关于伪压缩映象不动点迭代的一点注记

设$K$是自反的并且具有一致Gateaux可微范数的Banach空间$E$的非空有界闭凸子集.设$T:K\rightarrow K$是一致连续的伪压缩映象.假设$K$的每一非空有界闭凸子集对非扩张映象具有不动点性质.设$\{\lambda_n\}$是$(0,\frac{1}{2}]$中序列满足: (i) $\lim_{n\rightarrow \infty}\lambda_n=0$; (ii) $\sum_{n=0}^{\infty}\lambda_n=\infty$.任给$x_1\in K$,定义迭代序列$\{x_n\}$为:$x_{n+1}=(1-\lambda_n)x_n+\lambda_nTx_n-\lambda_n(x_n-x_1),n\geq 1.$若$\lim_{n\rightarrow \infty}\|x_n-Tx_n\|=0$, 则上述迭代产生的$\{x_n\}$强收敛到$T$的不动点.

A Note on Approximating Fixed Points of Pseudocontractive Mappings

Let $K$ be a nonempty bounded closed convex subset of a real reflexive Banach space $E$ with a uniformly Gateaux differentiable norm. Let $T:K\rightarrow K$ be a uniformly continuous pseudocontractive mapping. Suppose every closed convex and bounded subset of $K$ has the fixed point property for nonexpansive mappings. Let $\{\lambda_n\}\subset (0,\frac{1}{2}]$ be a sequence satisfying the conditions: (i) $\lim_{n\rightarrow \infty}\lambda_n=0$; (ii) $\sum_{n=0}^{\infty}\lambda_n=\infty$. Let the sequence $\{x_n\}$ be generated from arbitrary $x_1\in K$ by $x_{n+1}=(1-\lambda_n)x_n+\lambda_nTx_n-\lambda_n(x_n-x_1)$, $n\geq 1$. Suppose $\lim_{n\rightarrow \infty}\|x_n-Tx_n\|=0$. Then $\{x_n\}$ converges strongly to a fixed point of $T$.