Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings
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- by C. E. Chidume, Jinlu Li and A. Udomene PDF
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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, $\lim \limits _{n\to \infty }\|x_n - Tx_n\| = 0$ and $\lim \limits _{n\to \infty }\|z_n - Tz_n\| = 0$, then $\{z_n\}_n$ converges strongly to a fixed point of $T$.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
- 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
- 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
- © Copyright 2004 American Mathematical Society
- 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
- MathSciNet review: 2093070