×

zbMATH — the first resource for mathematics

Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems. (English) Zbl 1048.49004
Let \(X,M,\Lambda\) be Hausdorff topological spaces, \(Y\) be a topological vector space and \(C\subseteq Y\) be closed and such that \(\text{int}C\neq\emptyset\). Given two multifunctions \(K:X\times\Lambda\rightarrow2^{X}\) and \(F:X\times X\times M\rightarrow2^{Y}\), the “parametric vector quasiequilibrium problem” consists in finding, given \(\lambda\in\Lambda\) and \(\mu\in M\), some \(\bar{x}\in clK(\bar{x},\lambda)\) such that \(F(\bar{x} ,y,\mu)\cap(Y\backslash-\text{int}C)\neq\emptyset\) for every \(y\in K(\bar{x} ,\lambda)\).
Assuming that the solution set \(S_{1}(\lambda,\mu)\) is nonempty in a neighborhood of \((\lambda_{0},\mu_{0})\in\Lambda\times M\), the present paper gives necessary conditions for the multifunction \(S_{1}\) to be lower semicontinuous, or upper semicontinuous. Also, a “strong” version of the quasiequilibrium problem is investigated, and sufficient conditions are given for its solution set to be equal to \(S_{1}(\lambda,\mu)\). These results generalize and sometimes improve previously known results on quasivariational inequalities.

MSC:
49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
49J45 Methods involving semicontinuity and convergence; relaxation
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ansari, Q.H.; Bazán, F.F., Generalized vector quasiequilibrium problems with applications, J. math. anal. appl., 277, 246-256, (2003) · Zbl 1022.90023
[2] Ansari, Q.H.; Konnov, I.V.; Yao, J.C., Existence of a solution and variational principles for vector equilibrium problems, J. optim. theory appl., 110, 481-492, (2001) · Zbl 0988.49004
[3] Aubin, J.P.; Frankowska, H., Set-valued analysis, (1990), Birkhäuser Boston
[4] Bianchi, M.; Hadjisavas, N.; Schaible, S., Vector equilibrium problems with generalized monotone bifunctions, J. optim. theory appl., 92, 527-542, (1997) · Zbl 0878.49007
[5] Bianchi, M.; Schaible, S., Generalized monotone bifunctions and equilibrium problems, J. optim. theory appl., 90, 31-43, (1996) · Zbl 0903.49006
[6] Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math. student, 63, 123-145, (1994) · Zbl 0888.49007
[7] Chadli, O.; Chbani, Z.; Riahi, H., Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities, J. optim. theory appl., 105, 299-323, (2000) · Zbl 0966.91049
[8] Chadli, O.; Chbani, Z.; Riahi, H., Equilibrium problems and noncoercive variational inequalities, Optimization, 50, 17-27, (2001) · Zbl 1022.49013
[9] Chadli, O.; Riahi, H., On generalized vector equilibrium problems, J. global optim., 16, 33-41, (2000) · Zbl 0966.91023
[10] Y.H. Cheng, D.L. Zhu, Global stability results for the weak vector variational inequality, J. Global Optim., in press · Zbl 1097.49006
[11] Dafermos, S., Sensitivity analysis in variational inequalities, Math. oper. res., 13, 421-434, (1988) · Zbl 0674.49007
[12] Ding, X.P., Existence of solutions for quasiequilibrium problems in noncompact topological spaces, Comput. math. appl., 39, 13-21, (2000)
[13] Ding, X.P.; Luo, C.L., On parametric generalized quasivariational inequalities, J. optim. theory appl., 100, 195-205, (1999) · Zbl 0930.90080
[14] Domokos, A., Solution sensitivity of variational inequalities, J. math. anal. appl., 230, 382-389, (1999) · Zbl 0927.49005
[15] Fu, J.Y.; Wan, A.H., Generalized vector equilibrium problems with set-valued mappings, Math. methods oper. res., 56, 259-268, (2002) · Zbl 1023.90057
[16] N.X. Hai, P.Q. Khanh, Existence of solution to general quasiequilibrium problems and application, submitted for publication
[17] Kassay, G.; Kolumbán, J., Variational inequalities given by semi-pseudo-monotone mappings, Nonlinear anal., 5, 35-50, (2000) · Zbl 0981.49006
[18] Khanh, P.Q.; Luu, L.M., Upper semicontinuity of the solution set of parametric multivalued vector quasivariational inequalities and applications, J. global optim., (2004), in press · Zbl 1146.49006
[19] P.Q. Khanh, L.M. Luu, Lower and upper semicontinuity of the solution sets and approximate solution sets to parametric multivalued quasivariational inequalities, submitted for publication · Zbl 1146.49006
[20] Konnov, I., Combined relaxation methods for variational inequalities, (2001), Springer Berlin · Zbl 1044.49004
[21] Lancaster, K., Mathematical economic, (1968), Macmillan New York
[22] Levy, A.B., Sensitivity of solutions to variational inequalities on Banach spaces, SIAM J. control optim., 38, 50-60, (1999) · Zbl 0951.49031
[23] Li, S.J.; Chen, G.Y.; Teo, K.L., On the stability of generalized vector quasivariational inequality problems, J. optim. theory appl., 113, 283-295, (2002) · Zbl 1003.47049
[24] J.L. Lin, Q.H. Ansari, J.Y. Wu, Geometric properties and coincidence theorems with applications to generalized vector equilibrium problems, J. Optim. Theory Appl., in press · Zbl 1063.90062
[25] Lin, L.J.; Yu, Z.T.; Kassay, G., Existence of equilibria for multivalued mappings and its applications to vectorial equilibria, J. optim. theory appl., 114, 189-208, (2002) · Zbl 1023.49014
[26] Muu, L.D., Stability property of a class of variational inequalities, Math. operationsforsch. statist. ser. optim., 15, 347-351, (1984) · Zbl 0553.49007
[27] Noor, M.A., Generalized quasivariational inequalities and implicit wiener – hopf equations, Optimization, 45, 197-222, (1999) · Zbl 0939.49009
[28] Robinson, S.M., Sensitivity analysis of variational inequalities by normal-map techniques, () · Zbl 0861.49009
[29] Yen, N.D., Lipschitz continuity of solutions of variational inequalities with a parametric polyhedral constraint, Math. oper. res., 20, 695-708, (1995) · Zbl 0845.90116
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.