On some consequences of Nadler's fixed point problem

Abstract

Recently, N. Bunlue al. [N. Bunlue, Y.J. Cho, S. Suantai, Best proximity point theorems for proximal multi-valued contractions, Filomat, 35;6, (2021) 1889-1897] studied the existence of best proximity points for proximal multi-valued contractions as well as proximal multi-valued nonexpansive mappings in the framework of metric and Banach spaces, respectively. In this paper we show that the well-known Nadler's fixed point theorem implies the best proximity point results of such proximal multi-valued contractions and nonexpansive non-self mappings. Moreover, in the case that the considered non-self mapping is proximal multi-valued nonexpansive, we drop the conditions of semi-sharp proximinality as well as $q$-starshepedness which were assumed in a main result of aforementioned paper.