On best proximity points in metric and Banach spaces

TL;DR

This paper investigates the existence, uniqueness, and convergence of best proximity points for cyclic contractions across various spaces, including Banach, metric, and CAT(0) spaces, offering new insights and approaches.

Contribution

It introduces two different approaches to analyze best proximity points, extending results to Banach, geodesic, metric, and CAT(0) spaces, with improved or complementary findings.

Findings

Established conditions for existence and uniqueness of best proximity points.

Proved convergence of iterative methods to these points.

Extended analysis to CAT(0) spaces and other metric spaces.

Abstract

In this paper we study the existence and uniqueness of best proximity points of cyclic contractions as well as the convergence of iterates to such proximity points. We do it from two different approaches, leading each one of them to different results which complete, if not improve, other similar results in the theory. Results in this paper stand for Banach spaces, geodesic metric spaces and metric spaces. We also include an appendix on CAT(0) spaces where we study the particular behavior of these spaces regarding the problems we are concerned with.