On the oscillations of mixed neutral equations
type: Item
instance of: scholarly article
title: On the oscillations of mixed neutral equations
zbMATH Open document ID: 0845.34074
author: Said R. Grace
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-09-11T00:00:00Z
review text: The author considers neutral differential equations of odd order of the form \[ (x(t)+ cx(t- h)+ c^* x(t+ h^*))^{(n)}= qx(t- g)+ px(t+ g^*),\tag{1} \] where \(c\), \(c^*\), \(g\), \(g^*\), \(h\), \(h^*\), \(p\) and \(q\) are real constants. It is well-known that a necessary and sufficient condition for oscillation of all solutions of (1) is that the characteristic equation \(z^n(1+ ce^{- hz}+ c^* e^{h^* z})= qe^{- g z}+ pe^{g^* z}\) associated with (1) has no real roots. Since this is not easily verifiable, the author's aim is to obtain sufficient conditions for oscillation of (1) involving the coefficients and the arguments only. A typical result is the following theorem: ``Suppose that \(c^*\), \(g^*\), \(h^*\) and \(p\) are positive constants and \(c\), \(g\), \(h\) and \(q\) are nonnegative constants. Let \[ \Biggl({p\over 1+ c}\Biggr)^{1/n} \Biggl({g^*\over n}\Biggr) e> 1 \] and either \[ q> 0,\;\Biggl({q\over c^*}\Biggr)^{1/n} \Biggl({g+ h^*\over n}\Biggr) e> 1 \] or \[ h^*> g^*,\;\Biggl({p+ q\over c^*}\Biggr)^{1/n} \Biggl({h^*- g^*\over n}\Biggr) e> 1. \] Then the equation \((x(t)+ cx(t- h)- c^* x(t+ h^*))^{(n)}= qx(t- g)+ px(t+ g^*)\) is oscillatory.'' At the end of the paper, the author notes that his results are extendable to more general neutral and nonneutral equations.
Mathematics Subject Classification ID: 34K11
Mathematics Subject Classification ID: 34K40
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 859211
zbMATH Keywords: neutral differential equations of odd order
zbMATH Keywords: oscillation
reviewed by: V. Petrov
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1306
OpenAlex ID: W2034518495
DOI: 10.1006/JMAA.1995.1306
description: scientific article; zbMATH DE number 859211