Approximative compactness and continuity of the set-valued metric generalized inverse in Banach spaces

In this paper, continuity of the set-valued metric generalized inverse T^∂$T^{\partial }$ in approximatively compact Banach spaces is investigated by means of the methods of geometry of Banach spaces. Necessary and sufficient conditions for upper semicontinuity (continuity) for the set-valued metric generalized inverses T^∂$T^{\partial }$ are given. Moreover, authors also prove that if X is a nearly dentable space and H is a hyperplane of X, then H   is approximatively compact iff _PH(x)$P_H(x)$ is compact for any x∈X$x\in X$.