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

Published electronically:
May 3, 2001

MathSciNet review:
1860487

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Let be a Banach space, a weakly compact convex subset of and an asymptotically nonexpansive mapping. Under the usual assumptions on which assure the existence of fixed point for , we prove that the set of fixed points is a nonexpansive retract of . 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:
http://dx.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