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Structure of the fixed point set and common fixed points of asymptotically nonexpansive mappings


Authors: T. Domínguez Benavides and P. Lorenzo Ramírez
Journal: Proc. Amer. Math. Soc. 129 (2001), 3549-3557
MSC (2000): Primary 47H09, 47H10
DOI: https://doi.org/10.1090/S0002-9939-01-06141-X
Published electronically: May 3, 2001
MathSciNet review: 1860487
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Abstract:

Let $X$ be a Banach space, $C$ a weakly compact convex subset of $X$and $T:C\to C$ an asymptotically nonexpansive mapping. Under the usual assumptions on $X$ which assure the existence of fixed point for $T$, we prove that the set of fixed points is a nonexpansive retract of $C$. We use this result to prove that all known theorems about existence of fixed point for asymptotically nonexpansive mappings can be extended to obtain a common fixed point for a commuting family of mappings. We also derive some results about convergence of iterates.


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Additional Information

T. Domínguez Benavides
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, Sevilla 41080, Spain
Email: tomasd@cica.es

P. Lorenzo Ramírez
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, Sevilla 41080, Spain
Email: ploren@cica.es

DOI: https://doi.org/10.1090/S0002-9939-01-06141-X
Keywords: Nonexpansive mapping, asymptotically nonexpansive mapping, retraction, common fixed points, convergence of iterates
Received by editor(s): April 10, 2000
Published electronically: May 3, 2001
Additional Notes: This research is partially supported by D.G.I.C.Y.T. PB 96-1338-C01-C02 and J.A. FQM 0127.
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2001 American Mathematical Society

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