Skip to main content
SpringerLink
Log in

Stability of Solutions in Parametric Variational Relation Problems

  • Published:
Set-Valued Analysis Aims and scope Submit manuscript

Abstract

The purpose of this paper is to investigate topological properties and stability of solution sets in parametric variational relation problems. The results of the paper give a unifying way to treat these questions in the theory of variational inequalities, variational inclusions and equilibrium problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anh, L.Q., Khanh, P.Q.: Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems. J. Math. Anal. Appl. 294, 699–711 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Anh, L.Q., Khanh, P.Q.: On the stability of the solution set of general multivalued vector quasiequilibrium problems. J. Optim. Theory Appl. 135, 271–284 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Anh, L.Q., Khanh, P.Q.: Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks, I: Upper semicontinuities. Set-Valued Anal. 16, 267–279 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Anh, L.Q., Khanh, P.Q.: Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks, II: Lower semicontinuities. Applications. Set-Valued Anal. online first (2008)

  5. Anh, L.Q., Khanh, P.Q.: Various kinds of semicontinuity and the solution sets of parametric multivalued symmetric vector quasiequilibrium problems. J. Global Optim. 41, 539–558 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Springer, New York (1990)

    MATH  Google Scholar 

  7. Bianchi, M., Pini, R.: A note on stability for parametric equilibrium problems. Oper. Res. Lett. 31, 445–450 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cottle, R.W.: Nonlinear programs with positively bounded jacobians. SIAM J. Appl. Math. 14, 147–158 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  9. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementary Problems, I and II. Springer, New York (2003)

    Google Scholar 

  10. Flores-Bazan, F.: Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case. SIAM J. Optim. 11, 675–690 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria. Kluwer, London (2000)

    MATH  Google Scholar 

  12. Harker, P.T., Pang, J.S.: Finite-dimensional variational inequality and nonlinear complementary problems: a survey of theory, algorithms and applications. Math. Programming 48, 169–220 (1990)

    MathSciNet  Google Scholar 

  13. Kimura, K., Yao, J.-C.: Sensitivity analysis of solution mappings of parametric vector quasiequilibrium problems. J. Global Optim. 41, 187–202 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Luc, D.T.: Frechet approximate jacobian and local uniqueness of solutions in variational inequalities. J. Math. Anal. Appl. 266, 629–646 (2002)

    Article  Google Scholar 

  15. Luc, D.T.: An abstract problem of variational analysis. J. Optim. Theory Appl. 138, 65–76 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Luc, D.T., Noor, M.A.: Local uniqueness of solutions of general variational inequalities. J. Optim. Theory Appl. 117, 103–119 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, International University of Hochiminh City, Linh Trung, Thu Duc, Hochiminh City, Vietnam

    P. Q. Khanh

  2. Laboratoire d’Analyse Nonlinéaire et Géométrie (EA 2151), Université d’Avignon et des Pays de Vaucluse, 84018, Avignon, France

    D. T. Luc

Authors
  1. P. Q. Khanh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. T. Luc
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. Q. Khanh.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Khanh, P.Q., Luc, D.T. Stability of Solutions in Parametric Variational Relation Problems. Set-Valued Anal 16, 1015–1035 (2008). https://doi.org/10.1007/s11228-008-0101-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-008-0101-0

Keywords

Mathematics Subject Classification (2000)

Search

Navigation