Near-infinity concentrated norms and the fixed point property for nonexpansive maps on closed, bounded, convex sets

In this paper we define the concept of a near-infinity concentrated norm on a Banach space X with a boundedly complete Schauder basis. When k · k is such a norm, we prove that (X, k · k) has the fixed point property (FPP); that is, every nonexpansive self-mapping defined on a closed, bounded, convex...

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Autores: Castillo Santos, Francisco Eduardo, Dowling, Patrick N., Fetter Nathansky, Helga Andrea, Japón Pineda, María de los Ángeles, Lennard, Christopher J., Sims, Brailey, Turett, Barry
Tipo de recurso: artículo
Estado:Versión enviada para evaluación y publicación
Fecha de publicación:2018
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/80137
Acceso en línea:https://hdl.handle.net/11441/80137
https://doi.org/10.1016/j.jfa.2018.04.007
Access Level:acceso abierto
Palabra clave:Fixed point property
Nonexpansive mappings
Renorming theory
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spelling Near-infinity concentrated norms and the fixed point property for nonexpansive maps on closed, bounded, convex sets Castillo Santos, Francisco Eduardo Dowling, Patrick N. Fetter Nathansky, Helga Andrea Japón Pineda, María de los Ángeles Lennard, Christopher J. Sims, Brailey Turett, Barry Fixed point property Nonexpansive mappings Renorming theory In this paper we define the concept of a near-infinity concentrated norm on a Banach space X with a boundedly complete Schauder basis. When k · k is such a norm, we prove that (X, k · k) has the fixed point property (FPP); that is, every nonexpansive self-mapping defined on a closed, bounded, convex subset has a fixed point. In particular, P.K. Lin’s norm in l1 [P.K. Lin, There is an equivalent norm on l1 that has the fixed point property, Nonlinear Anal. 68 (8) (2008), 2303-2308] and the norm νp(·) (with p = (pn) and limn pn = 1) introduced in [P.N. Dowling, W.B. Johnson, C.J. Lennard and B. Turett, The optimality of James’s distortion theorems, Proc. Amer. Math. Soc. 124 (1) (1997), 167-174] are examples of near-infinity concentrated norms. When νp(·) is equivalent to the l1-norm, it was an open problem as to whether (l1, νp(·)) had the FPP. We prove that the norm νp(·) always generates a nonreflexive Banach space X = R ⊕p1(R ⊕p2(R ⊕p3. . . )) satisfying the FPP, regardless of whether νp(·) is equivalent to the l1-norm. We also obtain some stability results. Consejo Nacional de Ciencia y Tecnología (México) Ministerio de Ciencia, Innovación y Universidades Junta de Andalucía Elsevier https://hdl.handle.net/11441/80137 https://doi.org/10.1016/j.jfa.2018.04.007
title Near-infinity concentrated norms and the fixed point property for nonexpansive maps on closed, bounded, convex sets
spellingShingle Near-infinity concentrated norms and the fixed point property for nonexpansive maps on closed, bounded, convex sets
Castillo Santos, Francisco Eduardo
Fixed point property
Nonexpansive mappings
Renorming theory
title_short Near-infinity concentrated norms and the fixed point property for nonexpansive maps on closed, bounded, convex sets
title_full Near-infinity concentrated norms and the fixed point property for nonexpansive maps on closed, bounded, convex sets
title_fullStr Near-infinity concentrated norms and the fixed point property for nonexpansive maps on closed, bounded, convex sets
title_full_unstemmed Near-infinity concentrated norms and the fixed point property for nonexpansive maps on closed, bounded, convex sets
title_sort Near-infinity concentrated norms and the fixed point property for nonexpansive maps on closed, bounded, convex sets
author Castillo Santos, Francisco Eduardo
author_facet Castillo Santos, Francisco Eduardo
Dowling, Patrick N.
Fetter Nathansky, Helga Andrea
Japón Pineda, María de los Ángeles
Lennard, Christopher J.
Sims, Brailey
Turett, Barry
author_role author
author2 Dowling, Patrick N.
Fetter Nathansky, Helga Andrea
Japón Pineda, María de los Ángeles
Lennard, Christopher J.
Sims, Brailey
Turett, Barry
author2_role author
author
author
author
author
author
topic Fixed point property
Nonexpansive mappings
Renorming theory
topic_facet Fixed point property
Nonexpansive mappings
Renorming theory
description In this paper we define the concept of a near-infinity concentrated norm on a Banach space X with a boundedly complete Schauder basis. When k · k is such a norm, we prove that (X, k · k) has the fixed point property (FPP); that is, every nonexpansive self-mapping defined on a closed, bounded, convex subset has a fixed point. In particular, P.K. Lin’s norm in l1 [P.K. Lin, There is an equivalent norm on l1 that has the fixed point property, Nonlinear Anal. 68 (8) (2008), 2303-2308] and the norm νp(·) (with p = (pn) and limn pn = 1) introduced in [P.N. Dowling, W.B. Johnson, C.J. Lennard and B. Turett, The optimality of James’s distortion theorems, Proc. Amer. Math. Soc. 124 (1) (1997), 167-174] are examples of near-infinity concentrated norms. When νp(·) is equivalent to the l1-norm, it was an open problem as to whether (l1, νp(·)) had the FPP. We prove that the norm νp(·) always generates a nonreflexive Banach space X = R ⊕p1(R ⊕p2(R ⊕p3. . . )) satisfying the FPP, regardless of whether νp(·) is equivalent to the l1-norm. We also obtain some stability results.
publishDate 2018
format article
status_str submittedVersion
url https://hdl.handle.net/11441/80137
https://doi.org/10.1016/j.jfa.2018.04.007
eu_rights_str_mv openAccess
publisher Elsevier
institution Universidad de Sevilla (US)
collection idUS. Depósito de Investigación de la Universidad de Sevilla
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
instname_str Universidad de Sevilla (US)
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publishDateSort 2018
author_browse Castillo Santos, Francisco Eduardo
Dowling, Patrick N.
Fetter Nathansky, Helga Andrea
Japón Pineda, María de los Ángeles
Lennard, Christopher J.
Sims, Brailey
Turett, Barry
publisherStr Elsevier
score 6,9008884

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