation is written in the form \[ x''+\frac{1}{4t^2}x +\left[q(t)-\frac{1}{4t^2}\right]x=0, \tag{***} \] claims that this equation is oscillatory provided \[ \limsup_{t\to \infty} \int^t \left[q(s)-\frac{1}{4s^2}\right]s\,ds=\infty. \] The last condition is interesting from the following point of view. The transformation \(x=\sqrt t y\) transforms (***) into the equation \[ (ty')'+ \left[q(t)-\frac{1}{4t^2}\right]ty=0 \] and there exist counterexamples to the classical Leightnon-Wintner criterion (\(\int^\infty p^{-1}(t)\,dt=\infty=\int^\infty q(t)\,dt\) \(\implies\) \((p(t)x')'+q(t)x=0\) is ocillatory) that the co…
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 93C70
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 39A11
zbMATH DE Number: 2012680
zbMATH Keywords: time scale
zbMATH Keywords: Riccati transformation
zbMATH Keywords: nonlinear second order dynamic equation
zbMATH Keywords: Sturm-Liouville differential equation
zbMATH Keywords: Euler differential equation
zbMATH Keywords: Leightnon-Wintner criterion
author: S. H. Saker
reviewed by: Ondřej Došlý
MaRDI profile type: Publication
cites work: Dynamic equations on time scales: A survey
cites work: Q2723195
cites work: An oscillation theorem for discrete eigenvalue problems
cites work: A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales
cites work: Oscillation results for a dynamic equation on a time scale
cites work: Riccati equations on a measure chain
cites work: Positive solutions for a nonlinear differential equation on a measure chain
cites work: Oscillation criteria for second-order matrix dynamic equations on a time scale
cites work: On a disconjugacy criterion for second-order dynamic equations on time scales
cites work: Analysis on measure chains - a unified approach to continuous and discrete calculus
cites work: Q4167008
cites work: Q4526523
full work available at URL: s0096-3003(02)00829-9
OpenAlex ID: W1976148003
description: scientific article; zbMATH DE number 2012680