Skip to main content
SpringerLink
Log in

Approximate solutions of quasiequilibrium problems in Banach spaces

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

Abstract

In this note we show that a recent existence result on quasiequilibrium problems, which seems to improve deeply some well-known results, is not correct. We exhibit a counterexample and we furnish a generalization of a lemma about continuous \(\varepsilon \)-minimizers of quasiconvex functions depending on a parameter. This allows to establish an existence result of approximate solutions of quasiequilibrium 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. Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. A Hitchhikers Guide, 3rd edn. Springer, Berlin (2006)

    Google Scholar 

  2. Berge, C.: Espaces Topologiques: Fonctions Multivoques. Dunod, Paris (1959)

    MATH  Google Scholar 

  3. Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Existence and solution methods for equilibria. Eur. J. Oper. Res. 227, 1–11 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. Cubiotti, P.: Existence of Nash equilibria for generalized games without upper semicontinuity. Int. J. Game Theory 26, 267–273 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequalities III, pp. 103–113. Academic Press, New York (1972)

    Google Scholar 

  7. Georgiev, P.G.: Parametric Borwein–Preiss variational principle and applications. Proc. Am. Math. Soc. 133, 3211–3226 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Georgiev, P.G., Pardalos, P.M.: Generalized Nash equilibrium problems for lower semi-continuous strategy maps. J. Glob. Optim. 50, 119–125 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Michael, E.: Continuous selections. I. Ann. Math. 63, 361–382 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mosco, U.: Implicit variational problems and quasi variational inequalities. In: Lecture Notes in Mathematics, vol. 543, pp. 83–156. Springer, Berlin (1976)

Download references

Author information

Authors and Affiliations

  1. Department of Information Engineering, Computer Science and Mathematics, University of L’Aquila, L’Aquila, Italy

    M. Castellani & M. Giuli

Authors
  1. M. Castellani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Giuli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Giuli.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Castellani, M., Giuli, M. Approximate solutions of quasiequilibrium problems in Banach spaces. J Glob Optim 64, 615–620 (2016). https://doi.org/10.1007/s10898-015-0386-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-015-0386-0

Keywords

Search

Navigation