Some remarks on convergence of best proximity points and Semi‑cyclic contractions

Abstract

We show that the main conclusions of the recent paper by R. Suparatulatorn et al. [R. Suparatulatorn, W. Cholamjiak and S. Suantai, Existence and convergence theorems for global minimization of best proximity points in Hilbert spaces, Acta Appl. Math., 165, 81-90 (2020)] are not real generalizations but particular cases of convergence of Mann’s iteration scheme to a fxed point of a nonexpansive self mapping. As well as the main results of an article by G.K. Jacob et al. [G.K. Jacob, M. Postolache, M. Marudai and V. Raja, Norm convergence iterations for best proximity points of non-self nonexpansive mappings, U.P.B. Sci. Bull., Series A, 79, 49-56 (2017)] which are related to study of convergence of best proximity points for nonexpansive non-self mappings can be concluded, directly, from the convergence results of fxed points for nonexpansive self mappings and so they are not real generalizations. These techniques leads us to introduce a semi-cyclic contractions and therein prove the existence of best proximity points.