A unified approach to stability in integrodifferential equations via Liapunov functions

Abstract

In this paper we investigate the problems of stability, uniform stability, uniform Lipschitz stability, asymptotic stability, and uniform asymptotic stability in three classes of integrodifferential systems. These are linear nonautonomous systems, nonlinear systems, and nonlinearly perturbed nonlinear systems. A simple method for constructing Liapunov functions will be used to unify the treatment of all of the above-mentioned stability notions relative to the various classes of integrodifferential systems. In our investigation of linear systems we benefited from the work in [4-8, 123. For nonlinear systems and their perturbations, we have used some of the results obtained recently in [IS, 101. In particular, we have used an extension of Alekseev’s [l] nonlinear variation of constants formula which was developed in [lo]. The approach we use here was first formulated in [3] and utilized to study uniform Lipschitz stability [2] of nonlinear differential systems. The method has been extended in [12] to cover all other stability notions in nonlinear differential systems. We now give a description of the contents of the paper. Section 1 includes all the basic definitions and preliminaries. In Section 2, we investigate the various stability notions of nonautonomous integrodifferential systems. This extends the work in [6-81, which mainly dealt with autonomous systems. We recall here from [4] that, in linear systems, the notions of uniform stability and uniform Lipschitz stability are equivalent. It was shown in [4], however, that, for nonlinear systems, uniform