Equivalent conditions on stability of functional differential equations
type: Item
instance of: scholarly article
title: Equivalent conditions on stability of functional differential equations
zbMATH Open document ID: 0779.34062
DOI: 10.1016/0022-247X(92)90010-B
author: Tingxiu Wang
published in: Journal of Mathematical Analysis and Applications
publication date: 1993-04-01T00:00:00Z
review text: The paper deals with stability conditions (in various senses) for the equations \(x'(t)=F(t,X_ t)\), where \(X_ t(s)=X(t+s)\), \(-h\leq s\leq 0\), by means of Lyapunov's second method. After a brief review of the known results in this regard (Krasovskij, Burton, Hatvani), the author provides some improvements. The main concern is related to the equivalence of two types of conditions intervening on the ``wedge'' appearing in the limitation of the derivative of a Lyapunov functional \(V(t,\varphi)\) with respect to the system. Namely, usual wedges for systems with delay render the same services as the wedges of the form \(W\Bigl(\int^ t_{t- h}| X(s)| ds\Bigr)\). Several examples are considered, related to special equations, such as \(x'(t)=b(t)f(x(t-h))\), \(x'(t)=- a(t)x(t)+b(t)\int^ t_{t-h} x(u)du\).
Mathematics Subject Classification ID: 34K20
zbMATH DE Number: 150644
zbMATH Keywords: wedge
zbMATH Keywords: stability
zbMATH Keywords: Lyapunov's second method
zbMATH Keywords: Lyapunov functional
reviewed by: Constantin Corduneanu
MaRDI profile type: Publication
cites work: Functional differential equations and Jensen's inequality
cites work: Stability and periodic solutions of ordinary and functional differential equations
cites work: Q4205782
cites work: Stability theorems for nonautonomous functional differential equations by Lyapunov functionals
cites work: On nonuniform asymptotic stability for nonautonomous functional differential equations
cites work: Stability conditions for linear nonautonomous delay differential equations
cites work: Existence and stability of solutions of a delay-differential system
cites work: Theory of functional differential equations. 2nd ed
cites work: Q3845518
cites work: Q3265777
cites work: Q5520199
description: scientific article; zbMATH DE number 150644