On best proximity pair theorems and fixed-point theorems

Abstract

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 ${\text{min}}_{x\in A}d\left(x,Tx\right)$ 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.