An asymptotic stability theorem for nonautonomous functional differential equations
type: Item
instance of: scholarly article
title: An asymptotic stability theorem for nonautonomous functional differential equations
zbMATH Open document ID: 0534.34069
author: Carlotta Maffei
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1983-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/107925
review text: This paper is concerned with the problem of extending the classical Lyapunov and Matrosov theorems on the asymptotic stability of the stationary solutions of functional differential equations. The main result of this note gives, in fact, an asymptotic stability theorem for functional differential equations with unbounded right-hand side. (It is wellknown that in this case some difficulties arise in investigating the asymptotic behavior of solutions.) Here the aforesaid problem is approached by introducing not only an auxiliary Lyapunov functional, say V, but also a second auxiliary function, with suitable properties, which ''checks'' the behavior of solutions close to the sets in which the upper right-hand derivative of the functional V is no longer sign definite. This method allows us to bypass the study of the asymptotic behavior of limiting equations, moreover it is not necessary to require any precompactness and regularity hypotheses on the given equation. An example is presented to illustrate this result.
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34D05
zbMATH DE Number: 3847841
zbMATH Keywords: functional differential equations
zbMATH Keywords: asymptotic stability
zbMATH Keywords: Lyapunov functionals
zbMATH Keywords: Matrosov theorems
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3847841