Revisiting Best Proximity Results of Relatively Meir-Keeler Condensing Operators in Hyperconvex Spaces

Abstract

We first prove that if $(\mathcal G, \mathcal H)$ is a nonempty, compact and hyperconvex pair of subsets of a hyperconvex metric space $(\mathcal M,d)$, then every cyclic relatively $u$-continuous mapping $T$ defined on $\mathcal G\cup\mathcal H$ has a best proximity point. A same result is valid for the case that $T$ is the noncyclic relatively $u$-continuous map and $(\mathcal G, \mathcal H)$ is a semi-sharp proximinal pair to obtain the existence of best proximity pairs. We then consider the class of relatively Meir-keeler condensing operators by applying a concept of measure of noncompactness in the framework of hyperconvex spaces and in a special case in the nonreflexive Banach space $\ell_\infty$ and revisit the previous best proximity point (pair) results of the paper by M. Gabeleh and C. Vetro [M. Gabeleh, C. Vetro, A new extension of Darbo's fixed point theorem using relatively Meir-Keeler condensing operators, Bull. Aust. Math. Soc., 98, (2018) 286-297]. Examples are given to support our main discussions.