Abstract:

A differential system [see pdf for notation] where [see pdf for notation] has asymptotic equilibrium if 1) for any initial condition [see pdf for notation] the system has a solution [see pdf for notation] existing on and such that [see pdf for notation] exists and is
finite, and 2) for any v e B there exists [see pdf for notation] and a solution x(t) of (1)(2) with [see pdf for notation] Several papers have appeared dealing with asymptotic equilibrium of (1)(2) when [see pdf for notation], and f is majorized by a scalar function g(t,u) which is monotone in u for each t, [1,3]. However, when B is an arbitrary Banach space additional restrictions must be placed on f, (see [p.161,4; 5]). In [7] a set of sufficient conditions for local existence of solutions of (1)(2) in an arbitrary Banach space is given. These conditions include the use of the Kuratowski measure of noncompactness of bounded sets, denoted throughout this paper by a (see [2,7]). Since our goal will be to give sufficient conditions for asymptotic equilibrium of (1)(2) using [see pdf for notation], the first lemma
incorporates some known properties of [see pdf for notation] (see [7]). 