An asymptotic stability theorem for nonautonomous functional differential equations
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.
scientific article; zbMATH DE number 3847841