Banach空间中有限族渐进伪压缩映象的迭代程序

Let E be a real Banach space and K be a nonempty closed convex and bounded subset of E. Let Ti ： K→ K, i=1, 2,... ,N, be N uniformly L-Lipschitzian, uniformly asymptotically regular with sequences {ε^（i）n} and asymptotically pseudocontractive mappings with sequences {κ^（i）n}, where {κ^（i）n} and {ε^（i）n}, i = 1, 2,... ,N, satisfy certain mild conditions. Let a sequence {xn} be generated from x1 ∈ K by zn：= （1-μn）xn＋μnT^nnxn, xn＋1 ：= λnθnx1＋ 1 - λn（1 ＋ θn）]xn ＋ λnT^nnzn for all integer n ≥ 1, where Tn = Tn（mod N）, and {λn}, {θn} and {μn} are three real sequences in 0, 1] satisfying appropriate conditions. Then ｜｜xn- Tixn｜｜→ 0 as n→∞ for each l ∈ {1, 2,..., N}. The results presented in this paper generalize and improve the corresponding results of Chidume and Zegeye, Reinermann, Rhoades and Schu.

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

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$^{13]}$.