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Zbl 0993.47037
Shimoji, Kazuya; Takahashi, Wataru
Strong convergence to common fixed points of infinite nonexpansive mappings and applications.
(English)
[J] Taiwanese J. Math. 5, No.2, 387-404 (2001). ISSN 1027-5487

This article deals with iterations $x_{n+1}= \beta_n x+(1- \beta_n) W_nx_n$ $(n= 0,1,\dots)$, where $W_n$ $(n=1,2,\dots)$ are mappings generated by the scheme $$W_n= U_{n,1},\quad U_{n,k}= \alpha_k T_k U_{n,k+1}+(1- \alpha_k)I\quad (k= 1,\dots, n),\quad U_{n,n+1}= I,$$ $T_1,T_2,\dots$ are nonexpansive mappings of a convex subset of a Banach space $E$ into itself, $\bigcap^\infty_{n=1} \text{Fix }T_n\ne \emptyset$, $\alpha_n$ satisfy the condition $0< \alpha_n\le b< 1$, $\beta_n$ satisfy the conditions $0\le \beta_n\le 1$, $\lim_{n\to\infty} \beta_n= 0$, $\sum^\infty_{n=1} |\beta_{n+1}- \beta_n|< \infty$, $\sum^\infty_{n=1} \beta_n= \infty$. The basic result is the convergence of $x_n$ to $Px$, where $P$ is the unique sunny nonexpansive retraction from $C$ onto $\bigcap^\infty_{n=1} \text{Fix }T_n$; it is assumed that the norm in $E$ is uniformly convex and uniformly Gâteaux differentiable.
[Peter Zabreiko (Minsk)]
MSC 2000:
*47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47J25 Methods for solving nonlinear operator equations (general)
46B04 Isometric theory of Banach spaces

Keywords: sunny nonexpansive retraction; uniformly convex; uniformly Gâteaux differentiable

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