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Fixed points of nonexpansive mappings in spaces of continuous functions


Authors: T. Domínguez Benavides and María A. Japón Pineda
Journal: Proc. Amer. Math. Soc. 133 (2005), 3037-3046
MSC (2000): Primary 47H09, 47H10, 46B20, 46B42, 46E05
DOI: https://doi.org/10.1090/S0002-9939-05-08149-9
Published electronically: April 20, 2005
MathSciNet review: 2159783
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Abstract: Let $K$ be a compact metrizable space and let $C(K)$ be the Banach space of all real continuous functions defined on $K$ with the maximum norm. It is known that $C(K)$ fails to have the weak fixed point property for nonexpansive mappings (w-FPP) when $K$ contains a perfect set. However the space $C(\omega ^{n}+1)$, where $n\in \mathbb{N}$ and $\omega $ is the first infinite ordinal number, enjoys the w-FPP, and so $C(K)$ also satisfies this property if $K^{(\omega )}=\emptyset $. It is unknown if $C(K)$ has the w-FPP when $K$ is a scattered set such that $K^{(\omega )}\not =\emptyset $. In this paper we prove that certain subspaces of $C(K)$, with $K^{(\omega )}\not = \emptyset $, satisfy the w-FPP. To prove this result we introduce the notion of $\omega $-almost weak orthogonality and we prove that an $\omega $-almost weakly orthogonal closed subspace of $C(K)$ enjoys the w-FPP. We show an example of an $\omega $-almost weakly orthogonal subspace of $C(\omega ^{\omega }+1)$ which is not contained in $C(\omega ^{n}+1)$ for any $n\in \mathbb{N}$.


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Additional Information

T. Domínguez Benavides
Affiliation: Departamento de Análisis Matemático, University of Seville, P.O. Box 1160, 41080-Seville, Spain
Email: tomasd@us.es

María A. Japón Pineda
Affiliation: Departamento de Análisis Matemático, University of Seville, P.O. Box 1160, 41080-Seville, Spain
Email: japon@us.es

DOI: https://doi.org/10.1090/S0002-9939-05-08149-9
Received by editor(s): May 30, 2004
Published electronically: April 20, 2005
Additional Notes: This research was partially supported by the DGES (research project BMF2000-0344-C02-C01) and the Junta de Andalucia (project 127)
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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