Continuability of solutions of the generalized Liénard system with time delay
type: Item
instance of: scholarly article
title: Continuability of solutions of the generalized Liénard system with time delay
zbMATH Open document ID: 0567.34061
author: Jitsuro Sugie
published in: Proceedings of the Japan Academy. Series A
publication date: 1984-01-01T00:00:00Z
review text: The author considers the system of differential equations with time delay \[ (1)\quad x'(t)=y(t)-F(x(t)),\quad y'(t)=-g(t,x(t-r(t)), \] where \(F: {\mathbb{R}}\to {\mathbb{R}}\), g: [0,\(\infty)\in {\mathbb{R}}\to {\mathbb{R}}\), r: [0,\(\infty)\to [0,\infty)\) are continuous functions. It is proved that all solutions of (1) are continuable in the future if and only if \((A^+)\) and \((A^-)\) hold, where \((A^{\pm})\) is defined by \((A_ 1^{\pm})\) or \((A_ 2^{\pm})\), \((A_ 1^{\pm})\) means that \(\int^{\infty}_{0}(1+F_{\mp}(x))^{-1}dx=\pm \infty\) and \((A_ 2^{\pm})\) means that there exists a sequence \(\{x_ n\}\) such that \(x_ n\to \pm \infty\) and \(F(x_ n)\to \pm \infty\) as \(n\to \infty\); \(F_+(F_-)\) denotes a positive (negative) part of F.
Mathematics Subject Classification ID: 34K05
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 3905036
zbMATH Keywords: Liénard system with time delay
reviewed by: R. R. Akhmerov
MaRDI profile type: Publication
full work available at URL: pjaa.60.357
OpenAlex ID: W2086676003
cites work: Q5332758
cites work: Q5343704
cites work: Theory of functional differential equations. 2nd ed
cites work: Necessary and Sufficient Conditions for the Continuability of Solutions of the System
DOI: 10.3792/PJAA.60.357
description: scientific article; zbMATH DE number 3905036