Continuous and Lipschitzian selections from a measurable set-valued map

Abstract

In this paper we construct for a given measurable set-valued map K: (0, T) -+ R” with convex and compact values a lower semicontinuous set-valued map H( .) with convex compact values such that H(t) c K(t) for almost all t E (0, T). Moreover, H( ) contains every local continuous selection from K( ), i.e., every continuous function r: (a, b) H R”, (a, 6) c (0, T) such that r(t) E K(t) for almost all t E (a, b). Then we construct a set-valued map f( .) with convex compact values such that L(t) c K(t) for almost all t E (0, T), and L( .) contains every Lipschitzian selection from K( .) defined on the previously given open set PC (0, T) with the Lipschitz constant less or equal to k 3 0. If L( .) is not identically equal to the empty set then it is continuous on P. In both cases we define the maps H( .) and L( .) using the support function. One motivation for construction of these regularizations of the setvalued map K( .) comes from viability theory (see [ 1, 21). If a set-valued map F: Graph(K) c-f R” is given we may regard K( .) as a viability map and we consider the following viability problem