Near dentability and continuity of the set-valued metric generalized inverse in Banach spaces

Abstract

In this paper, upper semicontinuity and continuity of the set-valued metric generalized inverse $T^{\partial }$ in nearly dentable spaces are investigated using the methods of Banach space geometry. Moreover, it is proved that if $X$ is a nearly dentable space and if $C$ is a closed convex set of $X$, then $C$ is approximatively compact if and only if $P_C\left(x\right)$ is compact for any $x\in X$.