SEMIGROUPS OF MIDPOINT CONVEX SET-VALUED FUNCTIONS

A set-valued function F from a convex subset D of a real vector space X into the set n(Y) of all non-empty subsets of a real vector space Y is said to be midpoint convex iff 1/2[F(x)+F(x)]subset of F[1/2(x+y)] for all x, y is an element of D. This note deals with iteration semigroups of midpoint convex set-valued functions on a whole linear space.