Existence and convergence of best proximity points

Abstract

Consider a self map T defined on the union of two subsets A and B of a metric space and satisfying $T(A)\subseteq B$ and $T(B)\subseteq A$. We give some contraction type existence results for a best proximity point, that is, a point x such that $d(x,Tx)=\mathrm{dist}(A,B)$. We also give an algorithm to find a best proximity point for the map T in the setting of a uniformly convex Banach space.