Skip to main content
SpringerLink
Log in

Counterexamples on some articles on quasi-variational inclusion problems

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this paper, by counterexamples, we show that Proposition 3.1 in [5] which has a crucial role for proving the main results is not correct. Also, we give counterexamples, which show that some claims in the proof of the main results in [7, 12] are not valid. Finally, by applying some slightly modifications, we claim that these results can be proved in a similar manner.

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. Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 64, 1–23 (1993)

    Google Scholar 

  2. Chan D., Pang J.S.: The generalized quasi-variational inequality problem. Math. Oper. Res. 7, 211–222 (1982)

    Article  Google Scholar 

  3. Fan, K.: In: Shisha, O. A minimax inequality and application in inequalities III, Academic Press, New York (1972)

  4. Gurraggio A., Tan N.X.: On general vector quasi-optimization problems. Math. Meth. Oper. Res. 55, 347–358 (2002)

    Article  Google Scholar 

  5. Lin L.J., Tan N.X.: On quasi-variational inclusion problems of type I and related problems. J. Glob. Optim. 39(3), 393–407 (2007)

    Article  Google Scholar 

  6. Lin L.J., Yu Z.T., Kassay G.: Existence of equilibria for monotone multi-valued mappings and its applications to vectorial equilibria. J. Optim. Theory Appl. 114, 189–208 (2002)

    Article  Google Scholar 

  7. Minh N.B., Tan N.X.: On the existence of solutions of quasi-equilibrium problems with constraints. Math. Meth. Oper. Res. 64, 17–31 (2006)

    Article  Google Scholar 

  8. Minh N.B., Tan N.X.: Some sufficient conditions for the existence of equilibrium points concerning multi-valued mappings. Vietnam J. Math. 28, 295–310 (2000)

    Google Scholar 

  9. Park S.: Fixed points and quasi-equilibrium problems. Nonlinear operator theory. Math. Comput. Model. 32, 1297–1304 (2000)

    Article  Google Scholar 

  10. Parida J., Sen A.: A variational-like inequality for multi-functions with applications. J. Math. Anal. Appl. 124, 73–81 (1987)

    Article  Google Scholar 

  11. Schaefer H.H.: Topological vector spaces. Springer-Verlag, New York (1986)

    Google Scholar 

  12. Tan N.X.: On the existence of solutions of quasi-variational inclusion problems. J. Optim. Theory Appl. 123, 619–638 (2004)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mahani Mathematical Research Center and Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran

    H. Mohebi & M. Vatandoost

Authors
  1. H. Mohebi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Vatandoost
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Mohebi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mohebi, H., Vatandoost, M. Counterexamples on some articles on quasi-variational inclusion problems. J Glob Optim 49, 707–712 (2011). https://doi.org/10.1007/s10898-010-9559-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-010-9559-z

Keywords

Mathematics Subject Classification (2000)

Search

Navigation