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Journal of Optimization Theory and Applications

, Volume 147, Issue 3, pp 607–620 | Cite as

Generalized Vector Quasivariational Inclusion Problems with Moving Cones

  • P. H. Sach
  • L. J. LinEmail author
  • L. A. Tuan
Article

Abstract

This paper deals with the generalized vector quasivariational inclusion Problem (P1) (resp. Problem (P2)) of finding a point (z 0,x 0) of a set E×K such that (z 0,x 0)∈B(z 0,x 0A(z 0,x 0) and, for all ηA(z 0,x 0),
$$\begin{array}{l}F(z_0,x_0,\eta)\subset G(z_0,x_0,x_0)+C(z_0,x_0)\cr \mathrm{[resp.}F(z_0,x_0,x_0)\subset G(z_0,x_0,\eta)+C(z_0,x_0)],\end{array}$$
where A:E×K→2 K , B:E×K→2 E , C:E×K→2 Y , F,G:E×K×K→2 Y are some set-valued maps and Y is a topological vector space. The nonemptiness and compactness of the solution sets of Problems (P1) and (P2) are established under the verifiable assumption that the graph of the moving cone C is closed and that the set-valued maps F and G are C-semicontinuous in a new sense (weaker than the usual sense of semicontinuity).

Keywords

Generalized vector quasivariational inclusion problem, Set-valued maps Existence theorems Moving cones Generalized concavity 

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References

  1. 1.
    Li, S.L., Teo, K.L., Yang, X.Q.: Generalized vector quasi-equilibrium problems. Math. Methods Oper. Res. 61, 385–397 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Li, S.L., 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) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    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) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Luc, D.T., Penot, J.P.: Convergence of asymptotic directions. Trans. Am. Math. Soc. 353, 4095–4121 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Lin, L.J., Tan, N.X.: On quasivariational inclusion problems of type I and related problems. J. Glob. Optim. 39, 393–407 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Tuan, L.A., Sach, P.H.: Existence theorems for some generalized equilibrium problems with set-valued maps. Vietnam J. Math. 33(1), 111–122 (2005) zbMATHMathSciNetGoogle Scholar
  7. 7.
    Lin, L.J., Chen, H.L.: The study of KKM theorems with applications to vector equilibrium problems and implicit vector variational inequalities problems. J. Glob. Optim. 32, 135–157 (2005) zbMATHCrossRefGoogle Scholar
  8. 8.
    Lin, L.J., Wan, W.P.: KKM type theorems and coincidence theorems with applications to the existence of equilibria. J. Optim. Theory Appl. 123, 105–122 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fu, J.Y., Wang, S.H., Huang, Z.D.: New type of generalized vector quasiequilibrium problem. J. Optim. Theory Appl. 135, 643–652 (2007) zbMATHCrossRefMathSciNetGoogle 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) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Anh, L.Q., Khanh, P.Q.: Semicontinuity of the solution sets to parametric quasivariational inclusions with applications to traffic networks, II: lower semicontinuities applications. Set-Valued Anal. 16, 943–960 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Sach, P.H., Tuan, L.A.: Existence results for generalized vector quasi-equilibrium problems. J. Optim. Theory Appl. 133, 229–240 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Luc, D.T.: An abstract problem in variational analysis. J. Optim. Theory Appl. 138, 65–76 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Khanh, P.Q., Luc, D.T.: Stability of solutions in parametric variational relation problems. Set-Valued Anal. 16, 1015–1035 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Aubin, J.P.: Mathematical Methods of Game and Economic Theory. North-Holland, Amsterdam (1979) zbMATHGoogle Scholar
  16. 16.
    Park, S.: Some coincidence theorems on an acyclic multifunctions and applications to KKM theory. In: Tan, K.-K. (ed.) Fixed Point Theory and Applications, pp. 248–277. World Scientific, Singapore (1992) Google Scholar
  17. 17.
    Fan, K.: A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142, 305–310 (1961) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Sach, P.H., Tuan, L.A.: Generalizations of vector quasivariational inclusion problems with set-valued maps. J. Glob. Optim. 43, 23–45 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Tuan, L.A., Sach, P.H.: Existence of solutions of generalized quasivariational inequalities with set-valued maps. Acta Math. Vietnam. 29(3), 309–316 (2004) zbMATHMathSciNetGoogle Scholar
  20. 20.
    Sach, P.H.: On a class of generalized vector quasiequilibrium problems with set-valued maps. J. Optim. Theory Appl. 139, 337–350 (2008) CrossRefMathSciNetGoogle Scholar
  21. 21.
    Ding, X.P.: Generalized GKKM theorems in generalized convex spaces and their applications. J. Math. Anal. Appl. 226, 21–23 (2002) CrossRefGoogle Scholar
  22. 22.
    Ding, X.P., Park, J.Y.: Generalized vector equilibrium problems in generalized convex spaces. J. Optim. Theory Appl. 120, 327–353 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Anh, L.Q., Khanh, P.Q.: Semicontinuity of the solution sets to parametric quasivariational inclusions with applications to traffic networks, I: upper semicontinuities. Set-Valued Anal. 16, 267–279 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Hai, N.X., Khanh, P.Q.: The solution existence of general variational inclusion problems. J. Math. Anal. Appl. 328, 1268–1277 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Hai, N.X., Khanh, P.Q.: Systems of set-valued quasivariational inclusion problems. J. Optim. Theory Appl. 135, 55–67 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Gopfert, A., Riahi, H., Tammer, C., Zaninescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003) Google Scholar
  27. 27.
    Ding, X.P.: Existence of solutions for quasi-equilibrium problem in noncompact topological spaces. Comput. Math. Appl. 39, 13–21 (2000) zbMATHCrossRefGoogle Scholar
  28. 28.
    Klein, E., Thompson, A.C.: Theory of Correspondences. Wiley, New York (1984) zbMATHGoogle Scholar
  29. 29.
    Luc, D.T., Tan, N.X.: Existence conditions in variational inclusions with constraints. Optimization 53, 505–515 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Tan, N.X.: On the existence of solution of quasivariational inclusion problems. J. Optim. Theory Appl. 123, 619–638 (2004) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Institute of MathematicsHanoiVietnam
  2. 2.Department of MathematicsNational Changhua University of EducationChanghuaTaiwan
  3. 3.Ninh Thuan College of PedagogyNinh ThuanVietnam

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