A Characterization of Weak Proximal Normal Structure and Best Proximity Pairs

TL;DR

This paper characterizes weak proximal normal structure via best proximity pairs, introduces pointwise cyclic contractions, and proves the existence of best proximity pairs in reflexive Banach spaces.

Contribution

It provides a new characterization of weak proximal normal structure and establishes existence results for best proximity pairs using pointwise cyclic contractions.

Findings

Characterization of weak proximal normal structure using best proximity pairs

Introduction of pointwise cyclic contraction concept

Existence of best proximity pairs in reflexive Banach spaces

Abstract

The aim of this paper is to address an open problem given in [Kirk, W. A., Shahzad, Naseer, Normal structure and orbital fixed point conditions, J. Math. Anal. Appl. {\bf{vol 463(2)}}, (2018) 461--476]. We give a characterization of weak proximal normal structure using best proximity pair property. We also introduce a notion of pointwise cyclic contraction wrt orbits and therein prove the existence of a best proximity pair in the setting of reflexive Banach spaces.