Fixed Point Theory in Hyperconvex Metric Spaces

Rafael Espínola () and Aurora Fernández-León ()
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Rafael Espínola: Universidad de Sevilla, Dpto. de Análisis Matemático-IMUS
Aurora Fernández-León: Universidad de Sevilla, Dpto. de Análisis Matemático-IMUS

Chapter Chapter 4 in Topics in Fixed Point Theory, 2014, pp 101-158 from Springer

Abstract: Abstract In this chapter we propose a review of some of the most fundamental facts and properties on metric hyperconvexity in relation to Metric and Topological Fixed Point Theory. Hyperconvex metric spaces were introduced by Aronszajn and Panitchpakdi in 1956 in relation to the problem of extending uniformly continuous mappings defined between metric spaces. It was obvious from the very beginning that the structure given by the hyperconvexity of the metric to the space was a very rich one. As a consequence of that richness, a very profound and exhaustive Fixed Point Theory has been developed on hyperconvex metric spaces, especially from late eighties of the Twentieth Century by pioneering works due to Baillon, Sine and Soardi. This theory applies for single and multivalued mappings as well as for best-approximation results. Along 9 sections, we expose in a detailed and self-contained way the foundations of this theory. A final additional section, however, has been included to describe some of the newest trends on hyperconvexity and existence of fixed points.

Keywords: Nonexpansive Mapping; Multivalued Mapping; Fixed Point Theory; Nonempty Intersection; Ultrametric Space (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-01586-6_4

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DOI: 10.1007/978-3-319-01586-6_4

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