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April 2016 Nonconvex proximal normal structure in convex metric spaces
Moosa Gabeleh, Olivier Olela Otafudu
Banach J. Math. Anal. 10(2): 400-414 (April 2016). DOI: 10.1215/17358787-3495759


Given that A and B are two nonempty subsets of the convex metric space (X,d,W), a mapping T:ABAB is noncyclic relatively nonexpansive, provided that T(A)A, T(B)B, and d(Tx,Ty)d(x,y) for all (x,y)A×B. A point (p,q)A×B is called a best proximity pair for the mapping T if p=Tp, q=Tq, and d(p,q)=dist(A,B). In this work, we study the existence of best proximity pairs for noncyclic relatively nonexpansive mappings by using the notion of nonconvex proximal normal structure. In this way, we generalize a main result of Eldred, Kirk, and Veeramani. We also establish a common best proximity pair theorem for a commuting family of noncyclic relatively nonexpansive mappings in the setting of convex metric spaces, and as an application we conclude a common fixed-point theorem.


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Moosa Gabeleh. Olivier Olela Otafudu. "Nonconvex proximal normal structure in convex metric spaces." Banach J. Math. Anal. 10 (2) 400 - 414, April 2016.


Received: 28 May 2015; Accepted: 13 July 2015; Published: April 2016
First available in Project Euclid: 19 April 2016

zbMATH: 1335.54042
MathSciNet: MR3489646
Digital Object Identifier: 10.1215/17358787-3495759

Primary: 46B20
Secondary: 47H09

Keywords: $T$-regular reflexive pair , best proximity pair , nonconvex proximal normal structure , strictly convex metric space

Rights: Copyright © 2016 Tusi Mathematical Research Group


Vol.10 • No. 2 • April 2016
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