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

Assume that AA and BB are non-void subsets of a metric space, and that S:A⟶BS:A⟶B and T:A⟶BT:A⟶B are given non-self-mappings. In light of the fact that SS and TT are non-self-mappings, it may happen that the equations Sx=xSx=x and Tx=xTx=x have no common solution, named a common fixed point of the mappings SS and TT. Subsequently, in the event that there is no common solution of the preceding equations, one speculates about finding an element xx that is in close proximity to SxSx and TxTx in the sense that d(x,Sx)d(x,Sx) and d(x,Tx)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 SS and TT, to the equations Sx=xSx=x and Tx=xTx=x when there is no common solution. Moreover, it is emphasized that the real valued functions x⟶d(x,Sx)x⟶d(x,Sx) and x⟶d(x,Tx)x⟶d(x,Tx) evaluate the degree of the error involved for any common approximate solution of the equations Sx=xSx=x and Tx=xTx=x. Owing to the fact that the distance between xx and SxSx, and the distance between xx and TxTx are at least the distance between AA and BB for all xx in AA, a common best proximity point theorem accomplishes the global minimum of both functions x⟶d(x,Sx)x⟶d(x,Sx) and x⟶d(x,Tx)x⟶d(x,Tx) by postulating a common approximate solution of the equations Sx=xSx=x and Tx=xTx=x for meeting the condition that d(x,Sx)=d(x,Tx)=d(A,B)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.