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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps


Authors: C. E. Chidume and H. Zegeye
Journal: Proc. Amer. Math. Soc. 132 (2004), 831-840
MSC (2000): Primary 47H06, 47H09, 47J05, 47J25
Published electronically: August 19, 2003
MathSciNet review: 2019962
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Abstract: Let $K$ be a nonempty closed convex subset of a real Banach space $E$ and $T$ be a Lipschitz pseudocontractive self-map of $K$ with $F(T):=\{x\in K:Tx=x\}\neq \emptyset$. An iterative sequence $\{x_n\}$ is constructed for which $\vert\vert x_n-Tx_n\vert\vert\rightarrow 0$ as $n\rightarrow \infty$. If, in addition, $K$ is assumed to be bounded, this conclusion still holds without the requirement that $F(T)\neq \emptyset.$ Moreover, if, in addition, $E$ has a uniformly Gâteaux differentiable norm and is such that every closed bounded convex subset of $K$ has the fixed point property for nonexpansive self-mappings, then the sequence $\{x_n\}$ converges strongly to a fixed point of $T$. Our iteration method is of independent interest.


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Additional Information

C. E. Chidume
Affiliation: The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
Email: chidume@ictp.trieste.it

H. Zegeye
Affiliation: The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
Email: habz@ictp.trieste.it

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07101-6
PII: S 0002-9939(03)07101-6
Keywords: Normalized duality maps, uniformly G\^{a}teaux differentiable norm, pseudocontractive maps
Received by editor(s): May 27, 2002
Received by editor(s) in revised form: November 4, 2002
Published electronically: August 19, 2003
Additional Notes: The second author undertook this work when he was visiting the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, as a postdoctoral fellow.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2003 American Mathematical Society