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Continuity of the Solution Set to Parametric Weak Vector Equilibrium Problems

  • X. H. GongEmail author
Article

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.

Keywords

Parametric weak vector equilibrium problems Upper semicontinuity Lower semicontinus Monotone mapping 

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References

  1. 1.
    Ansari, Q.H., Oettli, W., Schläger, D.: A generalization of vector equilibria. Math. Methods Oper. Res. 46, 147–152 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bianchi, M., Hadjisavvas, N., Schaible, S.: Vector equilibrium problems with generalized monotone bifunctions. J. Optim. Theory Appl. 92, 527–542 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    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) Google Scholar
  4. 4.
    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) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Yang, X.Q.: Vector variational inequality and its duality. Nonlinear Anal. Theory Methods Appl. 21, 869–877 (1993) zbMATHCrossRefGoogle Scholar
  6. 6.
    Yu, S.J., Yao, J.C.: On vector variational inequalities. J. Optim. Theory Appl. 89, 749–769 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Lee, G.M., Lee, B.S., Chang, S.S.: On vector quasivariational inequalities. J. Math. Anal. Appl. 203, 626–638 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Konnov, I.V., Yao, J.C.: On the generalized vector variational inequality problem. J. Math. Anal. Appl. 206, 42–58 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fu, J.Y.: Generalized vector quasi-equilibrium problems. Math. Methods Oper. Res. 52, 57–64 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    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) Google Scholar
  11. 11.
    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) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Chiang, C., Chadli, O., Yao, J.C.: Generalized vector equilibrium problems with trifunctions. J. Glob. Optim. 30, 135–154 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ding, X.P., Park, J.Y.: Generalized vector equilibrium problems in generalized convex spaces. J. Optim. Theory Appl. 120, 327–353 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Huang, N.J.: On vector variational inequalities in reflexive Banach spaces. J. Glob. Optim. 32, 495–505 (2005) zbMATHCrossRefGoogle Scholar
  15. 15.
    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) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Cheng, Y.H., Zhu, D.L.: Global stability results for the weak vector variational inequality. J. Glob. Optim. 32, 543–550 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Khanh, P.Q., Luu, L.M.: Upper semicontinuity of the solution set to parametric vector quasivariational inequalities. J. Glob. Optim. 32, 569–580 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Anh, L.Q., Khanh, P.Q.: Semicontinuity of the solution sets to parametric quasiequilibrium problems. J. Math. Anal. Appl. 294, 699–711 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    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) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    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 Google Scholar
  21. 21.
    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) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Bianchi, M., Pini, R.: Sensitivity for parametric vector equilibria. Optimization 55, 221–230 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Huang, N.J., Li, J., Thompson, H.B.: Stability for parametric implicit vector equilibrium problems. Math. Comput. Model. 43, 1267–1274 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Ait Mansour, M., Riahi, H.: Sensitivity analysis for abstract equilibrium problems. J. Math. Anal. Appl. 306, 648–691 (2005) MathSciNetGoogle Scholar
  25. 25.
    Bianchi, M., Pini, R.: A note on stability for parametric equilibrium problems. Oper. Res. Lett. 31, 445–450 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    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) CrossRefMathSciNetGoogle Scholar
  27. 27.
    Gong, X.H.: Efficiency and Henig efficiency for vector equilibrium problems. J. Optim. Theory Appl. 108, 139–154 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Gong, X.H.: Connectedness of the solution sets and scalarization for vector equilibrium problems. J. Optim. Theory Appl. 133, 151–161 (2007) CrossRefMathSciNetGoogle Scholar
  29. 29.
    Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Academic Press, New York (1985) zbMATHGoogle Scholar
  30. 30.
    Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984) zbMATHGoogle Scholar
  31. 31.
    Muselli, E.: Upper and lower semicontinuity for set-valued mappings involving constraints. J. Optim. Theory Appl. 106, 527–550 (2000) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsNanchang UniversityNanchangChina

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