Approximations, Expansions and Univalued Representations of Multifunctions

Abstract

We consider multifunctions acting between two linear normed spaces and having closed convex images. Approximations are considered which serve as an expansion of it. Generalized delta theorems for random sets in infinite dimensions are shown using those approximations. Furthermore, univalued, resp. Castaing representations of the multifunction are constructed with (higher order) differentiability properties at certain points whose directional derivative form a univalued, resp. Castaing representation of the corresponding “derivative” of the multifunction. The construction yields higher order information about the asymptotic distributions of measurable selections forming the Castaing representation of the multifunction.