Abstract and Applied Analysis
Volume 2003 (2003), Issue 1, Pages 33-47
doi:10.1155/S1085337503209064

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 minx∈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.