Common Best Proximity Points: Global Optimal Solutions

Let S : A→ B and T : A→ B be given non-self mappings, where A and B are non-empty subsets of a metric space. As S and T are non-self mappings, the equations Sx = x and T x = x do not necessarily have a common solution, called a common fixed point of the mappings S and T . Therefore, in such cases of nonexistence of a common solution, it is attempted to find an element x that is closest to both Sx and T x in some sense. Indeed, common best proximity point theorems explore the existence of such optimal solutions, known as common best proximity points, to the equations Sx = x and T x = x when there is no common solution. It is remarked that the functions x → d(x,Sx) and x → d(x,T x) gauge the error involved for an approximate solution of the equations Sx = x and T x = x. In view of the fact that, for any element x in A, the distance between x and Sx, and the distance between x and T x are at least the distance between the sets A and B, a common best proximity point theorem achieves global minimum of both functions x → d(x,Sx) and x → d(x,T x) by stipulating a common approximate solution of the equations Sx = x and T x = x to fulfill the condition that d(x,Sx)= d(x,T x)= d(A,B). The purpose of this article is to elicit common best proximity point theorems for pairs of contractive non-self mappings and for pairs of contraction non-self mappings, yieldCommunicated by J.-C. Yao. N. Shahzad Department of Mathematics, King AbdulAziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia e-mail: [email protected] S. Sadiq Basha ( ) Department of Mathematics, Anna University, Chennai 600 025, India e-mail: [email protected] R. Jeyaraj St. Joesph’s College Higher Secondary School, Trichy 620 002, India e-mail: [email protected]