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Komlós' theorem and the fixed point property for affine mappings


Authors: Tomás Domínguez Benavides and Maria A. Japón
Journal: Proc. Amer. Math. Soc. 146 (2018), 5311-5322
MSC (2010): Primary 47H09, 47H10
DOI: https://doi.org/10.1090/proc/14201
Published electronically: September 17, 2018
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Abstract: Assume that $ X$ is a Banach space of measurable functions for which Komlós' Theorem holds. We associate to any closed convex bounded subset $ C$ of $ X$ a coefficient $ t(C)$ which attains its minimum value when $ C$ is closed for the topology of convergence in measure and we prove some fixed point results for affine Lipschitzian mappings, depending on the value of $ t(C)\in [1,2]$ and the value of the Lipschitz constants of the iterates. As a first consequence, for every $ L<2$, we deduce the existence of fixed points for affine uniformly $ L$-Lipschitzian mappings defined on the closed unit ball of $ L_1[0,1]$. Our main theorem also provides a wide collection of convex closed bounded sets in $ L^1([0,1])$ and in some other spaces of functions which satisfy the fixed point property for affine nonexpansive mappings. Furthermore, this property is still preserved by equivalent renormings when the Banach-Mazur distance is small enough. In particular, we prove that the failure of the fixed point property for affine nonexpansive mappings in $ L_1(\mu )$ can only occur in the extremal case $ t(C)=2$. Examples are given proving that our fixed point theorem is optimal in terms of the Lipschitz constants and the coefficient $ t(C)$.


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Additional Information

Tomás Domínguez Benavides
Affiliation: Departamento de Análisis Matemático, Universidad de Sevilla, P.O. Box 1160, 41080-Sevilla, Spain
Email: tomasd@us.es

Maria A. Japón
Affiliation: Departamento de Análisis Matemático, Universidad de Sevilla, P.O. Box 1160, 41080-Sevilla, Spain
Email: japon@us.es

DOI: https://doi.org/10.1090/proc/14201
Keywords: Measurable function spaces, fixed point, affine Lipschitzian mappings, equivalent norms, $\mu$-a.e. convergence, convergence in measure
Received by editor(s): May 31, 2017
Received by editor(s) in revised form: April 24, 2018
Published electronically: September 17, 2018
Additional Notes: The authors were partially supported by MINECO, Grant MTM2015-65242-C2-1-P and Andalusian Regional Government Grant FQM-127.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2018 American Mathematical Society

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