Asymptotic behavior of volterra integrodifferential equations

Abstract

where I " ] is any vector or matrix norm and d(t, s) > 0 is a continuous function. The perturbation often occurs in dynamical systems due to modelling errors, measurement errors, linearization approximations, and so on. Asymptotic behavior of differential equations with bounded delays and nonlinear perturbation have been intensively investigated by numerous authors [i] . However to the best of the present author's knowledge, most of the known works on integrodifferential equations only deal with ordinary differential equations with perturbation integral terms[2]. Moreover, the results for stability and boundedness require that the matrix A is stable[ 1-5] and bounded [3,a] because the stability of A is used to control the above-norm bound of the integral term. An at tempt has been made in this paper to study the boundedness and asymptotic behavior of solutions of (1) when the coefficient matrix A(t) in (1) is not necessarily stable and bounded. Our technique is new and the main results unify, improve and extend the major results in [2-6]. In order to make full use of the information of the linear part of (1), we shall develop an equivalent equation by the following transformation. Let D(t, s) be an n x n continuous matrix function on 0 < s < t < oo, and