OICC Press Mathematical Sciences 2251-7456 2026 12 31 A Survey: Kannan Fixed Point Theorem in Hyperconvex Spaces 10.57647/mathsci.jdwx.q7r2.0034 EN Moosa Gabeleh Department of Mathematics, Faculty of Basic Sciences, Ayatollah Boroujerdi University, Boroujerd, Iran https://orcid.org/0000-0001-5439-1631 Jack Markin 1440 8th St. Golden, Co 80401, Colorado, United States https://orcid.org/0000-0002-2339-4095 Journal Article 2026 12 31 We prove that every closed ball in a non-reflexive Banach space ℓ∞ is a proximinal set. We also define a concept of ℏ-proximal quasi-normal structure on a pair of nonempty and admissible subsets of a metric space and prove that every nonempty and admissible pair of subsets of a hyperconvex metric space has the ℏ-proximal quasi-normal structure. We then discuss best proximity theory by considering two family of cyclic maps entitled cyclic Kannan contractions and cyclic relatively Kannan nonexpansive mappings and show that every cyclic Kannan contraction defined on a union of two nonempty and admissible subsets of a hyperconvex metric space has a best proximity point and by using the geometric notion of ℏ-proximal quasi-normal structure, we conclude a similar result for cyclic relatively Kannan nonexpansive maps which preserves distance. In a special case, we obtain Baillon’s fixed point theorem for Kannan nonexpansive self-mapping.