10.57647/mathsci.jdwx.q7r2.0034

A Survey: Kannan Fixed Point Theorem in Hyperconvex Spaces

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  • Moosa Gabeleh*1,
  • Jack Markin2,
  1. Department of Mathematics, Faculty of Basic Sciences, Ayatollah Boroujerdi University, Boroujerd, Iran
  2. 1440 8th St. Golden, Co 80401, Colorado, United States

Received: 2025-09-01

Revised: 2025-11-27

Accepted: 2026-01-27

Published in Issue 2026-12-31

Published Online: 2026-07-06

Copyright (c) 2026 Moosa Gabeleh, Jack Markin (Author)

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This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Gabeleh, M., & Markin, J. (2026). A Survey: Kannan Fixed Point Theorem in Hyperconvex Spaces. Mathematical Sciences. https://doi.org/10.57647/mathsci.jdwx.q7r2.0034
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Abstract

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.

Keywords

  • Best proximity point,
  • Hyperconvex space,
  • Relatively Kannan nonexpansive,
  • Prox- iminal pair
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