Global attractivity for a delay logistic equation with instantaneous terms
type: Item
instance of: scholarly article
title: Global attractivity for a delay logistic equation with instantaneous terms
zbMATH Open document ID: 1069.34111
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-11-22T00:00:00Z
review text: Let \(g:[0,\infty)\rightarrow {\mathbb R}\) be a non-decreasing continuous function such that \(g(t)\leq t\) for all \(t\geq 0\) and \(g(t)\rightarrow \infty\) as \(t\rightarrow \infty\). Let \(BC\) be the Banach space consisting of bounded right-hand continuous functions \(\phi:(-\infty,0]\rightarrow {\mathbb R}\) with the sup norm \(\| \phi\| =\sup_{s\in (-\infty,0]}| \phi(s)| \). If \(a>0, \psi\in C([g(0),a],{\mathbb R})\) and \(0\leq t\leq a\), then \(\psi_t\in BC\) is defined by \(\psi_t(s)=\psi(t+s)\), for \(g(t)-t\leq s\leq 0\) and \(\psi_t(s)=0\), for \(-\infty<s<g(t)-t.\) In this paper, the author is concerned with the following one-dimensional delayed logistic differential equation \[ x^{\prime}(t)=-r(t)[1+x(t)][cx(t)+F(t,x_t)],\quad t\geq 0,\tag{1} \] with initial condition \[ x(t)=\phi(t),\text{ for } g(0)\leq t\leq 0,\quad \phi\in C([g(0),0],[-1,\infty)),\quad \phi(0)>-1, \] where \(c\geq 0, r(t)\in C([0,\infty),(0,\infty))\), \(F:[0,\infty)\times BC\rightarrow {\mathbb R}\) and for any \(\psi\in C([g(0),\infty),{\mathbb R})\) with \(\psi_t\in BC\) the function \(t\mapsto F(t,\psi_t)\) is continuous on \([0,\infty)\), and satisfies \(-M_t(-\phi)\leq F(t,\phi)\leq M_t(\phi)\) for \(t\geq 0\), \(\phi\in BC\), where \(M_t(\phi)=\max\{0,\sup_{s\in [g(t)-t,0]}\phi(s)\}\). By using the inequality technique, sufficient conditions are derived for the global attractivity of the zero solution to equation (1). Existing related results in the literature are extended and improved.
reviewed by: Rui Xu
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34K25
zbMATH DE Number: 2117571
zbMATH Keywords: logistic differential equation
zbMATH Keywords: global attractivity
zbMATH Keywords: instantaneous term
author: X. H. Tang
MaRDI profile type: Publication
full work available at URL: j.na.2004.07.011
OpenAlex ID: W2090373118
cites work: Q4003237
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DOI: 10.1016/J.NA.2004.07.011
description: scientific article; zbMATH DE number 2117571