Means on commutative semigroups and nonlinear ergodic theorems
Means on commutative semigroups and nonlinear ergodic theorems
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交换半群和非线性遍历定理的均值
DOI：
10.1016/0022-247x(85)90237-9
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发表时间：
1985
影响因子：
1.3
通讯作者：
W. Takahashi
中科院分区：
文献类型：
--
作者：
Kazuo Kido;W. Takahashi
设C是真实的Banach空间E的非空闭凸子集。则称T：C-+ C在C上是非扩张的，如果对所有x，y ∈ C，有/ITX-Tyli d//x-yll.我们用F（T）表示C上映射T的不动点集。令Y=（S（t）：I30）是一类C到自身的非扩张映射族，使得S（0）= Z，S（t+ s）= S（t）S（s）对所有t，SE [0，ω），且S（t）x在t E [0，ω）中连续. Baillon [1]在Hilbert空间的框架下证明了非扩张映射的第一非线性遍历定理。Baillon [a]、Bruck [S]、Hirano [7]和赖希[13]将这个定理推广到了Banach空间。特别是Bruck [S]研究了T的轨道与初值无关的渐近性，并给出了Banach空间中遍历定理的一个简单证明。布鲁克的证明是优雅的，并介绍了一些高度原创的想法，这是一定要找到进一步的应用。另一方面，Brezis和Browder [3]，Rode [141]，Takahashi [151]等研究了Hilbert空间中非扩张映射半群的非线性遍历定理。特别是罗德发现了一个序列的手段半群，推广的塞萨罗手段正整数，使相应的序列映射收敛到一个投影到一组共同的不动点。最近，Hirano，Kido和Takahashi [9]建立了Banach空间中非扩张映射半群的非线性Rode型遍历定理。本文利用Bruck [5]的思想，研究了Banach空间中非扩张映射半群轨道的渐近性态。我们首先在半群上找到一个平均序列（，u=），推广了塞萨罗平均，使得对于公共不动点集的每个弱邻域W，对应的平均向量X1包含在W中，且cr足够大;见定理1。这是585的概括
Let C be a nonempty closed convex subset of a real Banach space E. Then, T: C-+ C is called nonexpansive on C, if/I TX-Tyli d//x-yll for all x, y E C. We denote by F (T) the set of fixed points of a mapping T on C. Let, Y=(S (t): I 3 0) be a family of nonexpansive mappings of C into itself such that S (0)= Z, S (t+ s)= S (t) S (s) for all t, SE [0, co), and S (t) x is continuous in t E [0, cc) for each x E C. Then,, Y is said to be a nonexpansive semigroup on C.Baillon [1 J proved the first nonlinear ergodic theorem for nonexpansive mappings in the framework of Hilbert spaces. This theorem was extended to Banach spaces by Baillon [a], Bruck [S], Hirano [7], and Reich [13]. Especially, Bruck [S] studied the asymptotic behavior of orbits of T independent of initial values and gave a simple proof of the ergodic theorem in Banach spaces. Bruck’s proof is elegant and introduces a number of highly original ideas which are certain to find further applications. On the other hand, nonlinear ergodic theorems for a semigroup of nonexpansive mappings in a Hilbert space were studied by Brezis and Browder [3], Rode [141, Takahashi [151, and the others. Especially Rode found a sequence of means on the semigroup, generalizing the Cesaro means on positive integers, such that the corresponding sequence of mappings converges to a projection onto the set of common fixed points. Recently, Hirano, Kido, and Takahashi [9] established a nonlinear ergodic theorem of Rode’s type for semigroups of nonexpansive mappings in Banach spaces. In this paper, by using ideas of Bruck [5], we study the asymptotic behavior of orbits of a semigroup of nonexpansive mappings in a Banach space. We first find a sequence (, u=} of means on the semigroup, generalizing the Cesaro means, such that, for each weak neighborhood W of the set of common fixed points, the corresponding mean vectors X, are contained in W for sufhciently large cr; see Theorem 1. This is a generalization of 585