Continuity of generalized metric projections in Banach spaces

Abstract

Let $$\mathcal {P}$$ be the family of all proximinal subsets of a Banach space X. Let $$P:(X,\mathcal {P})\rightarrow 2^{X}$$ be the generalized metric projection defined as $$P(x,A)=P_{A}(x)=\{a\in A:\Vert x-a\Vert =d(x,A)\}$$ for any $$(x,A)\in (X,\mathcal {P})$$ , where $$P_{A}$$ is the usual metric projection on X. The mapping P is said to be (resp. weakly) upper semi-continuous at $$(x,A)\in (X,\mathcal {P})$$ in the Hausdorff sense if, for any (resp. weakly) open set $$W\supset P_A(x)$$ , any $$\{x_n\}_{n=1}^{\infty }\subset X$$ with $$x_n\rightarrow x$$ and any sequence $$\{A_n\}_{n=1}^{\infty }\subset \mathcal {P}$$ with $$A_n\xrightarrow {H}A$$ , there exists a $$N\in \mathbb {N}$$ such that $$P_{A_n}(x_n)\subset W$$ for any $$n>N$$ . In this paper, the continuity of $$P:(X,\mathcal {P})\rightarrow P_{A}(x)$$ are discussed. We prove that: (1) if X is a nearly strongly convex (resp. nearly very convex) space, then for any $$x\in X$$ and any convex subset $$A\in \mathcal {P}$$ the mapping $$P:(X,\mathcal {P})\rightarrow 2^{X},P(x,A)=P_{A}(x)$$ is (resp. weakly) upper semi-continuous at (x, A) in the Hausdorff sense; (2) if X has the property S (resp. property WS), then for any $$x^*\in X^*$$ and any convex subset $$A^*\in \mathcal {P}^*$$ with non-empty $$w^*$$ -interior points the mapping $$P:(X^*,\mathcal {P}^*)\rightarrow 2^{X^*},P(x^*,A^*)=P_{A^*}(x^*)$$ is (resp. weakly) upper semi-continuous at $$(x^*, A^*)$$ in the Hausdorff sense, where $$\mathcal {P}^*$$ stands for the family of all proximinal subsets of the dual space $$X^*$$ . Our results are generalizations of some known results concerning the continuity of the classical metric projection.