Common best proximity points: Global optimization of multi- objective functions

Abstract

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 S x = x and T x = 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 about finding an element x that is in close proximity to S x and T x in the sense that d ( x , S x ) and d ( x , T x ) 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 S x = x and T x = x when there is no common solution. Moreover, it is emphasized that the real valued functions x ⟶ d ( x , S x ) and x ⟶ d ( x , T x ) evaluate the degree of the error involved for any common approximate solution of the equations S x = x and T x = x . Owing to the fact that the distance between x and S x , and the distance between x and T x 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 , S x ) and x ⟶ d ( x , T x ) by postulating a common approximate solution of the equations S x = x and T x = x for meeting the condition that d ( x , S x ) = d ( x , T x ) = d ( A , B ) . This work is devoted to an interesting common best proximity point theorem for pairs of non-self-mappings satisfying a contraction-like condition, thereby producing common optimal approximate solutions of certain simultaneous fixed point equations.