Iterative Schemes for a Family of Finite Asymptotically Pseudocontractive Mappings in Banach Spaces
Received:June 07, 2007  Revised:October 30, 2007
Key Word: approximated fixed point sequence   uniformly asymptotically regular mapping   asymptotically pseudocontractive mapping.  
Fund ProjectL:the National Natural Science Foundation of China (No.10771141); the Natural Science Foundation of Zhejiang Province (No.Y605191); the Natural Science Foundation of Heilongjiang Province (No.A0211) and the Scientific Research Foundation from Zhejiang Provi
Author NameAffiliation
GU Feng Institute of Applied Mathematics, Hangzhou Normal University, Zhejiang 310036, China
Department of Mathematics, Hangzhou Normal University, Zhejiang 310036, China 
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      Let $E$ be a real Banach space and $K$ be a nonempty closed convex and bounded subset of $E$. Let $T_i: K\rightarrow K$, $i=1,2,\ldots,N$, be $N$ uniformly $L$-Lipschitzian, uniformly asymptotically regular with sequences $\{\varepsilon_n^{(i)}\}$ and asymptotically pseudocontractive mappings with sequences $\{k_n^{(i)}\}$, where $\{k_n^{(i)}\}$ and $\{\varepsilon_n^{(i)}\}$, $i=1,2,\ldots,N$, satisfy certain mild conditions. Let a sequence $\{x_n\}$ be generated from $x_1\in K$ by $z_n:=(1-\mu_n)x_n \mu_nT_{n}^{n}x_n,\;x_{n 1}:=\lambda_n\theta_nx_1 [1-\lambda_n(1 \theta_n)]x_n \lambda_nT_{n}^nz_n $ for all integer $n\geqslant1$, where $T_{n}=T_{n({\rm mod}\,N)}$, and $\{\lambda_n\}$, $\{\theta_n\}$ and $\{\mu_n\}$ are three real sequences in $[0, 1]$ satisfying appropriate conditions. Then $||x_n-T_lx_n||\rightarrow 0$ as $n\rightarrow\infty$ for each $l\in\{1,2,\ldots,N\}$. The results presented in this paper generalize and improve the corresponding results of Chidume and Zegeye$^{[1]}$, Reinermann$^{[10]}$, Rhoades$^{[11]}$ and Schu$^{[13]}$.
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