Oscillation of second-order differential equations with mixed argument
type: Item
instance of: scholarly article
title: Oscillation of second-order differential equations with mixed argument
zbMATH Open document ID: 0824.34073
author: J. Džurina
published in: Journal of Mathematical Analysis and Applications
publication date: 1995-11-09T00:00:00Z
review text: The author considers the second order differential equation with mixed argument \[ \Biggl({1\over r(t)} u'(t)\Biggr)'+ p(t) u(\tau(t))+ q(t) u(\sigma(t))= 0,\tag{1} \] where \(r, p, q, \tau, \sigma\in C([t_ 0, \infty))\), \(r(t)\) and \(q(t)\) are positive, \(p(t)\) is nonnegative, \[ \tau(t)\leq t,\;\lim_{t\to \infty} \tau(t)= \infty\quad\text{and}\quad\sigma(t)\geq t. \] A sufficient condition for oscillation of all solutions of (1) is obtained. As is customary a solution is called oscillatory if the set of its zeros is unbounded. It should be noted that despite the numerous results on oscillation of equations with deviating arguments there are comparatively few results for equations with mixed argument (they involve both advanced and delay argument). The main idea of the proof of the author's oscillation criteria is by using a comparison result (Theorem 1) and appropriate transformations to deduce the oscillatory properties of equation (1) from known oscillation criteria for simpler equations (with either delayed or advanced argument). Examples are also given that illustrate the author's results.
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 762193
zbMATH Keywords: second order differential equation with mixed argument
zbMATH Keywords: oscillation
reviewed by: V. Petrov
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1114
OpenAlex ID: W2073640825
DOI: 10.1006/JMAA.1995.1114
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description: scientific article; zbMATH DE number 762193