The viscosity approximation forward-backward splitting method for solving quasi inclusion problems in Banach spaces
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Authors
Fu Hai Zhao
- School of Science, South West University of Science and Technology, Mianyang, Sichuan 621010, P. R. China.
Li Yang
- School of Science, South West University of Science and Technology, Mianyang, Sichuan 621010, P. R. China.
Abstract
In this paper, we introduce viscosity approximation forward-backward splitting method for an accretive operator and an
m-accretive operator in Banach spaces. The strong convergence of this viscosity method is proved under certain assumptions
imposed on the sequence of parameters. Applications to the minimization optimization problem and the linear inverse problem
are included. The results presented in the paper extend and improve some recent results announced in the current literature.
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ISRP Style
Fu Hai Zhao, Li Yang, The viscosity approximation forward-backward splitting method for solving quasi inclusion problems in Banach spaces, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 1, 130--140
AMA Style
Zhao Fu Hai, Yang Li, The viscosity approximation forward-backward splitting method for solving quasi inclusion problems in Banach spaces. J. Nonlinear Sci. Appl. (2017); 10(1):130--140
Chicago/Turabian Style
Zhao, Fu Hai, Yang, Li. "The viscosity approximation forward-backward splitting method for solving quasi inclusion problems in Banach spaces." Journal of Nonlinear Sciences and Applications, 10, no. 1 (2017): 130--140
Keywords
- Accretive operator
- viscosity approximation
- Banach space
- splitting method
- forward-backward algorithm.
MSC
References
-
[1]
H. Attouch, Viscosity solutions of minimization problems, SIAM J. Optim., 6 (1996), 769–806.
-
[2]
J. B. Baillon, G. Haddad , Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones, (French) Israel J. Math., 26 (1977), 137–150.
-
[3]
D. P. Bertsekas, J. N. Tsitsiklis, Parallel and distributed computation: numerical methods, Prentice-Hall, Englewood Cliffs, NJ, (1989); republished in 1997 by Athena Scientific, MA (1997)
-
[4]
H. Brézis, P.-L. Lions, Produits infinis de résolvantes, (French) Israel J. Math., 29 (1978), 329–345.
-
[5]
C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103–120.
-
[6]
G. H.-G. Chen, R. T. Rockafellar, Convergence rates in forward-backward splitting, SIAM J. Optim., 7 (1997), 421–444.
-
[7]
C. Chidume , Geometric properties of Banach spaces and nonlinear iterations, Lecture Notes in Mathematics, Springer- Verlag London, Ltd., London (2009)
-
[8]
P. Cholamjiak, A generalized forward-backward splitting method for solving quasi inclusion problems in Banach spaces, Numer. Algorithms, 71 (2016), 915–932.
-
[9]
I. Cioranescu, Geometry of Banach spaces, duality mappings and nonlinear problems, Kluwer Academic Publishers GroupMathematics and its Applications,, Dordrecht (1990)
-
[10]
P. L. Combettes, Iterative construction of the resolvent of a sum of maximal monotone operators, J. Convex Anal., 16 (2009), 727–748.
-
[11]
P. L. Combettes, V. R.Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), 1168–1200.
-
[12]
J. Douglas Jr., H. H. Rachford, On the numerical solution of heat conduction problems in two and three space variables, Trans. Amer. Math. Soc., 82 (1956), 421–439.
-
[13]
J. C. Dunn, Convexity, monotonicity, and gradient processes in Hilbert space, J. Math. Anal. Appl., 53 (1976), 145–158.
-
[14]
O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim., 29 (1991), 403–419.
-
[15]
S.-N. He, C.-P. Yang, Solving the variational inequality problem defined on intersection of finite level sets, Abstr. Appl. Anal., 2013 (2013 ), 8 pages.
-
[16]
P.-L. Lions, B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964–979.
-
[17]
G. López, V. Martín-Márquez, F.-H. Wang, H.-K. Xu, Forward-backward splitting methods for accretive operators in Banach spaces, Abstr. Appl. Anal., 2012 (2012 ), 25 pages.