IN THE VERY RECENT PAPERS, AUTHORS PROVED SOME RESULT ABOUT THE EXISTENCE AND UNIQUENESS OF BEST proximity POINTS BY THE CYCLIC CONTRACTION MAPPING THAT IT OBTAINED FROM THE VERSIONS OF ASSOCIATED EXISTING RESULTS IN THE FIXED POINT THEORY.ALONG THE SAME LINE, IN THIS PAPER, WE PROVE THAT THESE RESULTS CAN BE OBTAINED UNDER A WEAKER MAPPING, NAMELY, CYCLIC ASYMPTOTIC CONTRACTION AND CYCLIC NEARLY CONTRACTION. ALSO, WE GIVE AN APPLICATION IN THE EQUILIBRIUM PROBLEM FOR MAPPING THAT HAS THE BEST proximity POINT.