Global optimal approximate solutions of best proximity points

Abstract

In this paper, we introduce the concept of contractive pair maps and give some necessary and sufficient conditions for existence and uniqueness of best proximity points for such pairs. In our approach, some conditions have been weakened. An application has been presented to demonstrate the usability of our results. Also, we introduce the concept of cyclic $\psi$-contraction and cyclic asymptotic $\psi$-contraction and give some existence and convergence theorems on best proximity point for cyclic $\psi$-contraction and cyclic asymptotic $\psi$-contraction mappings. The presented results extend, generalize and improve some known results from best proximity point theory and fixed-point theory.