The significance of fixed-point theory stems from the fact that it
furnishes a unified approach and constitutes an important tool in
solving equations which are not necessarily linear. On the other
hand, if the fixed-point equation $Tx=x$ does not possess a
solution, it is contemplated to resolve a problem of finding an
element $x$
such that $x$
is in proximity to $Tx$
in some sense.
Best proximity pair theorems analyze the conditions under which
the optimization problem, namely $\min_{x\in A}d(x,Tx)$
has a
solution. In this paper, we discuss the difference between best
approximation theorems and best proximity pair theorems. We also
discuss an application of a best proximity pair theorem to the
theory of games.