FASCICULI MATHEMAT ICI Nr 63 2020 DOI: 10.21008/j.0044-4413.2020.0002 Rakesh Batra COMMON BEST PROXIMITY POINTS FOR PROXIMALLY F -DOMINATED MAPPINGS Abstract. The principal aim of this work is to formulate an extension and improvement of the common best proximity point theorem for a pair of non-self mappings, one of which is dom- inated by the other as proved by Basha. The proposed exten- sion discusses a common best proximity point theorem for a pair of non-self mappings, one of which is F -dominated by the other proximally, for a function F as defined by Wardowski. Key words: global optimal approximate solution, common best proximity point, common fixed point, proximally F -dominated mappings. AMS Mathematics Subject Classification: 47H10, 46B40, 54H25, 55M20. 1. Introduction Banach’s fixed point theory deals with finding a solution to an equation fx = x, wherein f is a self map defined on a complete metric space X . The equation fx = x may not have a solution if f is not a self mapping. In that case, one may focus on the problem of searching an element x that is in close proximity to fx in some sense. Best approximation theorems and best proximity point theorems have been developed in recent times to solve above mentioned problem. The following best approximation theorem was established by Fan [6] in 1969. Theorem 1 ([6]). For a nonempty convex compact subset K of a Haus- dorff locally convex topological vector space X with a continuous semi-norm p and for a non self continuous map f : K → X , there exists an element x ∈ K called a best approximant in K such that p(x − fx)= d P (fx,K)= inf {p(fx − y): y ∈ K}. The preceding best approximation theorem has been generalized in vari- ous directions by many authors.