A necessary and sufficient condition for the solutions of a functional differential equation to be oscillatory or tend to zero
type: Item
instance of: scholarly article
title: A necessary and sufficient condition for the solutions of a functional differential equation to be oscillatory or tend to zero
zbMATH Open document ID: 0874.34058
author: O. Diekmann
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-11-13T00:00:00Z
review text: It is shown that every solution of the nonhomogeneous functional differential equation \[ {d\over dt} \bigl(x(t) -px(t-\tau) \bigr)+ Q(t)G \bigl(x(t-\sigma) \bigr)=f(t), \] where \(f,Q \in C([T,\infty)\), \((0,\infty))\), \(\sigma,\tau \in(0, \infty)\), \(0\leq p<1\), \(G:\mathbb{R}\to \mathbb{R}\) such that \(xG(x) >0\) for \(x\neq 0\), \(G\) is nondecreasing, Lipschitzian, and satisfies a sublinear condition \[ \int^{\pm k}_0 {dx\over G(x)} <\infty \quad\text{and} \quad\int^\infty f(x)ds <\infty, \] is either oscillatory or tends to zero asymptotically if and only if \[ \int^\infty_T Q(s)ds =\infty. \] Examples and counterexamples are given.
Mathematics Subject Classification ID: 34K11
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 1005701
zbMATH Keywords: oscillation
zbMATH Keywords: nonhomogeneous functional differential equation
Wikidata QID: Q115395409
reviewed by: Alexandra Rodkina
MaRDI profile type: Publication
full work available at URL: jmaa.1996.5143
OpenAlex ID: W2069476810
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DOI: 10.1006/JMAA.1996.5143
description: scientific article; zbMATH DE number 1005701