Banach空间中有限族渐进伪压缩映象的迭代程序

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

中文关键词:  渐进不动点序列  一致渐进正则映象  渐进伪压缩映象  条件  共振点.

英文关键词:approximated fixed point sequence  uniformly asymptotically regular mapping  asymptotically pseudocontractive mapping.

中文摘要:

      设$E$是一个实Banach空间, $K$是$E$的一个非空闭凸的有界子集. 设$T_i:K\rightarrow K$, $i=1,2,\cdots,N$, 是 $N$ 个一致 $L$-Lipschitzian,具有序列$\{\varepsilon_n^{(i)}\}$的一致渐进正则和具有序列$\{k_n^{(i)}\}$的渐进伪压缩映象, 其中 $\{k_n^{(i)}\}$ 和$\{\varepsilon_n^{(i)}\}$, $i=1,2,\cdots,N$满足某些适当的条件. 设序列 $\{x

英文摘要:

      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$^{[13]}$.