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

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.