具误差隐格式迭代逼近严格伪压缩映像族公共不动点 (英)

Implicit Iteration Process with Errors for Common Fixed Points of a Finite Family of Strictly Pseudocontractive Maps

中文关键词:  严格伪压缩映像  具误差隐格式迭代  公共不动点  收敛定理.

英文关键词:strictly pseudocontractive mappings  implicit iteration process with error  common fixed points  convergence theorems.

中文摘要:

      设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ $0<\alpha\leq\alpha_n\leq 1$; (ii)\ $\sum_{n=1}^{\infty}(1-\alpha_n)=+\infty$; (iii)\ $\sum_{n=1}^{\infty}\|u_n\|<+\infty.$\\ 设$x_0\in K$, $\{x_n\}$ 由下式定义$$x_n=\alpha_nx_{n-1}+(1-\alpha_n)T_nx_n+u_{n-1},\ \ n\geq 1,$$其中$T_n=T_n{\rm mod}N$, 则有下面结论 (i)\ $\lim_{n\rightarrow\infty}\|x_n-p\|$ 存在,对所有$p\in F$; (ii)\ $\lim_{n\rightarrow\infty}d(x_n, F)$存在, 当$d(x_n,F)=\inf_{p\in F}\|x_n-p\|$;(iii)\ $\liminf_{n\rightarrow\infty}\|x_n-T_nx_n\|=0$.\\文中另一个结果是, 如果$\{x_n\}\subset[1-2^{-n}, 1]$, 则$\{x_n\}$收敛. 文中结果改进与扩展了Osilike(2004)最近的结果, 证明方法也不同.

英文摘要:

      Let $E$ be a real Banach space and $K$ be a nonempty closed convex subset of $E$. Let $\{T_i\}_{i=1}^N$ be $N$ strictly pseudocontractive self-maps of $K$ such that $F=\bigcap_{i=1}^N F(T_i)\neq\emptyset$, where $F(T_i)=\{x \in K:T_ix=x\}$, $\{\alpha_n\}\subset [0,1]$ be a real sequence, and $\{u_n\}\subset K$ be a sequence satisfying the conditions: (i)~$0