Journal of Optimization Theory and Applications

, Volume 135, Issue 2, pp 271–284 | Cite as

On the Stability of the Solution Sets of General Multivalued Vector Quasiequilibrium Problems



We give sufficient conditions for the semicontinuity of solution sets of general multivalued vector quasiequilibrium problems. All kinds of semicontinuities are considered: lower semicontinuity, upper semicontinuity, Hausdorff upper semicontinuity, and closedness. Moreover, we investigate the weak, middle, and strong solutions of quasiequilibrium problems. Many examples are provided to give more insights and comparisons with recent existing results.


Quasiequilibrium problems Lower, upper and Hausdorff upper semicontinuity Closedness of solution multifunctions Quasivariational inequalities 


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  1. 1.
    Muu, L.D.: Stability property of a class of variational inequalities. Math. Oper. Stat. Ser. Optim. 15, 347–351 (1984) MATHGoogle Scholar
  2. 2.
    Bianchi, M., Pini, R.: A note on stability for parametric equilibrium problems. Oper. Res. Lett. 31, 445–450 (2003) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Anh, L.Q., Khanh, P.Q.: Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems. J. Math. Anal. Appl. 294, 699–711 (2004) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Mansour, M.A., Riahi, H.: Sensitivity analysis for abstract equilibrium problems. J. Math. Anal. Appl. 306, 684–691 (2005) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    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) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    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) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994) MATHMathSciNetGoogle Scholar
  8. 8.
    Fu, J.Y., Wan, A.H.: Generalized vector equilibrium problems with set-valued mappings. Math. Methods Oper. Res. 56, 259–268 (2002) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Lin, L.J., Yu, Z.T., Kassay, G.: Existence of equilibria for multivalued mappings and its application to vectorial equilibria. J. Optim. Theory Appl. 114, 189–208 (2002) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ansari, Q.H., Flores-Bazán, F.: Generalized vector quasiequilibrium problems with applications. J. Math. Anal. Appl. 277, 246–256 (2003) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Chen, M.P., Lin, L.J., Park, S.: Remarks on generalized quasiequilibrium problems. Nonlinear Anal. 52, 433–444 (2003) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hai, N.X., Khanh, P.Q.: Existence of solutions to general quasiequilibrium problems and applications. J. Optim. Theory Appl. 133 (2007, to appear, available online) Google Scholar
  13. 13.
    Hai, N.X., Khanh, P.Q.: Systems of multivalued quasiequilibrium problems. Adv. Nonlinear Var. Inequal. 9, 109–120 (2006) MathSciNetGoogle Scholar
  14. 14.
    Khanh, P.Q., Luc, D.T., Tuan, N.D.: Local uniqueness of solutions for equilibrium problems. Adv. Nonlinear Var. Inequal. 9, 1–11 (2006) MathSciNetGoogle Scholar
  15. 15.
    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 (2007) CrossRefMathSciNetGoogle Scholar
  16. 16.
    Bianchi, M., Schaible, S.: Equilibrium problems under generalized convexity and generalized monotonicity. J. Glob. Optim. 30, 121–134 (2004) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Ding, X.P.: Sensitivity analysis for generalized nonlinear implicit quasivariational inclusions. Appl. Math. Lett. 17, 225–235 (2004) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Agarwal, R.P., Huang, N.J., Tan, M.Y.: Sensitivity analysis for a new system of generalized nonlinear mixed quasivariational inclusions. Appl. Math. Lett. 17, 345–352 (2004) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Khanh, P.Q., Luu, L.M.: Upper semicontinuity of the solution set of parametric multivalued vector quasivariational inequalities and applications. J. Glob. Optim. 32, 569–580 (2005) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Khanh, P.Q., Luu, L.M.: Lower and upper semicontinuity of the solution sets and approximate solution sets to parametric multivalued quasivariational inequalities. J. Optim. Theory Appl. 133 (2007, to appear, available online) Google Scholar
  21. 21.
    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) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Cheng, Y.H., Zhu, D.L.: Global stability results for the weak vector variational inequality. J. Glob. Optim. 32, 543–550 (2005) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of MathematicsCantho UniversityCanthoVietnam
  2. 2.Department of MathematicsInternational University of Hochiminh CityHochiminh CityVietnam

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