A characterization of proximal normal structure via proximal diametral sequences

Abstract

We introduce a concept of pointwise cyclic relatively nonexpansive mapping involving orbits to investigate the existence of best proximity points using a geometric property defined on a nonempty and convex pair of subsets of a Banach space X, called weak proximal normal structure. Examples are given to support our main conclusions. We also introduce a notion of proximal diametral sequence and establish a characterization of proximal normal structure and show that every nonempty and convex pair in uniformly convex in every direction Banach spaces has weak proximal normal structure. As an application, we give a new existence theorem for cyclic contractions in reflexive Banach spaces without strictly convexity condition.