VISCOSITY APPROXIMATION METHODS FOR THE SPLIT EQUALITY COMMON FIXED POINT PROBLEM OF QUASI-NONEXPANSIVE OPERATORS

doi:10.1016/S0252-9602(16)30083-2

数学物理学报(英文版) ›› 2016, Vol. 36 ›› Issue (5): 1474-1486.doi: 10.1016/S0252-9602(16)30083-2

VISCOSITY APPROXIMATION METHODS FOR THE SPLIT EQUALITY COMMON FIXED POINT PROBLEM OF QUASI-NONEXPANSIVE OPERATORS

赵静¹, 王盛楠²

1. College of Science, Civil Aviation University of China, Tianjin 300300, China Tianjin Key Laboratory for Advanced Signal Processing, Tianjin 300300, China;
2. College of Science, Civil Aviation University of China, Tianjin 300300, China

收稿日期:2014-10-20 修回日期:2015-10-09 出版日期:2016-10-25 发布日期:2016-10-25
通讯作者: Jing ZHAO,zhaojing200103@163.com E-mail:zhaojing200103@163.com
作者简介:Shengnan WANG,shengnan19900507@163.com
基金资助:
The research was supported by National Natural Science Foundation of China (61503385) and Fundamental Research Funds for the Central Universities of China (3122016L002).

VISCOSITY APPROXIMATION METHODS FOR THE SPLIT EQUALITY COMMON FIXED POINT PROBLEM OF QUASI-NONEXPANSIVE OPERATORS

Jing ZHAO¹, Shengnan WANG²

1. College of Science, Civil Aviation University of China, Tianjin 300300, China Tianjin Key Laboratory for Advanced Signal Processing, Tianjin 300300, China;
2. College of Science, Civil Aviation University of China, Tianjin 300300, China

Received:2014-10-20 Revised:2015-10-09 Online:2016-10-25 Published:2016-10-25
Contact: Jing ZHAO,zhaojing200103@163.com E-mail:zhaojing200103@163.com
Supported by:
The research was supported by National Natural Science Foundation of China (61503385) and Fundamental Research Funds for the Central Universities of China (3122016L002).

摘要/Abstract

摘要：

Let H₁, H₂, H₃ be real Hilbert spaces, let A:H₁→H₃, B:H₂→H₃ be two bounded linear operators. The split equality common fixed point problem (SECFP) in the infinite-dimensional Hilbert spaces introduced by Moudafi (Alternating CQ-algorithm for convex feasibility and split fixed-point problems. Journal of Nonlinear and Convex Analysis) is to find x∈F(U), y∈F(T) such that Ax=By, (1) where U:H₁→H₁ and T:H₂→H₂ are two nonlinear operators with nonempty fixed point sets F(U)={x∈H₁:Ux=x} and F(T)={x∈H₂:Tx=x}. Note that, by taking B=I and H₂=H₃ in (1), we recover the split fixed point problem originally introduced in Censor and Segal. Recently, Moudafi introduced alternating CQ-algorithms and simultaneous iterative algorithms with weak convergence for the SECFP (1) of firmly quasi-nonexpansive operators. In this paper, we introduce two viscosity iterative algorithms for the SECFP (1) governed by the general class of quasi-nonexpansive operators. We prove the strong convergence of algorithms. Our results improve and extend previously discussed related problems and algorithms.

关键词: split equality common fixed point problems, quasi-nonexpansive operator, strong convergence, viscosity iterative algorithms, Hilbert space

Abstract:

Key words: split equality common fixed point problems, quasi-nonexpansive operator, strong convergence, viscosity iterative algorithms, Hilbert space

中图分类号:

47H09

赵静, 王盛楠. VISCOSITY APPROXIMATION METHODS FOR THE SPLIT EQUALITY COMMON FIXED POINT PROBLEM OF QUASI-NONEXPANSIVE OPERATORS[J]. 数学物理学报(英文版), 2016, 36(5): 1474-1486.

Jing ZHAO, Shengnan WANG. VISCOSITY APPROXIMATION METHODS FOR THE SPLIT EQUALITY COMMON FIXED POINT PROBLEM OF QUASI-NONEXPANSIVE OPERATORS[J]. Acta mathematica scientia,Series B, 2016, 36(5): 1474-1486.

