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

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

Abstract

Abstract:

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.

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

CLC Number:

47H09

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.

Trendmd

References

[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