Özet:

Abstract Aronszajn and Panitchpakdi developed hyperconvex metric spaces to expand Hahn-theorem Banach's beyond the real line to more generic spaces. The aim of this short article is to collect and combine basic notions and results in the fixed point theory in the context of hyperconvex metric spaces. In this paper, we first introduce the definitions of hyperconvex metric spaces, nonexpansive retract, externally hyperconvex and bounded subsets, and admissible subsets. We shall review and explore some fundamental characteristics of hyperconvexity. Next, we introduce the Knaster–Kuratowski and Mazurkiewicz (KKM) theory in hyperconvex metric spaces and related results. Furthermore, we find the relationship between extremal points and hyperconvexity and related properties. Furthermore, we have highlighted some known consequences of our main results.Finally, we prove the characterization of the generalized metric KKM mapping principle in hyperconvex metric spaces. It is also aimed at showing that there are still enough rooms for several researchers in this interesting direction and a huge application potential. In the concluding part of the article, we have finally reiterated the well-demonstrated fact that the results presented in this article can easily be rewritten as a nonexpansive retract by making some straightforward simplifications, and it will be an inconsequential exercise, simply because the additional properties are obviously unnecessary.

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