Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps
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- by C. E. Chidume and H. Zegeye PDF
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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 $||x_n-Tx_n||\rightarrow 0$ 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.References
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Additional Information
- C. E. Chidume
- Affiliation: The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
- MR Author ID: 232629
- Email: chidume@ictp.trieste.it
- H. Zegeye
- Affiliation: The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
- Email: habz@ictp.trieste.it
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 831-840
- MSC (2000): Primary 47H06, 47H09, 47J05, 47J25
- DOI: https://doi.org/10.1090/S0002-9939-03-07101-6
- MathSciNet review: 2019962