Strong convergence theorems for the general split common fixed point problem in Hilbert spaces

Volume 10, Issue 10, pp 5433--5444 http://dx.doi.org/10.22436/jnsa.010.10.27

Publication Date: October 27, 2017 Submission Date: March 12, 2017

Download PDF

Download XML

1913 Downloads
2932 Views

Authors

Rudong Chen - Department of Mathematics, Tian jin Polytechnic University, Tian jin 300387, China. Tao Sun - Department of Mathematics, Tian jin Polytechnic University, Tian jin 300387, China. Huimin He - School of Mathematics and Statistics, Xidian University, Xi'an 710071, China. Jen-Chih Yao - Center for General Education, China Medical University, Taichung 40402, Taiwan, ROC.

Abstract

In this paper, we propose and investigate a new iterative algorithm for solving the general split common fixed point problem in the setting of infinite-dimensional Hilbert spaces. We also prove the sequence generated by the proposed algorithm converge strongly to a common solution of the general split common fixed point problem. As application, some particular cases of directed operator and quasi-nonexpansive operator are also considered. Finally, we present several numerical results for general split common fixed point problem to demonstrate the efficiency of the proposed algorithm.

Share and Cite

ISRP Style

Rudong Chen, Tao Sun, Huimin He, Jen-Chih Yao, Strong convergence theorems for the general split common fixed point problem in Hilbert spaces, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 10, 5433--5444

AMA Style

Chen Rudong, Sun Tao, He Huimin, Yao Jen-Chih, Strong convergence theorems for the general split common fixed point problem in Hilbert spaces. J. Nonlinear Sci. Appl. (2017); 10(10):5433--5444

Chicago/Turabian Style

Chen, Rudong, Sun, Tao, He, Huimin, Yao, Jen-Chih. "Strong convergence theorems for the general split common fixed point problem in Hilbert spaces." Journal of Nonlinear Sciences and Applications, 10, no. 10 (2017): 5433--5444

Keywords

General split common fixed point problem
demicontractive operator
quasi-nonexpansive operator
directed operator

MSC

47J25
47H09
65J25

References

[1] C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18 (2002), 441–453.

[2] C. Byrne , A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103–120.

[3] Y. Censor, T. Bortfeld, B. Martin, A. Trofimov , A unified approach for inversion problems in intensity-modulated radiation therapy , Phys. Med. Biol., 51 (2005), 2353–2365.
- View Article
- Google Scholar

[4] Y. Censor, Y. Elfving, A multiprojection algorithm using Bregman projections in a product space , Numer. Algorithms, 8 (1994), 221–239.

[5] Y. Censor, Y. Elfving, N. Kopf, T. Bortfeld , The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071–2084.

[6] Y. Censor, A. Segal, The split common fixed point problem for directed operators, J. Convex Anal., 16 (2009), 587–600.
- MathSciNet
- Google Scholar

[7] R. D. Chen, Fixed point Theory and Applications, National Defence Industry Press, (2012)

[8] P. L. Combettes, V. R.Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), 1168–2000.

[9] H.-H. Cui, F.-H. Wang, Iterative methods for the split common fixed point problem in Hilbert spaces, Fixed Point Theory Appl., 2014 (2014), 8 pages.
- View Article
- Google Scholar

[10] P. E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899–912.

[11] A. Moudafi , The split common fixed point problem for demicontractive mappings, Inverse Problems, 26 (2010), 6 pages.
- View Article
- Google Scholar

[12] A. Moudafi, A note on the split common fixed-point problem for quasi-nonexpansive operators, Nonlinear Anal., 74 (2011), 4083–4087.

[13] W. Takahashi, Nonlinear functional analysis, Fixed point theory and its applications, Yokohama Publishers, Yokohama (2000)
- Google Scholar

[14] F.-H. Wang, H.-K. Xu , Cyclic algorithms for split feasibility problems in Hilbert spaces, Nonlinear Anal., 74 (2011), 4105–4111.

[15] H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240–256.
- View Article
- Google Scholar

[16] Y.-H. Yao, R. P. Agarwal, M. Postolache, Y.-C. Liou , Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem, Fixed Point Theory Appl., 2014 (2014), 14 pages.

[17] Y.-H. Yao, M. Postolache, Y.-C. Liou , Strong convergence of a self-adaptive method for the split feasibility problem, Fixed Point Theory Appl., 2013 (2013), 12 pages.