Iterative Schemes for a Family of Finite Asymptotically Pseudocontractive Mappings in Banach Spaces

Abstract:

      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$^{[13]}$.