Strong convergence results for the split common fixed point problem
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Authors
Huimin He
- School of Mathematics and Statistics, Xidian University, Xi'an 710071, China.
Sanyang Liu
- School of Mathematics and Statistics, Xidian University, Xi'an 710071, China.
Rudong Chen
- Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China.
Xiaoyin Wang
- Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China.
Abstract
Recently, Boikanyo [O. A. Boikanyo, Appl. Math. Comput., 265 (2015), 844-853] constructed an
algorithm for demicontractive operators and obtained the strong convergence theorem for the split common
fixed point problem. In this paper, we mainly consider the viscosity iteration algorithm of the algorithm
Boikanyo to approximate the split common fixed point problem, and we get the generated sequence strongly
converges to a solution of this problem. The main results in this paper extend and improve some results of
Boikanyo [O. A. Boikanyo, Appl. Math. Comput., 265 (2015), 844-853] and Cui and Wang [H. H. Cui, F.
H. Wang, Fixed Point Theory Appl., 2014 (2014), 8 pages]. The research highlights of this paper are novel
algorithms and strong convergence results.
Share and Cite
ISRP Style
Huimin He, Sanyang Liu, Rudong Chen, Xiaoyin Wang, Strong convergence results for the split common fixed point problem, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 9, 5332--5343
AMA Style
He Huimin, Liu Sanyang, Chen Rudong, Wang Xiaoyin, Strong convergence results for the split common fixed point problem. J. Nonlinear Sci. Appl. (2016); 9(9):5332--5343
Chicago/Turabian Style
He, Huimin, Liu, Sanyang, Chen, Rudong, Wang, Xiaoyin. "Strong convergence results for the split common fixed point problem." Journal of Nonlinear Sciences and Applications, 9, no. 9 (2016): 5332--5343
Keywords
- Split common fixed point problem
- demicontractive mapping
- explicit viscosity algorithm
- strong convergence.
MSC
References
-
[1]
O. A. Boikanyo, A strongly convergent algorithm for the split common fixed point problem, Appl. Math. Comput., 265 (2015), 844--853
-
[2]
C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems,, 18 (2002), 441--453
-
[3]
C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103--120
-
[4]
L.-C. Ceng, Q. H. Ansari, J.-C. Yao, Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem, Nonlinear Anal., 75 (2012), 2116--2125
-
[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, Y. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221--239
-
[7]
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
-
[8]
Y. Censor, A. Segal, The split common fixed point problem for directed operators, J. Convex Anal., 16 (2009), 587--600
-
[9]
H. H. Cui, F. H. Wang, Iterative methods for the split common fixed point problem in Hilbert spaces, Fixed Point Theory and Appl., 2014 (2014), 8 pages
-
[10]
Q. W. Fan, W. Wu, J. M. Zurada, Convergence of batch gradient learning with smoothing regularization and adaptive momentum for neural networks, SpringerPlus, 5 (2016), 1--17
-
[11]
K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1990)
-
[12]
R. Kraikaew, S. Saejung, On split common fixed point problems, J. Math. Anal. Appl., 415 (2014), 513--524
-
[13]
P. E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899--912
-
[14]
A. Moudafi, The split common fixed-point problem for demicontractive mappings, Inverse Problems, 26 (2010), 6 pages
-
[15]
A. Moudafi, A note on the split common fixed-point problem for quasi-nonexpansive operators, Nonlinear Anal., 74 (2011), 1083--1087
-
[16]
A. Moudafi, A relaxed alternating CQ-algorithm for convex feasibility problems, Nonlinear Anal., 79 (2013), 117--121
-
[17]
B. Qu, B. H. Liu, N. Zheng, On the computation of the step-size for the CQ-like algorithms for the split feasibility problem, Appl. Math. Comput., 262 (2015), 218--223
-
[18]
B. Qu, N. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problems, 21 (2005), 1655--1665
-
[19]
W. Takahashi, Nonlinear functional analysis, Fixed point theory and its applications, Yokohama Publishers, Yokohama (2000)
-
[20]
F. H. Wang, H.-K. Xu, Cyclic algorithms for split feasibility problems in Hilbert spaces, Nonlinear Anal., 74 (2011), 4105--4111
-
[21]
Z. W. Wang, Q. Z. Yang, Y. Yang, The relaxed inexact projection methods for the split feasibility problem, Appl. Math. Comput., 217 (2011), 5347--5359
-
[22]
H.-K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116 (2003), 659--678
-
[23]
H.-K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279--291
-
[24]
H.-K. Xu, A variable Krasonselskiĭ-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021--2034
-
[25]
Q. Z. Yang, The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, 20 (2004), 1261--1266
-
[26]
J. L. Zhao, Q. Z. Yang, Several solution methods for the split feasibility problem, Inverse Problems, 21 (2005), 1791--1799
-
[27]
J. L. Zhao, Y. J. Zhang, Q. Z. Yang, Modified projection methods for the split feasibility problem and the multiple- sets split feasibility problem, Appl. Math. Comput., 219 (2012), 1644--1653
-
[28]
Z. C. Zhu, R. Chen, Strong convergence on iterative methods of Cesáro means for nonexpansive mapping in Banach space, Abstr. Appl. Anal., 2014 (2014), 6 pages