Strong Convergence of Averaged Approximants for Asymptotically Nonexpansive Mappings in Banach Spaces

LetCbe a closed, convex subset of a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable and letTbe an asymptotically nonexpansive mapping fromCinto itself such that the setF(T) of fixed points ofTis nonempty. In this paper, we show thatF(T) is a sunny, nonexpansive retract o...

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Bibliographic Details
Published in:Journal of approximation theory 1999-03, Vol.97 (1), p.53-64
Main Authors: Shioji, Naoki, Takahashi, Wataru
Format: Article
Language:English
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Summary:LetCbe a closed, convex subset of a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable and letTbe an asymptotically nonexpansive mapping fromCinto itself such that the setF(T) of fixed points ofTis nonempty. In this paper, we show thatF(T) is a sunny, nonexpansive retract ofC. Using this result, we discuss the strong convergence of the sequence {xn} defined byxn=anx+(1−an)1/(n+1)∑nj=0Tjxnforn=0,1,2,…, wherex∈Cand {an} is a real sequence in (0, 1].
ISSN:0021-9045
1096-0430
DOI:10.1006/jath.1996.3251