Best proximity pairs in ultrametric spaces

TL;DR

This paper investigates the existence of best proximity pairs in ultrametric spaces, generalizing known approximation results and deriving fixed point theorems under certain conditions.

Contribution

It introduces new conditions for the existence of best proximity pairs in ultrametric spaces and extends classical approximation results.

Findings

Existence of best proximity pairs under suitable assumptions

Generalization of a classical best approximation theorem

Derivation of new fixed point theorems

Abstract

In the present paper, we study the existence of best proximity pairs in ultrametric spaces. We show, under suitable assumptions, that the proximinal pair $(A,B)$ has a best proximity pair. As a consequence we generalize a well known best approximation result and we derive some fixed point theorems. Moreover, we provide examples to illustrate the obtained results.