On the location of positive limit sets for autonomous functional differential equations with infinite delay

Abstract

In 1983, the authors published a paper [lo] on an invariance principle for finite delay functional differential equations (FDEs). In [9], T. Krisztin and the authors extended many of the ideas of [lo] to include FDEs with infinite delay. The basic results in both [9] and [lo] centered around the use of Liapunov-Razumikhin techniques and the location of positive limit sets of precompact orbits. The main purpose of this paper is to extend the results of [9] and to provide several examples to illustrate the main theorems. Of particular interest will be to establish asymptotic constancy of solutions of equations for which each constant function is itself a solution. This idea was one of the main themes in [lo] but was not examined in [9]. Likewise, we will see how a family of spaces (as opposed to a single phase space) can be useful in examining certain equations (cf. Example 3.4). In considering the use of limit sets in conjunction with Razumikhin techniques and invariance principles for infinite delay equations, there are several difficulties that are encountered. These include: (i) choice of space(s); (ii) precompactness of orbits; and (iii) appropriate comparison principles. These topics are discussed individually in [ 11, [7], and [8],