  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.
