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

We first prove that if (G, H) is a nonempty, compact and hyperconvex pair of subsets of a hyperconvex metric space (M, d), then every cyclic relatively u-continuous mapping T defined on G ∪ H has a best proximity point. The same result is valid for the case that T is the noncyclic relatively u-continuous map and (G, H) is a semi-sharp proximinal pair to obtain the existence of best proximity pairs. We then consider the class of relatively H-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 ℓ_∞ 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.