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# On Best Proximity Points in Metric and Banach Spaces

Published:2011-02-25
Printed: Jun 2011
• Rafa Espínola,
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, 41080-Sevilla, Spain
• Aurora Fernández-León,
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, 41080-Sevilla, Spain
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## Abstract

In this paper we study the existence and uniqueness of best proximity points of cyclic contractions as well as the convergence of iterates to such proximity points. We take two different approaches, each one leading to different results that complete, if not improve, other similar results in the theory. Results in this paper stand for Banach spaces, geodesic metric spaces and metric spaces. We also include an appendix on CAT$(0)$ spaces where we study the particular behavior of these spaces regarding the problems we are concerned with.
 MSC Classifications: 54H25 - Fixed-point and coincidence theorems [See also 47H10, 55M20] 47H09 - Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc.