Graph-approximation of set-valued maps on noncompact domains

In the paper we study the existence of the so-called graph-approximations of upper semicontinuous set-valued maps defined on noncompact domains. We prove, in particular, that an arbitrary neighborhood of the graph of a map whose values satisfy a certain UV-property contains the graph of a continuous single-valued map provided the domain is either a finite-dimensional metric space, a locally finite-dimensional polyhedron or an absolute neighborhood retract.