On the fundamental theory of multivalued differential equations

Abstract

Let&‘be a multivalued function defined on an open subset of Rn+‘containing 0, with compact but not necessarily convex values. We can consider the two problems k E F(t, N), x(0) = 0, (1) i E co F(t, x), x(0) = 0. (4 It is known [l, 5, 91 that if F satisfies the hypothesis of continuity or is of CarathPodory type, there exist solutions of problem 1 (and so of problem 2). The purpose of this article is to investigate the relations existing between the solutions of problems 1 and 2. In our Theorem l WC prove that if F satisfies the hypothesis of Kamke type, then the closure of the set of solutions of problem 1 is the set of solutions of problem 2. This generalizes a known theorem of Turowicz [ 1 I]. The continuity of F, by itself, is not sufficient to assure the same result, as is shown by the counterexample of Plis [9]. In Theorem 2 we prove that iff(t, X) is any continuous selection of co F(t, x), then at least one solution of * -5 f(t, x), x(0) -z 0, is the limit of solutions of problem 1. In particular, if f(t, x) is Lipschitzean or the problem 2 = j(t, x), x(O) : 0 has exactly one solution, then this solution belongs to the closure of the set of solutions of problem 1. Finally we consider the dependence of solutions on the initial data and we show by example that if F is assumed to be only continuous, then the dependence need not be upper semicontinuous. The techniques and notations used are those introducted in [I].