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Iterative approximation of fixed points of Lipschitz pseudocontractive maps


Author: C. E. Chidume
Journal: Proc. Amer. Math. Soc. 129 (2001), 2245-2251
MSC (2000): Primary 47H09, 47J05, 47J25
DOI: https://doi.org/10.1090/S0002-9939-01-06078-6
Published electronically: March 20, 2001
MathSciNet review: 1823906
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Abstract:

Let $E$ be a $q$-uniformly smooth Banach space possessing a weakly sequentially continuous duality map (e.g., $\ell_p, 1<p<\infty$). Let $T$ be a Lipschitzian pseudocontractive selfmapping of a nonempty closed convex and bounded subset $K$ of $E$ and let $\omega\in K$ be arbitrary. Then the iteration sequence $\{z_n\}$ defined by $z_0\in K, z_{n+1}=(1-\mu_{n+ 1})\omega + \mu_{n+1}y_n; y_n = (1-\alpha_n)z_n+\alpha_nTz_n$, converges strongly to a fixed point of $T$, provided that $\{\mu_n\}$ and $\{\alpha_n\}$ have certain properties. If $E$ is a Hilbert space, then $\{z_n\}$ converges strongly to the unique fixed point of $T$ closest to $\omega$.


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Additional Information

C. E. Chidume
Affiliation: The Abdus Salam International Centre for Theoretical Physics, P.O. Box 586, Trieste, Italy
Email: chidume@ictp.trieste.it

DOI: https://doi.org/10.1090/S0002-9939-01-06078-6
Keywords: Pseudocontractive operators, $q$-uniformly smooth spaces, duality maps, weak sequential continuity.
Received by editor(s): September 27, 1999
Published electronically: March 20, 2001
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2001 American Mathematical Society

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