A generalized forward-backward method for solving split equality quasi inclusion problems in Banach spaces
-
1841
Downloads
-
3605
Views
Authors
Shih-Sen Chang
- Center for General Education, China Medical University, Taichung, 40402, Taiwan.
- College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China.
Ching-Feng Wen
- Center for Fundamental Science; and Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University , Kaohsiung 80702, Taiwan.
- Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 80702, Taiwan.
Jen-Chih Yao
- Center for General Education, China Medical University, Taichung, 40402, Taiwan.
Jing-Qiang Zhang
- College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China.
Abstract
A generalized forward-backward method
for solving split equality quasi inclusion problems of accretive
operators in Banach spaces is studied. Some strong convergence theorems for the sequences generalized by the algorithm to a solution of quasi inclusion problems of accretive
operators are proved under certain assumptions. The results presented in this paper are new which extend and improve the corresponding results
announced in the recent literatures. At the end of the paper
some applications to monotone variational inequalities, convex minimization problem, and convexly constrained linear inverse problem are presented.
Share and Cite
ISRP Style
Shih-Sen Chang, Ching-Feng Wen, Jen-Chih Yao, Jing-Qiang Zhang, A generalized forward-backward method for solving split equality quasi inclusion problems in Banach spaces, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 4890--4900
AMA Style
Chang Shih-Sen, Wen Ching-Feng, Yao Jen-Chih, Zhang Jing-Qiang, A generalized forward-backward method for solving split equality quasi inclusion problems in Banach spaces. J. Nonlinear Sci. Appl. (2017); 10(9):4890--4900
Chicago/Turabian Style
Chang, Shih-Sen, Wen, Ching-Feng, Yao, Jen-Chih, Zhang, Jing-Qiang. "A generalized forward-backward method for solving split equality quasi inclusion problems in Banach spaces." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 4890--4900
Keywords
- Generalized forward-backward method
- accretive operator
- \(m\)-accretive operator
- maximal monotone operator
- split equality quasi inclusion problem
MSC
References
-
[1]
C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103–120.
-
[2]
S.-S. Chang, L. Wang, L.-J. Qin, Z.-L. Ma , Strongly convergent iterative methods for split equality variational inclusion problems in Banach spaces, Acta Math. Sci. Ser. B Engl. Ed., 36 (2016), 1641–1650.
-
[3]
G. H.-G. Chen, R. T. Rockafellar , Convergence rates in forward-backward splitting, SIAM J. Optim., 7 (1997), 421–444.
-
[4]
P. Cholamjiak, A generalized forward-backward splitting method for solving quasi inclusion problems in Banach spaces, Numer Algor, 71 (2016), 915–932.
-
[5]
I. Cioranescu , Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht (1990)
-
[6]
P. L. Combettes, Iterative construction of the resolvent of a sum of maximal monotone operators, J. Convex Anal., 16 (2009), 727–748.
-
[7]
P. L. Combettes, V. R.Wajs , Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), 1168–1200.
-
[8]
O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim., 29 (1991), 403–419.
-
[9]
S. Kamimura, W. Takahashi , Approximating solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory, 106 (2000), 226–240.
-
[10]
P.-L. Lions, B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964–979.
-
[11]
G. López, V. Martín-Márquez, F. Wang, H. K. Xu, Forward-Backward splitting methods for accretive operators in Banach spaces, Abstr. Appl. Anal., 2012 (2012), 25 pages.
-
[12]
G. Marino, H.-K. Xu, Convergence of generalized proximal point algorithm, Comm. Pure Appl. Anal., 3 (2004), 791–808.
-
[13]
A. Moudafi, Split monotone variational inclusions , J. Optim. Theory Appl., 150 (2011), 275–283.
-
[14]
A. Moudafi, A relaxed alternating CQ algorithm for convex feasibility problems, Nonlinear Anal, 79 (2013), 117–121.
-
[15]
A. Moudafi, E. Al-Shemas, Simultaneouss iterative methods for split equality problem, Trans. Math. Program Appl, 1 (2013), 1–11.
-
[16]
R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149 (1970), 75–88.
-
[17]
R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optimization, 14 (1976), 877–898.
-
[18]
S. Sra, S. Nowozin, S. J. Wright, Optimization for Machine Learning, MIT Press, England (2011)
-
[19]
W. Takahashi , Iterative methods for split feasibility problems and split common null point problems in Banach spaces, The 9th International Conference on Nonlinear Analysis and Convex Analysis. Thailand, Jan: Chiang Rai, (2015), 21–25.
-
[20]
W. Takahashi, N.-C. Wong, J.-C. Yao, Two generalized strong convergence theorems of Halpern’s type in Hilbert spaces and applications, Taiwanese J. Math., 16 (2012), 1151–1172.
-
[21]
P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431–446.
-
[22]
H. K. Xu , Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127–1138.