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Fixed Point Theory in Hyperconvex Metric Spaces

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Topics in Fixed Point Theory

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.

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Notes

  1. 1.

    As long as the authors know, the word proximinal was proposed by Robert Phelps as a mixture of the words proximinty and minimal to embrace in one word the meaning of both words separately. It is also usual to find the word proximal in the literature meaning the same as proximinal.

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Acknowledgments

The first author wishes to acknowledge The University of Tabuk for its kind invitation to take part in The International Mathematical Workshop (Fixed Point Theory and its Applications) May 26–27, 2012. This work is motivated by that event.

Rafa Espínola and Aurora Fernández León were supported by DGES, MTM2012-34847C02-01 and Junta de Andalucía, Grant FQM-127.

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Authors and Affiliations

  1. Dpto. de Análisis Matemático-IMUS, Universidad de Sevilla, 1160, 41080, Sevilla, Spain

    Rafael Espínola & Aurora Fernández-León

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  1. Rafael Espínola
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  2. Aurora Fernández-León
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Correspondence to Rafael Espínola .

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Editors and Affiliations

  1. Department of Mathematics, University of Tabuk, Tabuk, Saudi Arabia

    Saleh Almezel

  2. Department of Mathematics, Aligarh Muslim University, Aligarh, Uttar Pradesh, India

    Qamrul Hasan Ansari

  3. Department of Mathematical Sciences, University of Texas at El Paso, El Paso, Texas, USA

    Mohamed Amine Khamsi

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Espínola, R., Fernández-León, A. (2014). Fixed Point Theory in Hyperconvex Metric Spaces. In: Almezel, S., Ansari, Q., Khamsi, M. (eds) Topics in Fixed Point Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-01586-6_4

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