Advertisement

Optimization Letters

, Volume 7, Issue 1, pp 173–184 | Cite as

Henig proper generalized vector quasiequilibrium problems

  • Pham Huu SachEmail author
Original Paper

Abstract

In this paper, we give sufficient conditions for the existence of a Henig proper efficient solution of a general model in the theory of set-valued vector quasiequilibrium problems with moving cones. The main result of this paper is new, and is established under assumptions of existence of open lower sections and some properties of cone-semicontinuity and cone-concavity of set-valued maps. The moving cones are assumed to have bases which are Hausdorff lower semicontinuous and satisfy an additional suitable property.

Keywords

Existence result Equilibrium problem Proper efficiency Set-valued map Semicontinuity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aubin J.P.: Mathematical methods of game and economic theory. North-Holland, Amsterdam (1979)zbMATHGoogle Scholar
  2. 2.
    Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63, 123–145 (1994)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Borwein J.M., Zhuang D.: Super efficiency in vector optimization. Trans. Am. Math. Soc. 338, 105–122 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Browder F.E.: The fixed point theory of multivalued mappings in topological vector space. Math. Ann. 177, 283–301 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Fu J.Y.: Stampacchia generalized vector quasiequilibrium problems and vector saddle points. J. Optim. Theory Appl. 128, 605–619 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Gong X.H.: Efficiency and Henig efficiency for vector equilibrium problems. J. Optim. Theory Appl. 108, 139–154 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Gong X.H.: Connectedness of the solution sets and scalarization for vector equilibrium problems. J. Optim. Theory Appl. 133, 151–161 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Guerraggio A., Molho E., Zaffaroni A.: On the notion of proper efficiency in vector optimization. J. Optim. Theory Appl. 82, 1–21 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Klein E., Thompson A.C.: Theory of correspondences. Wiley, New York (1984)zbMATHGoogle Scholar
  10. 10.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Lin L.J., Park S.: On some generalized quasi-equilibrium problems. J. Math. Anal. Appl. 224, 167–181 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Lin L.J., Chuang C.S., Wang S.Y.: From quasivariational inclusion problems to Stampacchia vector quasiequilibrium problems, Stampacchia set-valued vector Ekeland’s variational principle and Caristi’s fixed point theorem. Nonlinear Anal. 71, 179–185 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Liu W., Gong X.H.: Proper efficiency for set-valued vector optimization problems and vector variational inequalities. Math. Meth. Oper. Res. 51, 443–457 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Michael E.: Continuous selections I. Ann. Math. 214, 361–382 (1956)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Massey W.S.: Singular homology theory. Springer, New York (1970)Google Scholar
  16. 16.
    Park S.: Some coincidence theorems on acyclic multifunctions and applications to KKM theory. In: Tan, K-K. (eds) Fixed point theory and applications, pp. 248–277. World Scientific Publishers, NJ (1992)Google Scholar
  17. 17.
    Sach P.H.: On a class of generalized vector quasiequilibrium problems with set-valued maps. J. Optim. Theory Appl. 139, 337–350 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Sach P.H., Lin L.J., Tuan L.A.: Generalized vector quasivariational inclusion problems with moving cones. J. Optim. Theory Appl. 147, 607–620 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Sach P.H., Tuan L.A.: Existence results for set-valued vector quasi-equilibrium problems. J. Optim. Theory Appl. 133, 229–240 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Sach P.H., Tuan L.A.: Generalizations of vector quasivariational inclusion problems with set-valued maps. J. Glob. Optim. 43, 23–45 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Sach P.H., Tuan L.A.: Sensitivity in mixed generalized vector quasiequilibrium problems with moving cones. Nonlinear Anal. 73, 713–724 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    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
  23. 23.
    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
  24. 24.
    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
  25. 25.
    Wang S.H., Fu J.Y.: Stampacchia generalized vector quasiequilibrium problem with set-valued mapping. J. Glob. Optim. 44, 99–110 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Yannelis N.C., Prabhakar N.D.: Existence of maximal element and equilibria in linear topological spaces. J. Math. Econ. 12, 233–245 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Zheng X.Y.: Proper efficiency in locally convex topological vector spaces. J. Optim. Theory Appl. 94, 469–486 (1997)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Hanoi Institute of MathematicsHanoiVietnam

Personalised recommendations