On the hierarchical variational inclusion problems in Hilbert spaces. (English) Zbl 1476.47044
Summary: The purpose of this paper is by using Maingé’s approach to study the existence and approximation problem of solutions for a class of hierarchical variational inclusion problems in the setting of Hilbert spaces. As applications, we solve the convex programming problems and quadratic minimization problems by using the main theorems. Our results extend and improve the corresponding recent results announced by many authors.
MSC:
| 47J22 | Variational and other types of inclusions |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
Keywords:
hierarchical variational inclusion problem; hierarchical variational inequality problem; hierarchical optimization problems; quasi-nonexpansive mapping; strongly quasi-nonexpansive mapping; demiclosed principleReferences:
| [1] | Noor MA, Noor KI: Sensitivity analysis of quasi variational inclusions .J. Math. Anal. Appl. 1999, 236:290-299. · Zbl 0949.49007 · doi:10.1006/jmaa.1999.6424 |
| [2] | Chang SS: Set-valued variational inclusions in Banach spaces .J. Math. Anal. Appl. 2000, 248:438-454. · Zbl 1031.49018 · doi:10.1006/jmaa.2000.6919 |
| [3] | Chang SS: Existence and approximation of solutions of set-valued variational inclusions in Banach spaces .Nonlinear Anal. 2001, 47:583-594. · Zbl 1042.49510 · doi:10.1016/S0362-546X(01)00203-6 |
| [4] | Demyanov VF, Stavroulakis GE, Polyakova LN, Panagiotopoulos PD: Quasidifferentiability and Nonsmooth Modeling in Mechanics, Engineering and Economics. Kluwer Academic, Dordrecht; 1996. · Zbl 1076.49500 |
| [5] | Zhang SS, Wang XR, Lee HW, Chan CK: Viscosity method for hierarchical fixed point and variational inequalities with applications .Appl. Math. Mech. 2011, 32:241-250. · Zbl 1296.47105 · doi:10.1007/s10483-011-1410-8 |
| [6] | Yao, YH; Liou, YC, An implicit extra-gradient method for hierarchical variational inequalities (2011) · Zbl 1222.49016 |
| [7] | Chang SS, Cho YJ, Kim JK: Hierarchical variational inclusion problems in Hilbert spaces with applications .J. Nonlinear Convex Anal. 2012, 13:503-513. · Zbl 1257.49009 |
| [8] | Xu HK: Viscosity methods for hierarchical fixed point approach to variational inequalities .Taiwan. J. Math. 2010, 14:463-478. · Zbl 1215.47099 |
| [9] | Cianciaruso F, Colao V, Muglia L, Xu HK: On a implicit hierarchical fixed point approach to variational inequalities .Bull. Aust. Math. Soc. 2009, 80:117-124. · Zbl 1168.49005 · doi:10.1017/S0004972709000082 |
| [10] | Cianciaruso, F.; Marino, G.; Muglia, L.; Yao, Y., On a two-steps algorithm for hierarchical fixed points and variational inequalities (2009) |
| [11] | Kim JK, Kim KS: A new system of generalized nonlinear mixed quasivariational inequalities and iterative algorithms in Hilbert spaces .J. Korean Math. Soc. 2007, 44:823-834. · Zbl 1140.49006 · doi:10.4134/JKMS.2007.44.4.823 |
| [12] | Kim JK, Kim KS: New system of generalized mixed variational inequalities with nonlinear mappings in Hilbert spaces .J. Comput. Anal. Appl. 2010, 12:601-612. · Zbl 1213.49013 |
| [13] | Kim JK, Kim KS: On new system of generalized nonlinear mixed quasivariational inequalities with two-variable operators .Taiwan. J. Math. 2007, 11:867-881. · Zbl 1139.49012 |
| [14] | Chang SS, Cho YJ, Kim JK: On the two-step projection methods and applications to variational inequalities .Math. Inequal. Appl. 2007, 10:755-760. · Zbl 1132.47047 |
| [15] | Mainge PE, Moudafi A: Strong convergence of an iterative method for hierarchical fixed point problems .Pac. J. Optim. 2007, 3:529-538. · Zbl 1158.47057 |
| [16] | Guo, G.; Wang, S.; Cho, YJ, Strong convergence algorithms for hierarchical fixed point problems and variational inequalities (2011) |
| [17] | Yao, Y.; Cho, YJ; Liou, YC, Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequality problems, No. 2011 (2011) · Zbl 1275.49018 |
| [18] | Yao, Y.; Cho, YJ; Yang, P., An iterative algorithm for a hierarchical problem (2012) · Zbl 1235.65051 |
| [19] | Chang SS, Joseph Lee HW, Chan CK: Algorithms of common solutions for quasi variational inclusion and fixed point problems .Appl. Math. Mech. 2008, 29:1-11. |
| [20] | Maingé PE: New approach to solving a system of variational inequalities and hierarchical problems .J. Optim. Theory Appl. 2008, 138:459-477. · Zbl 1159.49016 · doi:10.1007/s10957-008-9433-z |
| [21] | Kraikaew R, Saejung S: On Maingé’s approach for hierarchical optimization problem .J. Optim. Theory Appl. (2012). doi:10.1007/s10957-011-9982-4 · Zbl 1267.90154 |
| [22] | Kassay G, Kolumbán J, Páles Z: Factorization of Minty and Stampacchia variational inequality systems .Eur. J. Oper. Res. 2002, 143:377-389. · Zbl 1059.49015 · doi:10.1016/S0377-2217(02)00290-4 |
| [23] | Kassay G, Kolumbán J: System of multi-valued variational inequalities .Publ. Math. (Debr.) 2000, 56:185-195. · Zbl 0989.49010 |
| [24] | Verma RU: Projection methods, algorithms, and a new system of nonlinear variational inequalities .Comput. Math. Appl. 2001, 41:1025-1031. · Zbl 0995.47042 · doi:10.1016/S0898-1221(00)00336-9 |
| [25] | Yamada I, Ogura N: Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings .Numer. Funct. Anal. Optim. 2004, 25:619-655. · Zbl 1095.47049 · doi:10.1081/NFA-200045815 |
| [26] | Yamada, I.; Ogura, N.; Shirakawa, N.; Nashed, MZ (ed.); Scherzer, O. (ed.), A numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems, 269-305 (2002), Providence · Zbl 1039.47051 · doi:10.1090/conm/313/05379 |
| [27] | Takahashi W: Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama; 2009. · Zbl 1183.46001 |
| [28] | Maingé PE: The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces .Comput. Math. Appl. 2010, 59:74-79. · Zbl 1189.49011 · doi:10.1016/j.camwa.2009.09.003 |
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