Existence of periodic solutions for a state dependent delay differential equation
type: Item
instance of: scholarly article
title: Existence of periodic solutions for a state dependent delay differential equation
zbMATH Open document ID: 0977.34067
published in: Journal of Differential Equations
publication date: 2002-01-21T00:00:00Z
review text: The authors consider a scalar differential equation with delay of the form \[ \dot x(t)=- f\biggl(x\bigl(t-\tau(t) \bigr)\biggr), \quad\dot \tau (t)= h\bigl(x(t), \tau (t)\bigr),\tag{1} \] where \(f:\mathbb{R} \to\mathbb{R}\) and \(h:\mathbb{R} \times [\tau_1,\tau_2] \to\mathbb{R}\) (with \(0<\tau_1 <\tau_2)\) are \(C^1\) maps. By applying a semiejective fixed-point theorem (see theorem 1.1, A semiejective fixed-point theorem (1998), working paper by P. Magal and O. Arino) the authors prove the existence of a nontrivial slowly oscillating periodic solution to (1). This result stands an extension of the result by \textit{D. Arino}, \textit{K. P. Hadeler} and \textit{M. L. Hbid} [J. Differ. Equations 144, No. 2, 263-301 (1998; Zbl 0913.34057)] due to relaxation of the assumptions made in the above paper.
reviewed by: Stanislaw Walczak
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 34K11
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C15
zbMATH DE Number: 1538230
zbMATH Keywords: periodic solution
zbMATH Keywords: delay differential equation
author: Pierre Magal
author: Ovide Arino
MaRDI profile type: Publication
full work available at URL: jdeq.1999.3759
OpenAlex ID: W2011401245
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DOI: 10.1006/JDEQ.1999.3759
description: scientific article; zbMATH DE number 1538230