On Best Proximity Point Theorems without Ordering

Abstract

Recently, Basha (2013) addressed a problem that amalgamates approximation and optimization in the setting of a partially ordered set that is endowed with a metric. He assumed that if $A$ and $B$ are nonvoid subsets of a partially ordered set that is equipped with a metric and $S$ is a non-self-mapping from $A$ to $B$, then the mapping $S$ has an optimal approximate solution, called a best proximity point of the mapping $S$, to the operator equation $Sx=x$, when $S$ is a continuous, proximally monotone, ordered proximal contraction. In this note, we are going to obtain his results by omitting ordering, proximal monotonicity, and ordered proximal contraction on $S$.