A Darbo-type best proximity point theorem for proximal condensing multifunctions

In this paper, we study best proximity point problems for set-valued mappings in strictly convex Banach spaces. First, we establish an existence theorem for best proximity points of upper semicontinuous compact-valued multifunctions defined on compact convex pairs in strictly convex Banach spaces. This result is then extended to bounded, closed, and convex pairs by imposing a compactness condition on the multifunction. Motivated by these results, we introduce a new class of mappings, called proximal condensing multifunctions, which provides a natural extension of condensing operators to the framework of best proximity point theory. By using measures of noncompactness, we prove a Darbo-type best proximity point theorem for proximal condensing multifunctions. As consequences, several existence results are obtained for compact multifunctions satisfying appropriate proximal conditions. Finally, the developed theory is applied to establish existence results for a non-self Hammerstein integral inclusion and a system of differential inclusions, demonstrating the applicability of the proposed approach.