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Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings


Authors: C. E. Chidume, Jinlu Li and A. Udomene
Journal: Proc. Amer. Math. Soc. 133 (2005), 473-480
MSC (2000): Primary 47H06, 47H09, 47J05, 47J25
DOI: https://doi.org/10.1090/S0002-9939-04-07538-0
Published electronically: September 2, 2004
MathSciNet review: 2093070
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Abstract: Let $E$ be a real Banach space with a uniformly Gâteaux differentiable norm possessing uniform normal structure, $K$ be a nonempty closed convex and bounded subset of $E$, $T: K \longrightarrow K$ be an asymptotically nonexpansive mapping with sequence $\{k_n\}_n\subset [1, \infty)$. Let $u\in K$ be fixed, $\{t_n\}_n \subset (0, 1)$ be such that $\lim\limits_{n\to \infty}t_n = 1$, $t_nk_n < 1$, and $\lim\limits_{n\to \infty}\frac{k_n - 1}{k_n-t_n} =0$. Define the sequence $\{z_n\}_n$ iteratively by $z_0\in K$, $ z_{n+1}= (1-\frac{t_n}{k_n})u + \frac{t_n}{k_n}T^nz_n, \>n= 0, 1, 2, ..._. $ It is proved that, for each integer $n \geq 0$, there is a unique $x_n \in K$ such that $ x_n= (1-\frac{t_n}{k_n})u + \frac{t_n}{k_n}T^nx_n.$If, in addition, $\lim\limits_{n\to \infty}\Vert x_n - Tx_n\Vert = 0$ and $\lim\limits_{n\to \infty}\Vert z_n - Tz_n\Vert = 0$, then $\{z_n\}_n$ converges strongly to a fixed point of $T$.


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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

Jinlu Li
Affiliation: Department of Mathematics, Shawnee State University, Portsmouth, Ohio 45662
Email: jli@shawnee.edu

A. Udomene
Affiliation: Department of Mathematics, Statistics, Computer Science, University of Port Harcourt, Port Harcourt, Nigeria
Email: EpsilonAni@aol.com

DOI: https://doi.org/10.1090/S0002-9939-04-07538-0
Keywords: Asymptotically nonexpansive mappings, fixed points, uniformly Lipschitzian mappings
Received by editor(s): June 12, 2003
Received by editor(s) in revised form: October 6, 2003
Published electronically: September 2, 2004
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2004 American Mathematical Society

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