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Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces

Author(s): Naoki Shioji; Wataru Takahashi
Journal: Proc. Amer. Math. Soc. 125 (1997), 3641-3645.
MSC (1991): Primary 47H09, 49M05
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Abstract: In this paper, we study the convergence of the sequence defined by

$\begin{displaymath}% x_0\in C , \;\; x_{n+1} = \alpha _{n}x + (1-\alpha _{n})Tx_n , \;\; n=0,1,2, \ldots,\end{displaymath}$

where $0 \leq \alpha _n \leq 1$ and $T$ is a nonexpansive mapping from a closed convex subset of a Banach space into itself.

References:

1.: S. Banach, Théorie des opérations linéaires, Monografie Mat., PWN, Warszawa, 1932.
2.: B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967), 957-961. MR 36:2022
3.: G. G. Lorentz, A contribution to the theory of divergent series, Acta Math. 80 (1948), 167-190. MR 10:367e
4.: S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), 287-292. MR 82a:47050
5.: S. Reich, Some problems and results in fixed point theory, Contemp. Math. 21 (1983), 179-187. MR 85e:47082
6.: W. Takahashi and Y. Ueda, On Reich's strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl. 104 (1984), 546-553. MR 86c:47070
7.: R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992), 486-491. MR 93c:47069

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Additional Information:

Naoki Shioji
Affiliation: Faculty of Engineering, Tamagawa University, Tamagawa-Gakuen, Machida, Tokyo 194, Japan
Email: shioji@eng.tamagawa.ac.jp

Wataru Takahashi
Affiliation: Department of Information Science, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152, Japan
Email: wataru@is.titech.ac.jp

DOI: 10.1090/S0002-9939-97-04033-1
PII: S 0002-9939(97)04033-1
Keywords: Nonexpansive mappings, strong convergence, Banach limits, iteration
Received by editor(s): February 6, 1996
Received by editor(s) in revised form: July 15, 1996
Communicated by: Palle E. T. Jorgensen
Copyright of article: Copyright 1997, American Mathematical Society


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