Systems of variational inequalities with hierarchical variational inequality constraints in Banach spaces
-
1884
Downloads
-
2956
Views
Authors
Lu-Chuan Ceng
- Department of Mathematics, Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China.
Yeong-Cheng Liou
- Department of Healthcare Administration and Medical Informatics, and Research Center of Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 807, Taiwan.
- Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 807, Taiwan.
Ching-Fen Wen
- Center for Fundamental Science, and Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 807, Taiwan.
Abstract
Two implicit iterative algorithms are presented to solve a general system of variational inequalities with the hierarchical
variational inequality constraint for an infinite family of nonexpansive mappings. Strong convergence theorems are given in
a uniformly convex and 2-uniformly smooth Banach space. The results improve and extend the corresponding results in the
earlier and recent literature.
Share and Cite
ISRP Style
Lu-Chuan Ceng, Yeong-Cheng Liou, Ching-Fen Wen, Systems of variational inequalities with hierarchical variational inequality constraints in Banach spaces, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 6, 3136--3154
AMA Style
Ceng Lu-Chuan, Liou Yeong-Cheng, Wen Ching-Fen, Systems of variational inequalities with hierarchical variational inequality constraints in Banach spaces. J. Nonlinear Sci. Appl. (2017); 10(6):3136--3154
Chicago/Turabian Style
Ceng, Lu-Chuan, Liou, Yeong-Cheng, Wen, Ching-Fen. "Systems of variational inequalities with hierarchical variational inequality constraints in Banach spaces." Journal of Nonlinear Sciences and Applications, 10, no. 6 (2017): 3136--3154
Keywords
- Variational inequalities
- nonexpansive mapping
- fixed point
- implicit iterative algorithm.
MSC
References
-
[1]
Q. H. Ansari, J.-C. Yao, Systems of generalized variational inequalities and their applications, Appl. Anal., 76 (2000), 203–217.
-
[2]
K. Aoyama, H. Iiduka, W. Takahashi, Weak convergence of an iterative sequence for accretive operators in Banach spaces, Fixed Point Theory Appl., 2006 (2006), 13 pages.
-
[3]
F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A., 54 (1965), 1041–1044.
-
[4]
N. Buong, N. T. H. Phuong, Strong convergence to solutions for a class of variational inequalities in Banach spaces by implicit iteration methods, J. Optim. Theory Appl., 159 (2013), 399–411.
-
[5]
N. Buong, N. T. Quynh Anh, An implicit iteration method for variational inequalities over the set of common fixed points for a finite family of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2011 (2011), 10 pages.
-
[6]
L.-C. Ceng, Q. H. Ansari, J.-C. Yao, Mann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces, Numer. Funct. Anal. Optim., 29 (2008), 987–1033.
-
[7]
L.-C. Ceng, H. Gupta, Q. H. Ansari, Implicit and explicit algorithms for a system of nonlinear variational inequalities in Banach spaces, J. Nonlinear Convex Anal., 16 (2015), 965–984.
-
[8]
L.-C. Ceng, S.-M. Guu, J.-C. Yao, Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems, Fixed Point Theory Appl., 2012 (2012), 19 pages.
-
[9]
L.-C. Ceng, A. Petruşel, J.-C. Yao, Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of Lipschitz pseudocontractive mappings, J. Math. Inequal., 1 (2007), 243–258.
-
[10]
L.-C. Ceng, C.-Y. Wang, J.-C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper. Res., 67 (2008), 375–390.
-
[11]
S. Y. Cho, B. A. Bin Dehaish, X.-L. Qin, Weak convergence of a splitting algorithm in Hilbert spaces, J. Appl. Anal. Comput., 7 (2017), 427–438.
-
[12]
I. Cioranescu, Geometry of Banach spaces, duality mappings and nonlinear problems, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht (1990)
-
[13]
A. N. Iusem, B. F. Svaiter, A variant of Korpelevich’s method for variational inequalities with a new search strategy, Optimization, 42 (1997), 309–321.
-
[14]
S. Kamimura, W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim., 13 (2002), 938–945.
-
[15]
M. Kikkawa, W. Takahashi, Strong convergence theorems by the viscosity approximation method for a countable family of nonexpansive mappings, Taiwanese J. Math., 12 (2008), 583–598.
-
[16]
G. M. Korpelevič, An extragradient method for finding saddle points and for other problems, (Russian) Èkonom. i Mat. Metody, 12 (1976), 747–756.
-
[17]
X.-L. 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.
-
[18]
S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), 274–276.
-
[19]
M. V. Solodov, B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim., 37 (1999), 765–776.
-
[20]
W. Takahashi, Weak and strong convergence theorems for families of nonexpansive mappings and their applications, Proceedings of Workshop on Fixed Point Theory, Kazimierz Dolny, (1997), Ann. Univ. Mariae Curie-Skodowska Sect. A, 51 (1997), 277–292.
-
[21]
Y. Takahashi, K. Hashimoto, M. Kato, On sharp uniform convexity, smoothness, and strong type, cotype inequalities, J. Nonlinear Convex Anal., 3 (2002), 267–281.
-
[22]
R. U. Verma, On a new system of nonlinear variational inequalities and associated iterative algorithms, Math. Sci. Res. Hot-Line, 3 (1999), 65–68.
-
[23]
H.-K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127–1138.
-
[24]
Y.-H. Yao, R.-D. Chen, H.-K. Xu, Schemes for finding minimum-norm solutions of variational inequalities, Nonlinear Anal., 72 (2010), 3447–3456.
-
[25]
Y.-H. Yao, Y.-C. Liou, S. M. Kang, Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method, Comput. Math. Appl., 59 (2010), 3472–3480.
-
[26]
Y.-H. Yao, Y.-C. Liou, S. M. Kang, Y.-L. Yu, Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces, Nonlinear Anal., 74 (2011), 6024–6034.
-
[27]
Y.-H. Yao, Y.-.C Liou, J.-C. Yao, Iterative algorithms for the split variational inequality and fixed point problems under nonlinear transformations, J. Nonlinear Sci. Appl., 10 (2017), 843–854.
-
[28]
Y.-H. Yao, M. A. Noor, Y.-C. Liou, Strong convergence of a modified extragradient method to the minimum-norm solution of variational inequalities, Abstr. Appl. Anal., 2012 (2012), 9 pages.
-
[29]
Y.-H. Yao, M. A. Noor, Y.-C. Liou, S. M. Kang, Iterative algorithms for general multivalued variational inequalities, Abstr. Appl. Anal., 2012 (2012), 10 pages.
-
[30]
Y.-H. Yao, M. Postolache, Y.-C. Liou, Z.-S. Yao, Construction algorithms for a class of monotone variational inequalities, Optim. Lett., 10 (2016), 1519–1528.
-
[31]
H. Zegeye, N. Shahzad, Y.-H. Yao, Minimum-norm solution of variational inequality and fixed point problem in Banach spaces, Optimization, 64 (2015), 453–471.
-
[32]
L.-C. Zeng, J.-C. Yao, Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings, Nonlinear Anal., 64 (2006), 2507–2515.
