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

摘要

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 about finding 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 for meeting the condition that d(x,Sx)=d(x,Tx)=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.