An invariance principle of Lyapunov-Razumikhin type for neutral functional differential equations
type: Item
instance of: scholarly article
title: An invariance principle of Lyapunov-Razumikhin type for neutral functional differential equations
zbMATH Open document ID: 0796.34067
author: J. Terjéki
author: John R. Haddock
author: Tibor Krisztin
author: Jianhong Wu
published in: Journal of Differential Equations
publication date: 1994-05-12T00:00:00Z
review text: The problem \({d\over dt} (Dx_ t)= f(x_ t)\), \(x_ 0=\varphi\) is studied under the assumptions that \(C= C([-r,0],\mathbb{R}^ n)\), \(f: C\to \mathbb{R}^ n\) is completely continuous, \(D: C\to\mathbb{R}^ n\) is linear, continuous, atomic at zero and stable. Uniqueness, continuous dependence and continuation of solutions of that problem are assumed. The \(\omega\)- limit set \(\Omega(\varphi)\) of \(\varphi\) is located by using a Lyapunov function. The results are based on the lemma: If \(\{x_ t(\varphi): t\geq 0\}\) is bounded and there exists a stable subset \(S\subseteq C\) such that \(\Omega(\varphi)\cap S\) is nonempty, then \(x_ t(\varphi)\to S\) as \(t\to \infty\). By a special choice of the set \(S\) sufficient conditions for the asymptotic stability and the asymptotic constancy are derived.
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34K40
zbMATH DE Number: 553870
zbMATH Keywords: neutral functional differential equation
zbMATH Keywords: invariance principle
zbMATH Keywords: uniqueness
zbMATH Keywords: continuous dependence
zbMATH Keywords: continuation
zbMATH Keywords: Lyapunov function
zbMATH Keywords: asymptotic stability
MaRDI profile type: Publication
full work available at URL: jdeq.1994.1019
OpenAlex ID: W2018754548
DOI: 10.1006/JDEQ.1994.1019
description: scientific article; zbMATH DE number 553870