FURTHER INVESTIGATION INTO APPROXIMATION OF A COMMON SOLUTION OF FIXED POINT PROBLEMS AND SPLIT FEASIBILITY PROBLEMS

doi:10.1016/S0252-9602(16)30049-2

Abstract

Abstract:

The purpose of this paper is to study and analyze an iterative method for finding a common element of the solution set Ω of the split feasibility problem and the set F(T) of fixed points of a right Bregman strongly nonexpansive mapping T in the setting of p-uniformly convex Banach spaces which are also uniformly smooth. By combining Mann's iterative method and the Halpern's approximation method, we propose an iterative algorithm for finding an element of the set F(T)∩Ω; moreover, we derive the strong convergence of the proposed algorithm under appropriate conditions and give numerical results to verify the efficiency and implementation of our method. Our results extend and complement many known related results in the literature.

Key words: strong convergence, split feasibility problem, uniformly convex, uniformly smooth, fixed point problem, right Bregman strongly nonexpansive mappings

CLC Number:

49J53

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].Acta mathematica scientia,Series B, 2016, 36(3): 913-930.

Trendmd

References

[1] Alber Y I. Metric and generalized projection operator in Banach spaces: properties and applications//Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Vol 178 of Lecture Notes in Pure and Applied Mathematics. New York: Dekker, 1996: 15-50
[2] Byrne C. Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Problems, 2002, 18(2): 441-453
[3] Censor Y, Bortfeld T, Martin B, Trofimov A. A unified approach for inversion problem in intensitymodulated radiation therapy. Phys Med Biol, 2006, 51: 2353-2365
[4] Censor Y, Elfving T, Kopf N, Bortfeld T. The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Problems, 2005, 21(6): 2071-2084
[5] Censor Y, Elfving T. A multiprojection algorithm using Bregman projections in a product space. Numerical Algorithms, 1994, 8(2-4): 221-239
[6] Censor Y, Lent A. An iterative row-action method for interval convex programming. J Optim Theory Appl, 1981, 34: 321-353
[7] Censor Y, Motova A, Segal A. Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. J Math Anal Appl, 2007, 327(2): 1244-1256
[8] Censor Y, Segal A. The split common fixed point problem for directed operators. J Convex Anal, 2009, 16(2): 587-600
[9] Cioranescu I. Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Dordrecht: Kluwer, 1990
[10] Dunford N, Schwartz J T. Linear Operators I. New York: Wiley Interscience, 1958
[11] Maingé P E. Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal, 2008, 16: 899-912
[12] Martín-Márquez V, Reich S, Sabach S. Right Bregman nonexpansive operators in Banach spaces. Nonlinear Analy, 2012, 75: 5448-5465
[13] Martín-Márquez V, Reich S, Sabach S. Bregman strongly nonexpansive operators in reflexive Banach spaces. J Math Anal Appl, 2013, 400: 597-614
[14] Masad E, Reich S. A note on the multiple-set split convex feasibility problem in Hilbert space. J Nonlinear Convex Anal, 2007, 8(3): 367-371
[15] Moudafi A. The split common fixed-point problem for demicontractive mappings. Inverse Problems, 2010, 26(5): Article ID 055007
[16] Moudafi A. A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal, 2011, 74(12): 4083-4087
[17] Nakajo K, Takahashi W. Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J Math Anal Appl, 2003, 279: 372-379
[18] Phelps R P. Convex Functions, Monotone Operators, and Differentiability. 2nd ed. Berlin: Springer-Verlag, 1993
[19] Schöpfer F. Iterative Regularisation Method for the Solution of the Split Feasibility Problem in Banach Spaces[D]. Saabrü cken, 2007
[20] Schöpfer F, Schuster T, Louis A K. An iterative regularization method for the solution of the split feasibility problem in Banach spaces. Inverse Problems, 2008, 24(5): 055008
[21] Shehu Y, Iyiola O S, Enyi C D. Iterative algorithm for split feasibility problems and fixed point problems in Banach spaces. Numer Algor, DOI: 10.1007/s11075-015-0069-4
[22] Shehu Y, Ogbuisi F U, Iyiola O S. Convergence Analysis of an iterative algorithm for fixed point problems and split feasibility problems in certain Banach spaces. Optimization, DOI: 10.1080/02331934.2015.1039533
[23] Takahashi W. Nonlinear Functional Analysis-Fixed Point Theory and Applications. Yokohama: Yokohama Publishers Inc, 2000(in Japanese)
[24] Takahashi W. Nonlinear Functional Analysis. Yokohama: Yokohama Publishers, 2000
[25] Wang F. A new algorithm for solving the multiple sets split feasibility problem in Banach spaces. Numerical Functional Anal Opt, 2014, 35(1): 99-110
[26] Wang F, Xu H K. Cyclic algorithms for split feasibility problems in Hilbert spaces. Nonlinear Anal, 2011, 74(12): 4105-4111
[27] Xu H K. Iterative algorithms for nonlinear operators. J London Math Soc, 2002, 66(2): 240-256
[28] Xu H K. A variable Krasnosel'skii-Mann algorithm and the multiple-set split feasibility problem. Inverse Problems, 2006, 22(6): 2021-2034
[29] Xu H K. Iterative methods for the split feasibility problem in infinite dimensional Hilbert spaces.t Inverse Problems, 2010, 26(10): Article ID 105018
[30] Yang Q. The relaxed CQ algorithm solving the split feasibility problem. Inverse Problems, 2004, 20(4): 1261-1266
[31] Zegeye H, Shahzad N. Convergence Theorems for Right Bregman Strongly Nonexpansive Mappings in Reflexive Banach Spaces. Abstr Appl Anal, 2014, 2014: Article ID 584395
[32] Zhao J, Yang Q. Several solution methods for the split feasibility problem. Inverse Problems, 2005, 21(5): 1791-1799