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Fixed Point Theory in Hyperconvex Metric Spaces

In: Topics in Fixed Point Theory

Author

Listed:
  • Rafael Espínola

    (Universidad de Sevilla, Dpto. de Análisis Matemático-IMUS)

  • Aurora Fernández-León

    (Universidad de Sevilla, Dpto. de Análisis Matemático-IMUS)

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.

Suggested Citation

  • Rafael Espínola & Aurora Fernández-León, 2014. "Fixed Point Theory in Hyperconvex Metric Spaces," Springer Books, in: Saleh Almezel & Qamrul Hasan Ansari & Mohamed Amine Khamsi (ed.), Topics in Fixed Point Theory, edition 127, chapter 0, pages 101-158, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-01586-6_4
    DOI: 10.1007/978-3-319-01586-6_4
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