Advertisement

Set-Valued Analysis

, 16:1089 | Cite as

Lower Semicontinuity of the Solution Map to a Parametric Generalized Variational Inequality in Reflexive Banach Spaces

  • B. T. KienEmail author
Article

Abstract

This paper is concerned with the study of solution stability of a parametric generalized variational inequality in reflexive Banach spaces. Under the requirements that the operator of a unperturbed problem is of class (S) +  and operators under consideration are pseudo-monotone and demicontinuous, we show that the solution map of a parametric generalized variational inequality is lower semicontinuous. The obtained results are proved without conditions related to the degree theory and the metric projection.

Keywords

Parametric generalized variational inequality Generalized equation Lower semicontinuity Pseudo-monotonicity 

Mathematics Subject Classifications (2000)

47J20 49J40 49J53 90C33 

References

  1. 1.
    Bessis, D.N., Ledyaev, Yu.S.,Vinter, R.B.: Dualization of the Euler and Hamiltonian inclusions. Nonlinear Anal. 43, 861–882 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Browder, F.E., Hess, P.: Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 11, 251–294 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chang, D., Pang, J.S.: The generalized quasi-variational inequality problem. Math. Oper. Res. 2, 211–222 (1982)CrossRefGoogle Scholar
  4. 4.
    Cioranescu, I.: Geometry of Banach Spaces Duality Mappings and Nonlinear Problems. Kluwer, Dordrecht (1990)zbMATHGoogle Scholar
  5. 5.
    Dafermos, S.: Sensitivity analysis in variational inequalities. Math. Oper. Res. 13, 421–434 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Domokos, A.: Solution sensitivity of variational inequalities. J. Math. Math. Appl. 230, 382–389 (1999)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Dontchev, A.L., Hager, W.W.: Impicit functions, Lipschitz maps, and stability in optimization. Math. Oper. Res. 19, 753–768 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dontchev, A.L.: Implicit function theorems for generalized equations. Math. Programming 70, 91–106 (1995)MathSciNetGoogle Scholar
  9. 9.
    Kien, B.T., Wong, M.M., Wong, N.C., Yao, J.C.: Solution existence of variational inequalities with pseudomonotone operators in the sense of Brezis. J. Optim. Theory Appl. (2008, in press)Google Scholar
  10. 10.
    Kien, B.T., Yao, J.C.: Localization of generalized normal maps and stability of variational inequalities in reflexive Banach spaces. Set-Valued Anal. (2008, in press)Google Scholar
  11. 11.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and their Applications. Academic, London (1980)zbMATHGoogle Scholar
  12. 12.
    Levy, A.B., Rockafellar, R.T.: Sensitivity analysis of solutions to generalized equations. Trans. Amer. Math. Soc. 345, 661–671 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Levy, A.B.: Sensitivity of solutions to variational inequalities on Banach spaces. SIAM J. Control Optim. 38, 50–60 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Levy, A.B., Mordukhovich, B.S.: Coderivatives in parametric optimization. Math. Programming 99, 311–327 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Mangasarian, O.L., Shiau, T.-H.: Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems. SIAM J Control Optim. 25, 583–595 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Mansour, M.A., Aussel, D.: Quasimonotone variational inequalities and qusiconvex programming: qualitatve stability. Pac. J. Optim. 2, 611–626 (2006)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, New York (2006)Google Scholar
  18. 18.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Springer, New York (2006)Google Scholar
  19. 19.
    Robinson, S.M.: Regularity and stability for convex multivalued functions. Math. Oper. Res. 1, 130–143 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Robinson, S.M.: An implicit-function theorem for a class of nonsmooth functions. Math. Oper. Res. 16, 292–309 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Robinson, S.M.: Normal maps induced by linear transformations. Math. Oper. Res. 17, 691–714 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Robinson, S.M.: Constraint nondegeneracy in variational analysis. Math. Oper. Res. 28, 201–232 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Robinson, S.M.: Localized normal maps and the stability of variational conditions. Set-Valued Anal. 12, 259–274 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Robinson, S.M.: Solution continuity affine variational inequalities. SIAM. J. Optim. 18, 1046–1060 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Robinson, S.M., Lu, S.: Solution continuity in variational conditions. J. Glob. Optim. (2008, in press)Google Scholar
  26. 26.
    Rockafellar, R.T., Wets, R.J.: Variational Analysis. Springer, Berlin (1998)zbMATHGoogle Scholar
  27. 27.
    Sion, M.: On general minimax theorems. Pacific J. Math. 8, 171–176 (1958)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Yen, N.D.: Hölder continuity of solution to a parametric variational inequality. Appl. Math. Optim. 31, 245–255 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Yen, N.D.: Lipschitz continuity of solutions of variational inequalities with a parametric polyhedral constraint. Math. Oper. Res. 20, 695–707 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Yen, N.D., Lee, G.M.: Solution sensitivity of a class of variational inequalities. J. Math. Anal. Appl. 215, 48–55 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Zeidler, E.: Nonlinear Functional Analysis and its Application, II/B: Nonlinear Monotone Operators. Springer, Heidelberg (1990)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Information and TechnologyHanoi University of Civil EngineeringHanoiVietnam

Personalised recommendations