Existence of solutions for mixed variational inequalities with perturbation in Banach spaces. (English) Zbl 1442.49013
Summary: This paper is mainly devoted to the existence of solutions of generalized mixed variational inequalities (GMVI for short) with perturbation in reflexive Banach spaces. When the constraint set is weakly compact, we deduce two existence theorems for GMVI. Based on these two existence theorems, when the constraint set is unbounded, we obtain some existence theorems for GMVI perturbed by a nonlinear continuous mapping in finite dimensional spaces (resp. a multiplication of a scalar and a vector taken from the interior of the barrier cone of the constraint set in reflexive Banach spaces). The main results presented in this paper generalize some corresponding known results.
References:
| [1] | Aubin, JP, Optima and Equilibria (1993), Berlin: Springer, Berlin · Zbl 0781.90012 |
| [2] | Aubin, JP; Frankowska, H., Set-Valued Analysis (2009), Boston: Springer, Boston · Zbl 1168.49014 |
| [3] | Alleche, B., Multivalued mixed variational inequalities with locally Lipschitzian and locally cocorcive multivalued mappings, J. Math. Anal. Appl., 399, 625-639 (2013) · Zbl 1256.49020 · doi:10.1016/j.jmaa.2012.10.055 |
| [4] | Bonnans, JF; Shapiro, A., Perturbation Analysis of Optimization (2000), New York: Springer, New York · Zbl 0966.49001 |
| [5] | Facchinei, F.; Pang, JS, Finite-Dimensional Variational Inequalities and Complementary Problems (2003), New York: Springer, New York · Zbl 1062.90001 |
| [6] | Fan, JH; Zhong, RY, Stability analysis for variational inequality in reflexive Banach spaces, Nonlinear Anal. TMA, 69, 2566-2574 (2008) · Zbl 1172.49010 · doi:10.1016/j.na.2007.08.031 |
| [7] | Fan, K., Minimax theorems, Proc. Natl. Acad. Sci. USA, 39, 42-47 (1953) · Zbl 0050.06501 · doi:10.1073/pnas.39.1.42 |
| [8] | Fu, DM; He, YR, The Tikhonov regularization method for generalized mixed variational inequalities (in Chinese), J. Sichuan Norm. Univ. Nat. Sci., 37, 1, 12-17 (2014) · Zbl 1313.49009 |
| [9] | Hadjisavvas, N., Continuity and maximality properties of pseudomonotone operators, J. Convex Anal., 10, 465-475 (2003) · Zbl 1063.47041 |
| [10] | He, YR, Stable pseudomonotone variational inequality in reflexive Banach spaces, J. Math. Anal. Appl., 330, 352-363 (2007) · Zbl 1124.49005 · doi:10.1016/j.jmaa.2006.07.063 |
| [11] | He, Y.R.: The Tikhonov regularization method for set-valued variational inequalities. Abstr. Appl. Anal., Article ID 172061 (2012) · Zbl 1237.49013 |
| [12] | Kien, BT; Yao, JC; Yen, ND, On the solution existence of pseudomonotone variational inequalities, J. Glob. Optim., 41, 135-145 (2008) · Zbl 1149.49011 · doi:10.1007/s10898-007-9251-0 |
| [13] | Konnov, IV; Volotskaya, EO, Mixed variational inequalities and economics equilibrium problems, J. Appl. Math., 2, 289-314 (2002) · Zbl 1029.47043 · doi:10.1155/S1110757X02106012 |
| [14] | László, S., Multivalued variational inequalities and coincidence point results, J. Math. Anal. Appl., 404, 105-114 (2013) · Zbl 1304.49021 · doi:10.1016/j.jmaa.2013.03.007 |
| [15] | Li, FL; He, YR, Solvability of a perturbed variational inequality, Pac. J. Optim., 10, 1, 105-111 (2014) · Zbl 1288.90101 |
| [16] | Li, SJ; Chen, CR, Stability of weak vector variational inequality, Nonlinear Anal. TMA, 70, 1528-1535 (2009) · Zbl 1158.49018 · doi:10.1016/j.na.2008.02.032 |
| [17] | Luo, XP, Tikhonov regulaization methods for inverse variational inequalities, Optim. Lett., 8, 877-887 (2014) · Zbl 1310.90113 · doi:10.1007/s11590-013-0643-4 |
| [18] | Luo, X.P.: Solvability of some perturbed generalized variational inequalities in reflexive Banach spaces. J. Funct. Spaces, Article ID 3897495 (2018) · Zbl 1508.47107 |
| [19] | Peng, ZY; Yang, XM; Zhao, Y., On the semicontinuity of the solution set map to parametric set-valued weak vector equilibrium problems (in Chinese), Acta Math. Sin. Chin. Ser., 56, 3, 391-400 (2013) · Zbl 1329.62138 · doi:10.1007/s10255-013-0222-6 |
| [20] | Qi, HD, Tikhonov regularization methods for variational inequality problems, J. Optim. Theory Appl., 102, 1, 193-201 (1999) · Zbl 0939.90019 · doi:10.1023/A:1021802830910 |
| [21] | Shih, MH; Tan, KK, Browder-Hartman-Stampacchia variational inequalities for multi-valued monotone operators, J. Math. Anal. Appl., 134, 431-440 (1988) · Zbl 0671.47043 · doi:10.1016/0022-247X(88)90033-9 |
| [22] | Tang, GJ; Huang, NJ, Projected subgradient method for non-Lipschitz set-valued mixed variational inequalities, Appl. Math. Mech. Engl. Ed., 32, 10, 1345-1356 (2011) · Zbl 1258.47078 · doi:10.1007/s10483-011-1505-x |
| [23] | Tang, GJ; Huang, NJ, Existence theorems of the variational-hemivariational inequalities, J. Glob. Optim., 56, 605-622 (2013) · Zbl 1272.49019 · doi:10.1007/s10898-012-9884-5 |
| [24] | Wang, M.: Existence theorems for perturbed variational inequalities in Banach spaces. J. Fix. Point Theory Appl. (2018). 10.1007/s11784-018-0536-3. · Zbl 1395.58019 |
| [25] | Wang, YL; He, YR, Solvability of perturbed variational inequality in finite dimensional spaces (in Chinese), J. Sichuan Norm. Univ. Nat. Sci., 39, 625-629 (2016) · Zbl 1374.90386 |
| [26] | Wang, ZB; Chen, ZL, Existence of solutions for generalized mixed variational inequalities in reflexive Banach spaces, J. Nonlinear Sci. Appl., 9, 3299-3309 (2016) · Zbl 1341.49014 · doi:10.22436/jnsa.009.05.116 |
| [27] | Zhong, RY; Huang, NJ, Stability analysis for Minty mixed variational inequality in reflexive Banach spaces, J. Optim. Theory Appl., 147, 454-472 (2010) · Zbl 1218.49032 · doi:10.1007/s10957-010-9732-z |
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