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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 non-compactness 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]). |
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