Continuous selections of multivalued maps with non-convex non-closed decomposable values

Abstract

A class of multivalued maps with non-convex non-closed decomposable values is distinguished, and theorems are proved on the existence of continuous selections for such maps. This class contains multivalued maps whose values are extreme points of continuous multivalued maps with closed convex decomposable values in a Banach space of Bochner-integrable functions. The proofs are based on the Baire category theorem. It is known that the set of extreme points of a closed convex set is in general not closed. Hence the results or paper answer the question of the existence of continuous selections for multivalued maps with non-convex non-closed values.