Best Proximity Pair Results for Relatively Nonexpansive Mappings in Geodesic Spaces

Abstract

Given A and B two nonempty subsets in a metric space, a mapping T :In this work, we extend the results given in [A.A. Eldred, W.A. Kirk, P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math. 171, 283-293 (2005)] for relatively nonexpansive mappings in Banach spaces to more general metric spaces. Namely, we give existence results of best proximity points for cyclic and noncyclic relatively nonexpansive mappings in the context of Busemann convex reflexive metric spaces. Moreover, particular results are proved in the setting of CAT(0) and uniformly convex geodesic spaces. Finally, we show that proximal normal structure is a sufficient but not necessary condition for the existence in A × B of a pair of best proximity points.MSC: Primary 54E40, 47H10.

Existence and approximating of common best proximity points of relatively nonexpansive mappings via Ishikawa iteration method

“…Proposition 2.1. (Proposition 3.1 of [11]) If (A, B) is a nonempty, closed and convex pair in a reflexive and Busemann space X such that B is bounded, then (A 0 , B 0 ) is nonempty, bounded, closed and convex. Definition 2.4.…”

“…A concept of proximal normal structure was first introduced in [4]. It was then improved in [11] from Banach spaces to geodesic spaces as below.…”

“…For instance, every nonempty, compact and convex pair in a geodesic space with convex metric, has the PNS (see Proposition 3.10 of [11]). Also, every nonempty, bounded, closed and convex pair in a uniformly convex metric space X has the PNS (see Proposition 3.5 of [11]). We refer to [16] for some interesting characterization of PNS.…”

Best proximity point (pair) results via MNC in Busemann convex metric spaces

“…We mention that the proof of Theorem 2.3 is based on the fact that any compact and convex pair in a geodesic space with convex metric has a geometric notion, called proximal normal structure (see Proposition 3.10 of [11]). To state an extended version of Theorem 2.3 we recall the following concept.…”

Nonconvex proximal normal structure in convex metric spaces

“…We mention that Theorem 1.1 is based on the fact that every nonempty, bounded, closed, and convex pair of subsets of a uniformly convex Banach space X has a proximal normal structure, and so the result follows from Theorem 2.2 of [4] (see also [6] for a different approach to the same problem).…”