Approximate Best Proximity Pairs in Metric Space

Abstract

Let $A$ and $B$ be nonempty subsets of a metric space $X$ and also $T:A\cup B\to A\cup B$ and $T\left(A\right)\subseteq B$, $T\left(B\right)\subseteq A$. We are going to consider element $x\in A$ such that $d(x,Tx)\le d(A,B)+?$ for some $?>0$. We call pair $(A,B)$ an approximate best proximity pair. In this paper, definitions of approximate best proximity pair for a map and two maps, their diameters, $T$-minimizing a sequence are given in a metric space.