WEAK CONVERGENCE THEOREMS FOR NONEXPANSIVE MAPPINGS IN BANACH-SPACES
WEAK CONVERGENCE THEOREMS FOR NONEXPANSIVE MAPPINGS IN BANACH-SPACES
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DOI：
10.1016/0022-247x(79)90024-6
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发表时间：
1979-01-01
影响因子：
1.3
通讯作者：
REICH, S
中科院分区：
文献类型：
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作者：
REICH, S
令 C 为 Banach 空间 E 的闭凸子集，并令 T: C-C 为非扩张的（即，对于 C 中的所有 x 和 y，j TX-Ty 1< 1 N-y 1）。 J.-B。 Baillon [l] 最近表明，如果 E= D，1< p< co，并且 T 有一个固定点，则对于 C 中的每个 x，迭代 [T” s} 的 Cesaro 均值弱收敛到 T 的固定点。本文的目的是指出他的想法也导致以下结果。回想一下，序列 {x,} CE 是弱几乎收敛的（参见[9]） toy EE if (~~~~ xick)/n-y 均匀地在 k 中，并且如果 R (.Z+-4)= E 和/.xx~]<~~~-~~+ v (y,-y~)~ for ally, EAZx,, i= l, 2, andr> O. 算子 ACE x E 被称为 m 累加。 定理 1. 令 C 为具有 Frechet 可微范数的一致凸 Banach 空间 E 的闭凸子集。 C 是具有 $ 固定点的非扩张映射，则 (T* x} 弱几乎收敛于 T 的固定点。
Let C be a closed convex subset of a Banach space E, and let T: C-C be nonexpansive (that is, j TX-Ty 1< 1 N-y 1 for all x and y in C). J.-B. Baillon [l] has recently shown that if E= D, 1< p< co, and T has a fixed point, then for each x in C the Cesaro means of the iterates [T” s} convegre weakly to a fixed point of T. The purpose of this note is to point out that his ideas also lead to the following results. Recall that a sequence {x,} CE is weakly almost convergent (cf.[9]) toy EE if (~~~~ xick)/n-y uniformly in k, and that an operator ACE x E is said to be m-accretive if R (. Z+-4)= E and/. xx~]<~~~-~~+ v (y,-y~)~ for ally, EAZx,, i= l, 2, andr> O.THEOREM 1. Let C be a closed convex subset of a uniformly convex Banach space E with a Frechet differentiable norm. If T: C+ C is a nonexpansive mapping with a $ xed point, then (T* x} is weakly almosf convergent to a fixed point of T.