Abstract
In this paper, we obtain some stability results for parametric weak vector equilibrium problems in topological vector spaces. We provide sufficient conditions for the continuity of the solution set mapping in parametric weak monotone vector equilibrium problems.
Similar content being viewed by others
References
Ansari, Q.H., Oettli, W., Schläger, D.: A generalization of vector equilibria. Math. Methods Oper. Res. 46, 147–152 (1997)
Bianchi, M., Hadjisavvas, N., Schaible, S.: Vector equilibrium problems with generalized monotone bifunctions. J. Optim. Theory Appl. 92, 527–542 (1997)
Giannessi, F.: Theorem of the alternative, quadratic programs, and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds.) Variational Inequalities and Complementarity Problems, pp. 151–186. Wiley, New York (1980)
Chen, G.Y.: Existence of solution for a vector variational inequality: an extension of the Hartman-Stampacchia theorem. J. Optim. Theory Appl. 74, 445–456 (1992)
Yang, X.Q.: Vector variational inequality and its duality. Nonlinear Anal. Theory Methods Appl. 21, 869–877 (1993)
Yu, S.J., Yao, J.C.: On vector variational inequalities. J. Optim. Theory Appl. 89, 749–769 (1996)
Lee, G.M., Lee, B.S., Chang, S.S.: On vector quasivariational inequalities. J. Math. Anal. Appl. 203, 626–638 (1996)
Konnov, I.V., Yao, J.C.: On the generalized vector variational inequality problem. J. Math. Anal. Appl. 206, 42–58 (1997)
Fu, J.Y.: Generalized vector quasi-equilibrium problems. Math. Methods Oper. Res. 52, 57–64 (2000)
Song, W.: Vector equilibrium problems with set-valued mapping. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, pp. 403–418. Kluwer, Dordrecht (2000)
Lin, L.J., Ansari, Q.H., Wu, J.Y.: Geometric properties and coincidence theorems with applications to generalized vector equilibrium problems. J. Optim. Theory Appl. 117, 121–137 (2003)
Chiang, C., Chadli, O., Yao, J.C.: Generalized vector equilibrium problems with trifunctions. J. Glob. Optim. 30, 135–154 (2004)
Ding, X.P., Park, J.Y.: Generalized vector equilibrium problems in generalized convex spaces. J. Optim. Theory Appl. 120, 327–353 (2004)
Huang, N.J.: On vector variational inequalities in reflexive Banach spaces. J. Glob. Optim. 32, 495–505 (2005)
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)
Cheng, Y.H., Zhu, D.L.: Global stability results for the weak vector variational inequality. J. Glob. Optim. 32, 543–550 (2005)
Khanh, P.Q., Luu, L.M.: Upper semicontinuity of the solution set to parametric vector quasivariational inequalities. J. Glob. Optim. 32, 569–580 (2005)
Anh, L.Q., Khanh, P.Q.: Semicontinuity of the solution sets to parametric quasiequilibrium problems. J. Math. Anal. Appl. 294, 699–711 (2004)
Anh, L.Q., Khanh, P.Q.: On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems. J. Math. Anal. Appl. 321, 308–315 (2006)
Anh, L.Q., Khanh, P.Q.: On the stability of the solution sets of general multivalued vector quasiequilibrium problems. J. Optim. Theory Appl. 14 (2007). Available online
Anh, L.Q., Khanh, P.Q.: Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces. J. Glob. Optim. 37, 449–465 (2007)
Bianchi, M., Pini, R.: Sensitivity for parametric vector equilibria. Optimization 55, 221–230 (2006)
Huang, N.J., Li, J., Thompson, H.B.: Stability for parametric implicit vector equilibrium problems. Math. Comput. Model. 43, 1267–1274 (2006)
Ait Mansour, M., Riahi, H.: Sensitivity analysis for abstract equilibrium problems. J. Math. Anal. Appl. 306, 648–691 (2005)
Bianchi, M., Pini, R.: A note on stability for parametric equilibrium problems. Oper. Res. Lett. 31, 445–450 (2003)
Khanh, P.Q., Luu, L.M.: Lower semicontinuity and upper semicontinuity of the solution sets to parametric multivalued quasivariational inequalities. J. Optim. Theory Appl. 133, 329–339 (2007)
Gong, X.H.: Efficiency and Henig efficiency for vector equilibrium problems. J. Optim. Theory Appl. 108, 139–154 (2001)
Gong, X.H.: Connectedness of the solution sets and scalarization for vector equilibrium problems. J. Optim. Theory Appl. 133, 151–161 (2007)
Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Academic Press, New York (1985)
Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)
Muselli, E.: Upper and lower semicontinuity for set-valued mappings involving constraints. J. Optim. Theory Appl. 106, 527–550 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by X.Q. Yang.
This research was partially supported by the National Natural Science Foundation of China (Grant 10561007) and the Natural Science Foundation of Jiangxi Province, China.
Rights and permissions
About this article
Cite this article
Gong, X.H. Continuity of the Solution Set to Parametric Weak Vector Equilibrium Problems. J Optim Theory Appl 139, 35–46 (2008). https://doi.org/10.1007/s10957-008-9429-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-008-9429-8