On a class of generalized stochastic Browder mixed variational inequalities. (English) Zbl 1482.49011
Summary: In this paper, we introduce a class of stochastic variational inequalities generated from the Browder variational inequalities. First, the existence of solutions for these generalized stochastic Browder mixed variational inequalities (GS-BMVI) are investigated based on FKKM theorem and Aummann’s measurable selection theorem. Then the uniqueness of solution for GS-BMVI is proved based on strengthening conditions of monotonicity and convexity, the compactness and convexity of the solution sets are discussed by Minty’s technique. The results of this paper can provide a foundation for further research of a class of stochastic evolutionary problems driven by GS-BMVI.
MSC:
| 49J40 | Variational inequalities |
| 60H25 | Random operators and equations (aspects of stochastic analysis) |
Keywords:
GS-BMVI; Gwinner’s section theorem; Aummann’s measurable selection; Minty’s technique; solution setsReferences:
| [1] | Li, X.; Zou, Y. Z., Existence result and error bounds for a new class of inverse mixed quasi-variational inequalities, J. Inequal. Appl., 2016 (2016) · Zbl 1337.47080 |
| [2] | Tang, Y. X.; Guan, J. Y.; Xu, Y. C., A kind of system of multivariate variational inequalities and the existence theorem of solutions, J. Inequal. Appl., 2017 (2017) · Zbl 1375.49012 |
| [3] | Tang, G. J.; Li, Y. S., Uniqueness and stability of the solution for strongly pseudomonotone variational inequalities in Banach spaces, Adv. Math., 47, 3, 137-144 (2018) |
| [4] | Díaz, J. I.; Véron, L., Existence theory and qualitative properties of the solutions of some first order quasilinear variational inequalities, Indiana Univ. Math. J., 32, 3, 319-361 (2020) · Zbl 0488.35042 |
| [5] | Daniele, P.; Giuffré, S., Random variational inequalities and the random traffic equilibrium problem, J. Optim. Theory Appl., 167, 1, 1-19 (2014) |
| [6] | Jadamba, B.; Raciti, F., Variational inequality approach to stochastic Nash equilibrium problems with an application to Cournot oligopoly, J. Optim. Theory Appl., 165, 3, 1050-1070 (2015) · Zbl 1316.49012 |
| [7] | Bensoussan, A.; Li, Y.; Yam, S. C.P., Backward stochastic dynamics with a subdifferential operator and non-local parabolic variational inequalities, Stoch. Process. Appl., 128, 2, 644-688 (2017) · Zbl 1381.37063 |
| [8] | Barbagallo, A.; Scilla, G., Stochastic weighted variational inequalities in non-pivot Hilbert spaces with applications to a transportation model, J. Math. Anal. Appl., 457, 2, 1118-1134 (2017) · Zbl 1377.49008 |
| [9] | Chen, X. J.; Pong, T. K.; Wets, J. B., Two-stage stochastic variational inequalities: an ERM-solution procedure, Math. Program., 165, 1, 71-111 (2017) · Zbl 1386.90157 |
| [10] | Rockafellar, R. T.; Sun, J., Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging, Math. Program. (2018) |
| [11] | Ren, J. G.; Shi, Q.; Wu, J., Limit theorems for stochastic variational inequalities with non-Lipschitz coefficients, Potential Anal. (2018) |
| [12] | Zhang, S. S.; Huang, N. J., Random generalized set-valued quasi-complementarity problems, Acta Math. Appl. Sin., 16, 3, 396-405 (1993) · Zbl 0793.47063 |
| [13] | Ravat, U.; Shanbhag, U. V., On the existence of solutions to stochastic quasi-variational inequality and complementarity problems, Math. Program., 165, 1, 291-330 (2017) · Zbl 1375.90231 |
| [14] | Zhang, S. S.; Ma, Y. H., KKM technique and its applications, Appl. Math. Mech., 14, 1, 11-20 (1993) · Zbl 0788.54049 |
| [15] | Fan, K., A generalization of Tychonoff’s fixed point theorem, Math. Ann., 142, 3, 305-310 (1961) · Zbl 0093.36701 |
| [16] | Tan, K. K.; Yuan, X. Z., On deterministic and random fixed points, Proc. Am. Math. Soc., 119, 3, 849-856 (1993) · Zbl 0801.47044 |
| [17] | Aumann, R. J., Integrals of set-valued functions, J. Math. Anal. Appl., 12, 1, 1-12 (1965) · Zbl 0163.06301 |
| [18] | Himmelberg, C. J.; Parthasarathy, T.; Vleck, F. S.V., On measurable relations, Fundam. Math., 111, 2, 52-72 (1981) · Zbl 0465.28002 |
| [19] | Chuong, N. M.; Thuan, N. X., Random nonlinear variational inequalities for mappings of monotone type in Banach spaces, Stoch. Anal. Appl., 24, 3, 489-499 (2006) · Zbl 1093.60043 |
| [20] | Wu, D. P., On random variational inequalities and random complementarity problem, Sichuan Daxue Xuebao, 28, 5, 535-537 (2005) · Zbl 1095.47505 |
| [21] | Minty, G. J., On the generalization of a direct method of the calculus of variations, Bull. Am. Math. Soc., 73, 3, 315-322 (1967) · Zbl 0157.19103 |
| [22] | Zhou, W., A class of generalized random nonlinear variational inequalities in Hilbert spaces, J. Southwest Univ. Natur. Sci. Ed., 42, 4, 443-445 (2016) |
| [23] | Lee, B. S.; Salahuddin, Minty lemma for inverted vector variational inequalities, Optimization, 66, 3, 351-359 (2017) · Zbl 1358.49008 |
| [24] | Han, Y.; Huang, N. J.; Lu, J., Existence and stability of solutions to inverse variational inequality problems, Appl. Math. Mech., 38, 5, 137-152 (2017) |
| [25] | Liu, Z. H.; Zeng, S.; Motreanu, D., Evolutionary problems driven by variational inequalities, J. Differ. Equ., 260, 9, 6787-6799 (2016) · Zbl 1341.47088 |
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