两个平衡问题和可数无穷多非扩张映像公共不动点问题的新的迭代算法

数学物理学报 ›› 2013, Vol. 33 ›› Issue (5): 860-881.

两个平衡问题和可数无穷多非扩张映像公共不动点问题的新的迭代算法

胡慧英|曾六川

上海师范大学数学系上海 200234

收稿日期:2012-01-16 修回日期:2013-08-23 出版日期:2013-10-25 发布日期:2013-10-25
基金资助:
国家自然科学基金 (11071169)资助.

The New Iterative Scheme for two Equilibrium Problems and Fixed Point Problems of Infinitely Nonexpansive Mappings

HU Hui-Ying, ZENG Liu-Chuan

Department of Mathematics, Shanghai Normal University, Shanghai |200234

Received:2012-01-16 Revised:2013-08-23 Online:2013-10-25 Published:2013-10-25
Supported by:
国家自然科学基金 (11071169)资助.

摘要/Abstract

摘要：

最近, Hu C S和Cai G^[7]在Hilbert空间的框架下提出并分析了两个迭代算法用于寻求平衡问题, 可数无穷多非扩张映像公共不动点问题和可数无穷多α_i-逆-强单调映像组成的变分不等式问题的公共解. 该文在Hilbert空间中引入一个新的迭代算法用于寻求可数无穷多α_i-逆-强单调映像组成的两个平衡问题和可数无穷多非扩张映像公共不动点问题的公共解. 证明了一些强收敛性结果, 这些结果推广了Hu Changsong和Cai Gang 的结果(2010).

关键词: 强收敛性, 平衡问题, 不动点, 变分不等式, 拟-强单调映像

Abstract:

Very recently, Hu C S and Cai G^[7] suggested and analyzed two iterative schemes for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points for a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for an infinite family of α_i-inverse-strongly monotone mappings in the framework of Hilbert space. In this paper, we introduce a new iterative scheme for finding a common element of the sets of solutions of two equilibrium problems for two infinite families of inverse-strongly monotone mappings and the set of common fixed points for a family of infinitely nonexpansive mappings in a Hilbert space. Then we prove some strong convergence theorems which extend and improve Hu Changsong and Cai Gang's results(2010).

Key words: Strong convergence, Equilibrium problem, Fixed point, Variational , inequality, Inverse-strongly monotone mapping

中图分类号:

47H09

胡慧英, 曾六川. 两个平衡问题和可数无穷多非扩张映像公共不动点问题的新的迭代算法[J]. 数学物理学报, 2013, 33(5): 860-881.

HU Hui-Ying, ZENG Liu-Chuan. The New Iterative Scheme for two Equilibrium Problems and Fixed Point Problems of Infinitely Nonexpansive Mappings[J]. Acta mathematica scientia,Series A, 2013, 33(5): 860-881.

参考文献

[1] Browder F E, Petryshyn W V. Construction of fixed points of nonlinear mappings in Hilbert space. J Math Anal Appl, 1967, 20: 197--228

[2] Combettes P L, Hirstoaga A. Equilibrium programming in Hilbert spaces. J Nonlinear Convex Anal, 2005, 6: 117--136
[3] Moudafi A, Thera M. Proximal and Dynamical Approaches to Equilibrium Problems//Lecture Notes in Economics and Mathematical Systems. Vol 477. Springer, 1999: 187--201

[4] Tada A, Takahashi W. Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. J Optim Theory Appl, 2007, 133: 359--370

[5] Takahashi S, Takahashi W. Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert space. J Math Anal Appl, 2007, 331: 506--515

[6] Takahashi S, Takahashi W. Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal, 2008, 69: 1025--1033

[7] Hu C S, Cai G. Viscosity approximation schemes for fixed point problems and equilibrium problems and variational inequality problems. Nonlinear Anal, 2010, 72: 1792--1808

[8] Rockafellar R T. On the maximality of sums of nonlinear monotone operators. Trans Amer Math Soc, 1970, 149: 75--88

[9] Iiduka H, Takahashi W. Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Anal, 2005, 61: 341--350

[10] Suzuki T. Strong convergence of Krasnoselskii and Manns type sequences for oneparameter nonexpansive semigroups without Bochner integrals. J Math Anal Appl, 2005, 305: 227--229

[11] Xu H K. Interative algorithms for nonlinear operators. Journal of London Mathematical Society, 2002, 66: 240--256

[12] Shimoji K, Takahashi W. Strong convegence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese J Math, 2001, 5(2): 387--404

[13] Chang S S, et al. A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal, 2008, doi: 10.1016/j.na.2008.04.035

[14] Blum E, Oettli S. From optimation and variational inequalities to equilibrium problems. Math Student, 1994, 63(1C4): 123--145

[15] Marino G, Xu H K. A general iterative method for nonexpansive mappings in Hilbert spaces. J Math Anal Appl, 2006, 318: 43--52