无限簇非扩张非自映象公共不动点的黏性逼近法

数学物理学报 ›› 2010, Vol. 30 ›› Issue (4): 1144-1157.

无限簇非扩张非自映象公共不动点的黏性逼近法

赵良才¹, 张石生^{1, 2}

1.宜宾学院数学系重庆宜宾 644000|2.四川大学数学系成都 610064

收稿日期:2008-10-21 修回日期:2009-09-30 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(10701059)%、浙江省自然科学基金(101016)和浙江省哲学社会科学规划常规性课题(06CGYJ21YBQ)资助

Viscosity Approximation Methods for Common Fixed Points of Infinite Nonexpansive Nonself-mappings

ZHAO Liang-Cai¹, ZHANG Shi-Sheng^{1, 2}

1.Department of Mathematics, Yibin University, Chongqing Yibin |644000;
2.Department of Mathematics, Sichuan University, Chengdu |610064

Received:2008-10-21 Revised:2009-09-30 Online:2010-07-25 Published:2010-07-25
Supported by:
国家自然科学基金(10701059)%、浙江省自然科学基金(101016)和浙江省哲学社会科学规划常规性课题(06CGYJ21YBQ)资助

摘要/Abstract

摘要：

设E是具有一致Gâteaux可微范数的严格凸的自反的Banach空间, K是E的非空闭凸子集而且是$E$的sunny非扩张收缩核.设f: K→ K是一压缩映象, P: E→ K是一sunny非扩张保核收缩, {T_n}_n=1^∞: K → E是一可数无限簇非扩张非自映象且F:=∩^∞_n=1F(T_n)≠Φ, {λ_n}是[0,1]中的非负数列. 考虑下列迭代序列
           { x_n+1=(1-α_n-β_n)x_n+α_n f(y_n)+β_nW_ny_n,
             y_n=(1-γ_n-δ_n)x_n+γ_nW_nx_n+δ_nu_n, n ≥1.

其中W_n是由P, T_n, T_n_-1, …, T₁和λ_n, λ_n_-1, …, λ₁, ∨_n≥ 1生成的W -映象.该文在较弱条件下用黏性逼近方法证明了迭代序列{x_n}强收敛于p∈ F且p是下列变分不等式
           <(I-f)p, j(p-x*)>≤ 0, ∨_x* ∈ F
的唯一解.

关键词: 非扩张非自映象, 一致Gâteaux可微范数, 黏性逼近, 公共不动点

Abstract:

Let E be a real strictly convex and reflexive Banach space with a uniformly Gâteaux differentiable norm and K be a nonempty closed convex subset of E which is also a sunny nonexpansive retract of E. Let f : K→K be a contractive mapping, P be a sunny nonexpansive retraction of $E$ onto K and {T_n}_n=1^∞: K→E be a family of countable infinite nonexpansive nonself-mappings such that the common fixed point set F:=:=∩^∞_n=1F(T_n)≠Φ and {λ_n} be a seqence of nonnegative numbers in [0,1]. Consider the following
iterative sequence
              { x_n+1=(1-α_n-β_n)x_n+α_n f(y_n)+β_nW_ny_n,
                 y_n=(1-γ_n-δ_n)x_n+γ_nW_nx_n+δ_nu_n, n ≥1.

where W_n is the W-mapping generated by P, T_n, T_n_-1,  …, T₁and λ_n, λ_n_-1, …, λ₁for n ≥ 1. It is shown that under very mild conditions on the parameters, the sequence {x_n} converges strongly to p ∈ F, where p is the unique solution in F to the following variational inequality
               <(I-f)p, j(p-x*)>≤ 0, ∨_x* ∈ F.

Key words: Nonexpansive nonself-mapping, Uniformly Gâteaux differentiable norm, Viscosity approximation, Common fixed point

中图分类号:

47H10

赵良才, 张石生. 无限簇非扩张非自映象公共不动点的黏性逼近法[J]. 数学物理学报, 2010, 30(4): 1144-1157.

ZHAO Liang-Cai, ZHANG Shi-Sheng. Viscosity Approximation Methods for Common Fixed Points of Infinite Nonexpansive Nonself-mappings[J]. Acta mathematica scientia,Series A, 2010, 30(4): 1144-1157.

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