非线性算子具误差的隐迭代程序的强收敛性

数学物理学报 ›› 2010, Vol. 30 ›› Issue (4): 1062-1070.

非线性算子具误差的隐迭代程序的强收敛性

杨理平¹, 谢湘生²

1.广东工业大学应用数学学院广州 510090|2.广东工业大学系统工程研究所广州 510090

收稿日期:2008-04-24 修回日期:2009-02-23 出版日期:2010-07-25 发布日期:2010-07-25
基金资助:
国家自然科学基金 (70871028, 70671029) 资助

Strong Convergence of Implicit Iterative Scheme with Errors for Nonlinear Mappings

YANG Li-Ping¹, XIE Xiang-Sheng²

1.Faculty of Applied Mathematics, Guangdong University of Technology, Guangzhou 510090;
2.System Engineering Institute, Guangdong University of Technology, Guangzhou 510090

Received:2008-04-24 Revised:2009-02-23 Online:2010-07-25 Published:2010-07-25
Supported by:
国家自然科学基金 (70871028, 70671029) 资助

摘要/Abstract

摘要：

设 K 是一致凸 Banach 空间中的非空闭凸子集, T_i: K→K (i=1, 2, …, N) 是有限族完全渐近非扩张映象. 对任意的 x₀∈K, 具误差的隐迭代序列 {x_n} 为: x_n=α_nx_n_-1+β_nT_n^k x_n+γ_nu_n, n≥1, 其中{α_n}, {β_n}, {γ_n} ( [0, 1] 满足 α_n+β_n+γ_n=1, {u_n} 是K 中的有界序列. 在一定的条件下, 该文建立了隐迭代序列 {x_n} 的强收敛性. 得到隐迭代序列 {x_n} 强收敛于有限族完全渐近非扩张映象公共不动点的充要条件. 所得结果改进和推广了Shahzad 与 Zegeye, Zhou 与 Chang, Chang, Tan, Lee 与Chan 等人的相应结果.

关键词: 完全渐近非扩张映象, 具误差的隐迭代序列, 公共不动点, 一致凸 Banach 空间, 半紧

Abstract:

Let K be a nonempty closed convex subset of a real uniformly convex Banach space E and T_i: K → K~ (i=1, 2, …, N) be a finite family of total asymptotically nonexpansive mappings. Let the implicit iteration scheme {x_n} generated from arbitrary x₀∈K by x_n=α_nx_n_-1+β_nT_n^k x_n+γ_nu_n, n≥1, where {α_n}, {β_n}, {γ_n} ( [0, 1] and α_n+β_n+γ_n=1, {u_n} is bounded in K. The purpose of this paper is to study several strong convergence of the implicit iteration scheme {x_n} under certain conditions. It derives a necessary and sufficient condition for the strong convergence of this iteration scheme to a common fixed point of these mappings. The results of this paper improve and extend the corresponding results of Shahzad and Zegeye, Zhou and Chang, Chang, Tan, Lee and Chan.

Key words: Total asymptotically nonexpansive mappings, Implicit iterative sequence with errors, Common fixed point, Uniformly convex Banach spaces, Demi-compact

中图分类号:

47H09

杨理平, 谢湘生. 非线性算子具误差的隐迭代程序的强收敛性[J]. 数学物理学报, 2010, 30(4): 1062-1070.

YANG Li-Ping, XIE Xiang-Sheng. Strong Convergence of Implicit Iterative Scheme with Errors for Nonlinear Mappings[J]. Acta mathematica scientia,Series A, 2010, 30(4): 1062-1070.

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