参考文献

[1] Byrne C. Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Problems, 2002, 18:441-453
[2] Shehu Y, Mewomo O T, Ogbuisi F U. Further investigation into approximation of a common solution of fixed point problems and split feasibility problems. Acta Math Sci, 2016, 36B(3):913-930
[3] Byrne C, Censor Y, Gibali A, et al. The split common null point problem. J Nonl Convex Anal, 2012, 13:759-775
[4] Censor Y, Gibalin A, Reich S. Algorithms for the split variational inequality problem. Numerical Algorithms, 2012, 59:301-323
[5] Masad E, Reich S. A note on the multiple-set split convex feasibility problem in Hilbert space. J Nonl Convex Anal, 2007, 8:367-371
[6] Moudafi A. A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonl Anal Theory Meth Appl, 2011, 74:4083-4087
[7] Moudafi A. Alternating CQ-algorithm for convex feasibility and split fixed-point problems. J Nonl Convex Anal, 2014, 15(4):809-818
[8] Attouch H, Bolte J, Redont P, et al. Alternating proximal algorithms for weakly coupled minimization problems. Applications to dynamical games and PDEs. J Convex Anal, 2008, 15(3):485-506
[9] Censor Y, Bortfeld T, Martin B, et al. A unified approach for inversion problems in intensity-modulated radiation therapy. Physics in Medicine and Biology, 2006, 51:2353-2365
[10] Censor Y, Elfving T. A multiprojection algorithm using Bregman projections in a product space. Num Algor, 1994, 8:221-239
[11] Byrne C. A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems, 2004, 20:103-120
[12] Xu HK. A variable Krasnosel'skii-Mann algorithm and the multiple-set split feasibility problem. Inverse Problems, 2006, 22:2021-2034
[13] Qu B, Xiu N. A note on the CQ algorithm for the split feasibility problem. Inverse Problems, 2005, 21:1655-1665
[14] Yang Q. The relaxed CQ algorithm solving the split feasibility problem. Inverse Problems, 2004, 20:1261-1266
[15] Yang Q, Zhao J. Generalized KM theorems and their applications. Inverse Problems, 2006, 22:833-844
[16] Bauschke H H, Borwein J M. On projection algorithms for solving convex feasibility problems. SIAM Rev, 1996, 38(3):367-426
[17] Xu HK. Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Problems, 2010, 26:105018
[18] Censor Y, Segal A. The split common fixed point problem for directed operators. Journal of Convex Analysis, 2009, 16:587-600
[19] Moudafi A, Al-Shemas E. Simultaneous iterative methods for split equality problems and application. Tran Math Prog Appl, 2013, 1:1-11
[20] Zhao J, He S. Strong Convergence of the viscosity approximation process for the split common fixed-point problem of quasi-nonexpansive mappings. J Appl Math, 2012, 2012:Article ID 438023
[21] He S, Yang C. Solving the variational inequality problem defined on intersection of finite level sets. Abs Appl Anal, 2013, 2013:Article ID 942315
[22] Matinez-Yanes C, Xu H K. Strong convergence of the CQ method for fixed point processes. Nonlinear Anal-TMA, 2006, 64:2400-2411

VISCOSITY APPROXIMATION METHODS FOR THE SPLIT EQUALITY COMMON FIXED POINT PROBLEM OF QUASI-NONEXPANSIVE OPERATORS

VISCOSITY APPROXIMATION METHODS FOR THE SPLIT EQUALITY COMMON FIXED POINT PROBLEM OF QUASI-NONEXPANSIVE OPERATORS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 13

Metrics

本文评价

推荐阅读 10

[1]	Yekini SHEHU, Olaniyi. S. IYIOLA. CONVERGENCE OF HYBRID VISCOSITY AND STEEPEST-DESCENT METHODS FOR PSEUDOCONTRACTIVE MAPPINGS AND NONLINEAR HAMMERSTEIN EQUATIONS[J]. 数学物理学报(英文版), 2018, 38(2): 610-626.
[2]	Y. SHEHU, O. T. MEWOMO, F. U. OGBUISI. FURTHER INVESTIGATION INTO APPROXIMATION OF A COMMON SOLUTION OF FIXED POINT PROBLEMS AND SPLIT FEASIBILITY PROBLEMS[J]. 数学物理学报(英文版), 2016, 36(3): 913-930.
[3]	唐金芳, 张石生, 刘敏. GENERAL SPLIT FEASIBILITY PROBLEMS FOR TWO FAMILIES OF NONEXPANSIVE MAPPINGS IN HILBERT SPACES[J]. 数学物理学报(英文版), 2016, 36(2): 602-613.
[4]	Huaixin CAO, Zhengli CHEN, Li LI, Baomin YU. AN APPLICABLE APPROXIMATION METHOD AND ITS APPLICATION[J]. 数学物理学报(英文版), 2015, 35(5): 1189-1202.
[5]	H.R. SAHEBI, A. RAZANI. A SOLUTION OF A GENERAL EQUILIBRIUM PROBLEM[J]. 数学物理学报(英文版), 2013, 33(6): 1598-1614.
[6]	B. YOUSEFI, Sh. KHOSHDEL, Y. JAHANSHAHI. MULTIPLICATION OPERATORS ON INVARIANT SUBSPACES OF FUNCTION SPACES[J]. 数学物理学报(英文版), 2013, 33(5): 1463-1470.
[7]	刘三洋, 贺慧敏, 陈汝栋. APPROXIMATING SOLUTION OF 0 ∈T(x) FOR AN H-MONOTONE OPERATOR IN HILBERT SPACES[J]. 数学物理学报(英文版), 2013, 33(5): 1347-1360.
[8]	陈泽乾, 孙牧. AREA INTEGRAL FUNCTIONS AND H^∞ FUNCTIONAL CALCULUS FOR SECTORIAL OPERATORS ON HILBERT SPACES[J]. 数学物理学报(英文版), 2013, 33(4): 989-997.
[9]	B.Yousefi. UNIVERSAL INTERPOLATING SEQUENCES ON SPACES OF ANALYTIC FUNCTIONS[J]. 数学物理学报(英文版), 2011, 31(1): 68-72.
[10]	王瑞东. ON LINEAR EXTENSION OF 1-LIPSCHITZ MAPPING FROM HILBERT SPACE INTO A NORMED SPACE[J]. 数学物理学报(英文版), 2010, 30(1): 161-165.
[11]	周勇, 尤进红, 王晓婧. STRONG CONVERGENCE RATES OF SEVERAL ESTIMATORS IN SEMIPARAMETRIC VARYING-COEFFICIENT PARTIALLY LINEAR MODELS[J]. 数学物理学报(英文版), 2009, 29(5): 1113-1127.
[12]	丁夏畦, 罗佩珠. Generalized expansions in Hilbert space[J]. 数学物理学报(英文版), 1999, 19(3): 241-250.
[13]	喻小培, 蹇明. MEAN VALUE OF ANALYTIC OPERATOR FUNCTIONS[J]. 数学物理学报(英文版), 1995, 15(4): 468-473.