Existence and convergence of best proximity points

Consider a self map T defined on the union of two subsets A and B of a metric space and satisfying T (A) ⊆ B and T (B) ⊆ A. We give some contraction type existence results for a best proximity point, that is, a point x such that d(x,T x) = 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. © 2005 Elsevier Inc. All rights reserved.