Skip to main content
SpringerLink
Log in

Generalized Symmetric Vector Quasiequilibrium Problems

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We establish an existence result for scalar quasiequilibrium problems without any continuity requirement on noncompact subsets of locally convex topological vector spaces. As a consequence, we obtain a solution of symmetric scalar quasiequilibrium problem. Moreover, using a so-called nonlinear scalarization function, existence theorems for vector quasiequilibrium problems and general symmetric vector quasiequilibrium problems are obtained.

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. Giannessi, F.: Theorems of alternative quadratic programs and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds.) Variational Inequalities and Complementarity Problems, pp. 151–186. Wiley, Chicherster (1980)

    Google Scholar 

  2. Giannessi, F.: Vector Variational Inequalities and Vector Equilibrium. Kluwer, Dordrecht (1999)

    Google Scholar 

  3. Ansari, Q.H., Wu, S.Y.: Existence and duality of generalized vector equilibrium problems. J. Math. Anal. Appl. 259, 115–126 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ansari, Q.H., Yao, J.C.: An existence result for generalized vector equilibrium problems. Appl. Math. Lett. 12, 53–56 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bianchi, M., Hadjisavvas, N., Schaible, S.: Vector equilibrium problems with generalized monotone bifunctions. J. Optim. Theory Appl. 92, 527–542 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. Fakhar, M., Zafarani, J.: Generalized vector equilibrium problems for pseudomonotone multivalued bifunctions. J. Optim. Theory Appl. 126, 109–124 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Fakhar, M., Zafarani, J.: Equilibrium problems in the quasimonotone case. J. Optim. Theory Appl. 126, 125–136 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hadjisavvas, N., Schaible, S.: From scalar to vector equilibrium problems in the quasimonotone case. J. Optim. Theory Appl. 96, 297–309 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Konnov, I.V., Yao, J.C.: Existence of solutions for generalized vector equilibrium problems. J. Math. Anal. Appl. 233, 328–335 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Oettli, W.: A remarks on vector-valued equilibria and generalized monotonicity. Acta Math. Vietnam. (ii) 22, 213–221 (1997)

    MATH  MathSciNet  Google Scholar 

  11. Oettli, W., Schläger, D.: Generalized vectorial equilibria and generalized monotonicity. In: Brokate, M., Siddigi, A.H. (eds.) Functional Analysis with Current Applications in Science, Technology, and Industry, pp. 145–154. Longman, London (1998)

    Google Scholar 

  12. Oettli, W., Schläger, D.: Existence of equilibria for monotone multivalued mappings. Math. Method Oper. Res. 48, 219–228 (1998)

    Article  MATH  Google Scholar 

  13. Ansari, Q.H., Flores-Bazan, F.: Generalized quasiequilibrium problems with applications. J. Math. Anal. Appl. 277, 246–256 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ansari, Q.H., Yao, J.-C.: On vector quasiequilibrium problems in variational problems and equilibria. In: Maugeri, A., Giannessi, F. (eds.) Kluwer, Dordrecht (2003)

  15. Khanh, P.Q., Luu, L.M.: On the existence of solutions to vector quasivariational inequalities and quasicomplementarity problems with applications to traffic network equilibria. J. Optim. Theory Appl. 123, 533–548 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Giannessi, F., Maugeri, A. (eds.): Variational Analysis and Applications. Nonconvex Optimization and Its Applications, vol. 79. Springer, New York (2005)

    MATH  Google Scholar 

  17. Chadli, O., Riahi, H.: On generalized vector equilibrium problems. J. Glob. Optim. 16, 33–41 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Aubin, J.P.: Mathematical Methods of Game and Economic Theory. North Holland, Amsterdam (1979)

    MATH  Google Scholar 

  19. Cubiotti, P.: Generalized quasivariational inequalities without continuities. J. Optim. Theory Appl. 92, 477–495 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ding, X.P., Tan, K.K.: Generalized variational inequalities and generalized quasivariational inequalities. J. Math. Anal. Appl. 148, 497–508 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  21. El-Arni, A.: Generalized quasivariational inequalities with noncompact sets. J. Math. Anal. Appl. 241, 189–197 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Li, S.L., Teo, K.L., Yang, X.Q.: Generalized vector quasiequilibrium problems. Math. Methods Oper. Res. 61, 385–397 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Noor, M.A., Oettli, W.: On general nonlinear complementarity problems and quasiequilibria. Le Matematiche 49, 313–331 (1994)

    MATH  MathSciNet  Google Scholar 

  24. Fu, J.Y.: Symmetric vector quasiequilibrium problems. J. Math. Anal. Appl. 285, 708–713 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. El-Arni, A.: Generalized quasivariational inequalities with noncompact sets with pseudo-monotone operators. J. Math. Anal. Appl. 249, 515–526 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  26. Yao, J.C.: Generalized quasivariational inequality problems with discontinuous mappings. Math. Oper. Res. 20, 465–478 (1995)

    MATH  MathSciNet  Google Scholar 

  27. Chowdhury, M.S.R., Tan, K.K.: Applications of pseudomonotone operators with some kind of upper semicontinuty in generalized quasivariational inequalities on noncompact sets. Proc. Am. Math. Soc. 126, 2957–2968 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  28. Fang, Y.-P., Huang, N.-J.: Existence results for systems of strong implicit vector variational inequalities. Acta Math. Hung. 103, 265–277 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. Li, J., Huang, N.-J.: Implicit vector equilibrium problems via nonlinear scalarization. Bull. Aust. Math. Soc. 72, 161–172 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  30. Tanaka, T.: Generalized semicontinuity and existence theorems for cone saddle points. Appl. Math. Optim. 36, 313–322 (1997)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Isfahan, Isfahan, Iran

    M. Fakhar

  2. University of Isfahan and Sheikhbahaee University, Isfahan, Iran

    J. Zafarani

  3. Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran

    M. Fakhar

Authors
  1. M. Fakhar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Zafarani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Fakhar.

Additional information

Communicated by F. Giannessi.

The authors express their sincere gratitude to the referee for valuable comments and suggestions. This research was in part supported by a grant from IPM (No. 86470016) and the Center of Excellence for Mathematics, University of Isfahan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fakhar, M., Zafarani, J. Generalized Symmetric Vector Quasiequilibrium Problems. J Optim Theory Appl 136, 397–409 (2008). https://doi.org/10.1007/s10957-007-9310-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-007-9310-1

Keywords

Search

Navigation