Liapunov-Razumikhin pairs and the location of positive limit sets for precompact functional differential equations with infinite delay

Abstract

IN A RECENT paper [l], Haddock and Terjeki addressed the problem of locating positive limit sets for autonomous functional differential equations with infinite delay. The authors obtained invariance principles and other results concerning asymptotic behavior of solutions by employing Liapunov-Razumikhin techniques. The autonomous nature of the equations they considered was central to their work. The purpose of this paper is to extend their results to a class of nonautonomous equations. Such a task was undertaken for finite delay equations in [2]. Here, as in [2], the assumption of autonomy is replaced by a continuity condition, and our invariance principles involve limiting equations of our original equation. We feel that the setting for our theory is quite natural, and that our methods illustrate the role of limiting equations in extending results from the autonomous theory to a significantly larger class of equations. Several examples are given to illustrate our techniques.