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Zbl 1219.47135
Yao, Yonghong; Liou, Yeong-Cheng
(Yao, Yong-hong; Liou, Yeongcheng)
Strong convergence to common fixed points of a finite family of asymptotically nonexpansive mappings. (Strong convergence to common fixed points of a finite family of asymptotically nonexpansive map.) (English)
[J] Taiwanese J. Math. 11, No. 3, 849-865 (2007). ISSN 1027-5487

Summary: Suppose that $E$ is a real Banach space with uniform normal structure and suppose that $E$ has a uniformly Gâteaux differentiable norm. Let $C$ be a nonempty closed convex and bounded subset of $E$. Let $T_1,T_2,\dots,T_r:C\to C$ be a finite family of asymptotically nonexpansive mappings. In this paper, we suggest and analyze an iterative algorithm for $\{T_i\}_{i=1}^r$. We show the convergence of the proposed algorithm to a common fixed point $p\in\bigcap_{i=1}^r F(T_i)$ which is the unique solution of some variational inequality. Our results can be considered as an refinement and improvement of many known results.

MSC 2000:: *47J25 Methods for solving nonlinear operator equations (general)
47H05 Monotone operators (with respect to duality)
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: uniformly Gâteaux differentiable norm; finite family of asymptotically nonexpansive mappings; common fixed point; strong convergence

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