Convergence analysis of a Halpern-like iterative algorithm in Hilbert spaces

Volume 10, Issue 8, pp 4143--4149

Publication Date: 2017-08-10


Yunpeng Zhang - College of Electric Power, North China University of Water Resources and Electric Power, Zhengzhou 450011, China.
Sun Young Cho - Center for General Education, China Medical University, Taichung 40402, Taiwan.


In this paper, a Halpern-like iterative algorithm is investigated for finding a solution of a split feasibility problem and a solution to a nonexpansive operator equation. Strong convergence theorems are established in the framework of infinite dimensional Hilbert spaces.


Convergence analysis, Hilbert space, monotone mapping, split feasibility problem.


[1] B. A. Bin Dehaish, A. Latif, H. O. Bakodah, X. Qin, A regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces, J. Inequal. Appl., 2015 (2015), 14 pages.
[2] B. A. Bin Dehaish, X. Qin, A. Latif, H. Bakodah, Weak and strong convergence of algorithms for the sum of two accretive operators with applications, J. Nonlinear Convex Anal., 16 (2015), 1321–1336.
[3] F. E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Natl. Acad. Sci. U.S.A., 53 (1965), 1272–1276.
[4] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20 (2004), 103–120.
[5] Y. Censor, T. Bortfeld, B. Martin, A. Trofimov, A unified approach for inversion problems in intensity modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353–2365.
[6] Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221–239.
[7] Y. Censor, T. Elfving, N. Kopf, T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Probl., 21 (2005), 2071–2084.
[8] S. Y. Cho, B. A. Bin Dehaish, X. Qin, Weak convergence of a splitting algorithm in Hilbert spaces, J. Appl. Anal. Comput., 7 (2017), 427–438.
[9] S. Y. Cho, S. M. Kang, Approximation of common solutions of variational inequalities via strict pseudocontractions, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 1607–1618.
[10] S. Y. Cho, W. Li, S. M. Kang, Convergence analysis of an iterative algorithm for monotone operators, J. Inequal. Appl., 2013 (2013), 14 pages.
[11] N. Fang, Y. Gong, Viscosity iterative methods for split variational inclusion problems and fixed poit problems of a nonexpansive mappings, Commun. Optim. Theory, 2016 (2016), 15 pages.
[12] L. S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl., 194 (1995), 114–125.
[13] X. Qin, S. S. Chang, Y. J. Cho, Iterative methods for generalized equilibrium problems and fixed point problems with applications, Nonlinear Anal. Real World App., 11 (2010), 2963–2972.
[14] X. Qin, S. Y. Cho, Convergence analysis of a monotone projection algorithm in reflexive banach spaces, Acta Math. Sci. Ser. B Engl. Ed., 37 (2017), 488–502.
[15] X. Qin, J. C. Yao, Weak convergence of a Mann-like algorithm for nonexpansive and accretive operators, J. Inequal. Appl., 2016 (2016), 9 pages.
[16] W. Takahahsi, Weak and strong convergence theorems for families of nonlinear and nonself mappings in Hilbert spaces, J. Nonlinear Var. Anal., 1 (2017), 1–23.
[17] J. Tang, S. S. Chang, Strong convergence theorem of two-step iterative algorithm for split feasibility problems, J. Inequal. Appl., 2014 (2014), 13 pages.
[18] J. Tang, S. S. Chang, J. Dong, Split equality fixed point problems for two quasi-asymptotically pseudocontractive mappings, J. Nonlinear Funct. Anal., 2017 (2017), 15 pages.
[19] H. Y. Zhou, Y.Wang, Adaptively relaxed algorithms for solving the split feasibility problem with a new step size, J. Inequal. Appl., 2014 (2014), 22 pages.


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