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Henig proper generalized vector quasiequilibrium problems

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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.

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References

  1. Aubin J.P.: Mathematical methods of game and economic theory. North-Holland, Amsterdam (1979)

    MATH  Google Scholar 

  2. Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63, 123–145 (1994)

    MathSciNet  MATH  Google Scholar 

  3. Borwein J.M., Zhuang D.: Super efficiency in vector optimization. Trans. Am. Math. Soc. 338, 105–122 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Browder F.E.: The fixed point theory of multivalued mappings in topological vector space. Math. Ann. 177, 283–301 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fu J.Y.: Stampacchia generalized vector quasiequilibrium problems and vector saddle points. J. Optim. Theory Appl. 128, 605–619 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gong X.H.: Efficiency and Henig efficiency for vector equilibrium problems. J. Optim. Theory Appl. 108, 139–154 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gong X.H.: Connectedness of the solution sets and scalarization for vector equilibrium problems. J. Optim. Theory Appl. 133, 151–161 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Guerraggio A., Molho E., Zaffaroni A.: On the notion of proper efficiency in vector optimization. J. Optim. Theory Appl. 82, 1–21 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Klein E., Thompson A.C.: Theory of correspondences. Wiley, New York (1984)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lin L.J., Park S.: On some generalized quasi-equilibrium problems. J. Math. Anal. Appl. 224, 167–181 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Michael E.: Continuous selections I. Ann. Math. 214, 361–382 (1956)

    Article  MathSciNet  Google Scholar 

  15. Massey W.S.: Singular homology theory. Springer, New York (1970)

    Google Scholar 

  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. Sach P.H.: On a class of generalized vector quasiequilibrium problems with set-valued maps. J. Optim. Theory Appl. 139, 337–350 (2008)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sach P.H., Tuan L.A.: Existence results for set-valued vector quasi-equilibrium problems. J. Optim. Theory Appl. 133, 229–240 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sach P.H., Tuan L.A.: Generalizations of vector quasivariational inclusion problems with set-valued maps. J. Glob. Optim. 43, 23–45 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sach P.H., Tuan L.A.: Sensitivity in mixed generalized vector quasiequilibrium problems with moving cones. Nonlinear Anal. 73, 713–724 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wang S.H., Fu J.Y.: Stampacchia generalized vector quasiequilibrium problem with set-valued mapping. J. Glob. Optim. 44, 99–110 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yannelis N.C., Prabhakar N.D.: Existence of maximal element and equilibria in linear topological spaces. J. Math. Econ. 12, 233–245 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zheng X.Y.: Proper efficiency in locally convex topological vector spaces. J. Optim. Theory Appl. 94, 469–486 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Pham Huu Sach.

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Sach, P.H. Henig proper generalized vector quasiequilibrium problems. Optim Lett 7, 173–184 (2013). https://doi.org/10.1007/s11590-011-0406-z

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