Common best proximity point theorems under proximal $F$-weak dominance in complete metric spaces

TL;DR

This paper establishes new common best proximity point theorems in complete metric spaces under the framework of proximal $F$-weak dominance, improving previous results and providing applicable examples.

Contribution

It introduces the concept of proximally $F$-weakly dominated pairs of mappings and proves new theorems ensuring the existence of common best proximity points.

Findings

New theorems guarantee common best proximity points under proximal $F$-weak dominance.

Provides examples where new results apply but previous ones do not.

Improves upon earlier work in the area of proximity point theory.

Abstract

Suppose that $S_1$ and $S_2$ are nonempty subsets of a complete metric space $(\mathcal{M},d)$ and $\phi,\psi:S_1\to S_2$ are mappings. The aim of this work is to investigate some conditions on $\phi$ and $\psi$ such that the two functions, one that assigns to each $x\in S_1$ exactly $d(x,\phi x)$ and the other that assigns to each $x\in S_1$ exactly $d(x,\psi x)$, attain the global minimum value at the same point in $S_1$. We have introduced the notion of proximally $F$-weakly dominated pair of mappings and proved two theorems that guarantee the existence of such a point. Our work is an improvement of earlier work in this direction. We have also provided examples in which our results are applicable, but the earlier results are not applicable.