Continuity of generalized metric projections in Banach spaces

Original Paper
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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 (xA) 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.

Keywords

Metric projection Proximinal set Hausdorff (Wijsman) convergence Continuity Nearly strongly (very) convex space Property S (WS) 

Mathematics Subject Classification

Primary 41A65 Secondary 46B20 

Notes

Acknowledgements

The authors are indebted to the referees for their insightful comments and suggestions. Zihou Zhang is supported by National Natural Science Foundation of China (Grant no. 11671252) and Yu Zhou is supported by National Natural Science Foundation of China (Grant no. 11771278).

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© Springer-Verlag Italia S.r.l. 2017

Authors and Affiliations

  1. 1.School of Mathematics, Physics and StatisticsShanghai University of Engineering ScienceShanghaiPeople’s Republic of China

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