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New Scalarizing Approach to the Stability Analysis in Parametric Generalized Ky Fan Inequality Problems

  • Pham Huu SachEmail author
  • Le Anh Tuan
Article

Abstract

This paper gives sufficient conditions for the upper and lower semicontinuities of the solution mapping of a parametric mixed generalized Ky Fan inequality problem. We use a new scalarizing approach quite different from traditional linear scalarization approaches which, in the framework of the stability analysis of solution mappings of equilibrium problems, were useful only for weak vector equilibrium problems and only under some convexity and strict monotonicity assumptions. The main tools of our approach are provided by two generalized versions of the nonlinear scalarization function of Gerstewitz. Our stability results are new and are obtained by a unified technique. An example is given to show that our results can be applied, while some corresponding earlier results cannot.

Keywords

Nonlinear scalarization function Ky Fan inequality problem Set-valued map Semicontinuity Strict monotonicity 

Notes

Acknowledgements

The authors would like to thank the referees for their suggestions, which improved the paper.

The support of The National Foundation for Science and Technology Development, Vietnam, is gratefully acknowledged.

References

  1. 1.
    Fan, K.: A minimax inequality and its applications. In: Shisha, O. (ed.) Inequalities III, pp. 103–113. Academic Press, New York (1972) Google Scholar
  2. 2.
    Brezis, H., Nirenberg, L., Stampacchia, G.: A remark on Ky Fan’s minimax principle. Boll. Unione Mat. Ital. (III) 6, 129–132 (1972) MathSciNetGoogle Scholar
  3. 3.
    Sach, P.H., Tuan, L.A., Lee, G.M.: Sensitivity results for a general class of generalized vector quasi-equilibrium problems with set-valued maps. Nonlinear Anal. 71, 571–586 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Tuan, L.A., Lee, G.M., Sach, P.H.: Upper semicontinuity in a parametric general variational problem and application. Nonlinear Anal. 72, 1500–1513 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Tuan, L.A., Lee, G.M., Sach, P.H.: Upper semicontinuity result for the solution mapping of a mixed parametric generalized vector quasiequilibrium problem with moving cones. J. Glob. Optim. 47, 639–660 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Khanh, P.Q., Luc, D.T.: Stability of solutions in parametric variational relation problems. Set-Valued Anal. 16, 1015–1035 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Anh, L.Q., Khanh, P.Q.: Semicontinuity of the solutions sets to parametric quasivariational inclusions with applications to traffic networks. I. Upper semicontinuities. Set-Valued Anal. 16, 267–279 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Anh, L.Q., Khanh, P.Q.: Semicontinuity of the solutions sets to parametric quasivariational inclusions with applications to traffic networks. II. Lower semicontinuities applications. Set-Valued Anal. 16, 943–960 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kimura, K., Yao, J.C.: Sensitivity analysis of solution mappings of parametric vector-equilibrium problems. J. Glob. Optim. 41, 187–202 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Cheng, Y.H., Zhu, D.L.: Global stability results for the weak vector variational inequality. J. Glob. Optim. 32, 543–550 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Gong, X.H.: Continuity of the solution set to parametric vector equilibrium problem. J. Optim. Theory Appl. 139, 35–46 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Chen, C.R., Li, S.J., Teo, K.L.: Solution semicontinuity of parametric generalized vector equilibrium problems. J. Glob. Optim. 45, 309–318 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Li, S.J., Liu, H.M., Chen, C.R.: Lower semicontinuity of parametric generalized weak vector equilibrium problems. Bull. Aust. Math. Soc. 81, 85–95 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Luc, D.T.: Theory of Vector Opimization. Lect. Notes Econ. Math. Syst., vol. 319. Springer, Berlin (1989) CrossRefGoogle Scholar
  15. 15.
    Gerth, C., Weidner, P.: Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 297–320 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Aubin, J.P.: Mathematical Methods of Game and Economic Theory. North-Holland, Amsterdam (1979) zbMATHGoogle Scholar
  17. 17.
    Hernandez, E., Rodriguez-Marin, L.: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325, 1–18 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Chen, G.Y., Yang, X.Q., Yu, H.: A nonlinear scalarization function and generalized quasi-vector equilibrium problems. J. Glob. Optim. 32, 451–466 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Li, S.J., Teo, K.L., Yang, X.Q.: Generalized vector quasi-equilibrium problems. Math. Methods Oper. Res. 61, 385–397 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Li, S.H., Teo, K.L., Yang, X.Q., Wu, S.Y.: Gap functions and existence of solutions to generalized vector quasi-equilibrium problems. J. Glob. Optim. 34, 427–440 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Gopfert, A., Riahi, H., Tammer, C., Zalinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003) Google Scholar
  22. 22.
    Gong, X.H.: Scalarization and optimality conditions for vector equilibrium problems. Nonlinear Anal. 73, 3598–3612 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Giannessi, F., Mastroeni, G., Pellegrini, L.: On the theory of vector optimization and variational inequalities. Image space analysis and separation. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria, pp. 141–215. Kluwer, Dordrecht (2000) CrossRefGoogle Scholar
  24. 24.
    References on vector variational inequalities. J. Glob. Optim. 32, 529–536 (2005) Google Scholar
  25. 25.
    Ansari, Q.H., Florez-Bazan, F.: Generalized vector quasi-equilibrium problems with applications. J. Math. Anal. Appl. 277, 246–256 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Lin, L.J., Huang, Y.J., Ansari, Q.H.: Some existence results for solution of generalized vector quasi-equilibrium problems. Math. Methods Oper. Res. 65, 85–98 (2007) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Hanoi Institute of MathematicsHanoiVietnam
  2. 2.Nong Lam UniversityHo Chi Minh CityVietnam
  3. 3.Ninh Thuan College of PedagogyNinh ThuanVietnam

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