Common best proximity points: global minimization of multi-objective functions

Assume that A and B are non-void subsets of a metric space, and that S : A −→ B and T : A −→ B are given non-self mappings. In light of the fact that S and T are non-self mappings, it may happen that the equations Sx = x and Tx = x have no common solution, named a common fixed point of the mappings S and T . Subsequently, in the event that there is no common solution of the preceding equations, one speculates to find an element x that is in close proximity to Sx and Tx in the sense that d(x, Sx) and d(x, Tx) are minimum. Indeed, a common best proximity point theorem investigates the existence of such an optimal approximate solution, named a common best proximity point of the mappings S and T , to the equations Sx = x and Tx = x when there is no common solution. Moreover, it is emphasized that the real valued functions x −→ d(x, Sx) and x −→ d(x, Tx) evaluate the degree of the error involved for any common approximate solution of the equations Sx = x and Tx = x. Owing to the fact that the distance between x and Sx, and the distance between x and Tx are at least the distance between A and B for all x in A, a common best proximity point theorem accomplishes the global minimum of both functions x −→ d(x, Sx) and x −→ d(x, Tx) by postulating a common approximate solution of the equations Sx = x and Tx = x to meet the condition that d(x, Sx) = d(x, Tx) = d(A, B). This article is devoted to an interesting common best proximity point theorem for pairs of non-self mappings satisfying contraction-like condition, thereby producing common optimal approximate solutions of certain simultaneous fixed point equations.