Global attractivity and oscillation in a nonlinear delay equation
type: Item
instance of: scholarly article
title: Global attractivity and oscillation in a nonlinear delay equation
zbMATH Open document ID: 0987.34065
DOI: 10.1016/S0362-546X(99)00182-0
author: Jurang Yan
author: Qiuxiang Feng
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-06-12T00:00:00Z
review text: The authors consider the delay periodic equation \[ x^{'}(t)=x(t)[a(t)+b(t)x^p(t-mw)-c(t)x^q(t-mw)], \] where \(a,b,c\in C[[0,\infty),\mathbb{R}]\) with a common period \(w>0\), and \(a>0\), \(c>0 \), \(m\) is a positive integer, \(p\) and \(q\) are positive constants \(q>p\). This equation with the initial condition \[ x(t)=\phi (t)\text{ for }-mw\leq t\leq 0,\quad \phi \in C[[-mw,0],[0,\infty)], \] has a unique solution which is positive for all \(t\geq 0\). The authors investigate the global attractivity and the oscillatority of this unique solution. They provide a sufficient condition for the global attractivity of the unique positive \(w\)-periodic solution \(\widetilde{x}\) and for all other positive solutions to the equation in the non-delay case (\(m=0\)). The function \(\widetilde{x}\) is also a unique \(w\)-periodic solution to the equation for any positive integer \(m\). The paper establishes a necessary and sufficient condition for every solution to the equation to oscillate about \(\widetilde{x}\). A sufficient condition for the global attractivity of \(\widetilde{x}\) is provided, too.
Mathematics Subject Classification ID: 34K11
Mathematics Subject Classification ID: 34K25
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34K12
Mathematics Subject Classification ID: 34D45
zbMATH DE Number: 1539294
zbMATH Keywords: nonlinear delay equation
zbMATH Keywords: global attractivity
zbMATH Keywords: oscillation
MaRDI profile type: Publication
cites work: Q3262986
cites work: Q3667239
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full work available at URL: s0362-546x(99)00182-0
OpenAlex ID: W2060846613
Wikidata QID: Q126556732
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description: scientific article; zbMATH DE number 1539294