Periodic solutions of nonlinear equations generalizing logistic equations with delay
type: Item
instance of: scholarly article
title: Periodic solutions of nonlinear equations generalizing logistic equations with delay
zbMATH Open document ID: 1391.34116
DOI: 10.1134/S0001434617070203
published in: Mathematical Notes
publication date: 2018-01-05T00:00:00Z
review text: The following generalization of the logistic equation with delay is considered \[ u^{\,\prime}(t)=\gamma u(t)\, F_\alpha(1-u(t-\tau)), \eqno{(1)} \] where \(\alpha\geq0, \gamma>0, \tau>0\) are constants and the nonlinearity \(F_\alpha\) is defined as \(F_\alpha(v)=v|v|^{\alpha-1}\) with \(F_0(v)=\text{sig}\,(v)\) and \(F_1(v)=v\). The main result of the paper is the existence, shape, and asymptotic stability with the asymptotic phase of slowly oscillating periodic solutions of equation (1) for large value of parameters \(\gamma\) (and for normalized delay \(\tau=1\)). As an auxiliary result of author's considerations the existence of periodic solutions of period \(4\) is shown for the delay differential equation \[ v^{\,\prime}(t)=-F_\alpha(v(t-1)). \] The latter is a well known result due to \textit{J. L. Kaplan} and \textit{J. A. Yorke} [J. Math. Anal. Appl. 48, 317--324 (1974; Zbl 0293.34102)] and its numerous extensions by others.
reviewed by: Anatoli F. Ivanov
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34K25
zbMATH DE Number: 6822728
author: S. A. Kashchenko
MaRDI profile type: Publication
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full work available at URL: s0001434617070203
OpenAlex ID: W2766080575
description: scientific article; zbMATH DE number 6822728