COMMON BEST PROXIMITY POINTS AND GLOBAL OPTIMAL APPROXIMATE SOLUTIONS FOR NEW TYPES OF PROXIMAL CONTRACTIONS

Let (X, d) be a metric space, A and B be two non-empty subsets of X and S,T:A -> B be two non-self mappings. In view of the fact that, given any point x is an element of A, the distances between x and Sx and between x and Tx are at least d(A, B), which is the absolute infimum of d(x, Sx) and d(x, Tx), a common best proximity point theorem affirms the global minimum of both the functions x -> d(x, Sx) and x -> d(x, Tx) by imposing the common approximate solution of the equations Sr = x and Tx = x to satisfy the condition d(x, Sx) = d(x, Tx) = d(A, 8). In this paper, we present two new types of proximal contractions and develop a common best proximity point theorem for proximally commuting non-self mappings, thereby yielding the common optimal approximate solution of some fixed point equations when there is no common solution.