Oscillation of differential equation of neutral type
type: Item
instance of: scholarly article
title: Oscillation of differential equation of neutral type
zbMATH Open document ID: 0831.34076
author: Rudolf Oláh
published in: Hiroshima Mathematical Journal
publication date: 1995-07-17T00:00:00Z
review text: The author considers the neutral differential equation \[ \biggl[ x(t) - p(t)x \bigl( \sigma (t) \bigr) \biggr]' + q(t)x \bigl( \tau (t) \bigr) = 0,\;t \geq t_0, \tag{1} \] where \(p,q, \sigma, \tau\) are continuous functions, \(\sigma\) and \(\tau\) tend to \(\infty\) as \(t \to \infty\) and \(\sigma (t)\) is strictly increasing. Sufficient conditions are obtained for the oscillation of all solutions of (1) and also the asymptotic behaviour of the nonoscillatory solutions of (1) is studied. It should be noted that some of the results in the paper are already known (Theorem 1 for example) but most of them, especially those in which the author makes use of the inverse of \(\sigma (t)\), are new. A typical result of this type is the following theorem: ``Suppose that \(1 \leq p(t)\), \(0 \leq q(t)\), \(\sigma (t) < t\), \(\sigma^{-1} (\tau (t)) > t\) for \(t \geq t_0\) and \[ \liminf \int_t^{ \sigma^{-1} (\tau (t))} {q(s) \over p \biggl( \sigma^{-1} \bigl( \tau (s) \bigr) \biggr)} ds > {1 \over e}. \] Then every solution of (1) is oscillatory.'' In the last part of the paper the author studies the asymptotic behaviour of the nonoscillatory solutions. He proves that under certain conditions any nonoscillatory solution of (1) tends to zero as \(t \to \infty\).
Mathematics Subject Classification ID: 34K11
Mathematics Subject Classification ID: 34K40
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 774751
zbMATH Keywords: neutral differential equation
zbMATH Keywords: oscillation
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: nonoscillatory solutions
reviewed by: V. Petrov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206127821
OpenAlex ID: W1497288017
Wikidata QID: Q115222428
DOI: 10.32917/HMJ/1206127821
description: scientific article; zbMATH DE number 774751