Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Existence theorems for generalized vector variational inequalities with a variable ordering relation

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

Abstract

In this paper we study the solvability of the generalized vector variational inequality problem, the GVVI problem, with a variable ordering relation in reflexive Banach spaces. The existence results of strong solutions of GVVIs for monotone multifunctions are established with the use of the KKM-Fan theorem. We also investigate the GVVI problems without monotonicity assumptions and obtain the corresponding results of weak solutions by applying the Brouwer fixed point theorem. These results are also the extension and improvement of some recent results in the literature.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Brouwer L.E.J.: Über Abbildungen von Mannigfaltigkeiten. Math. Ann. 71, 97–115 (1912)

    Article  Google Scholar 

  2. Chen G.Y.: Existence of solutions for a vector variational inequality: An extension of Hartman-Stampacchia theorem. J. Optim. Theory Appl. 74, 445–456 (1992)

    Article  Google Scholar 

  3. Chen G.Y., Yang X.Q.: The vector complementarity problem and its equivalence with the weak minimal element. J. Math. Anal. Appl. 153, 136–158 (1990)

    Article  Google Scholar 

  4. Chen Y.Q.: On the semi-monotone operator theory and applications. J. Math. Anal. Appl. 231, 177–192 (1999)

    Article  Google Scholar 

  5. Fan K.: A generalization of Tychonoff’s fixed-point theorem. Math. Ann. 142, 305–310 (1961)

    Article  Google Scholar 

  6. 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, Wiley and Sons, New York (1980)

    Google Scholar 

  7. Giannessi F.: Vector variational inequalities and vector equilibria. Kluwer Academic Publisher, Dordrechet Holland (2000)

    Google Scholar 

  8. Giannessi F., Maugeri A.: Variational inequalities and network equilibrium problems. Plenum Press, New York (1995)

    Google Scholar 

  9. Huang N.J., Fang Y.P.: On vector variational inequalities in reflexive Banach spaces. J. Glob. Optim. 32, 495–505 (2005)

    Article  Google Scholar 

  10. Konnov I.V., Yao J.C.: On the generalized vector variational inequality problems. J. Math. Anal. Appl. 206, 42–58 (1997)

    Article  Google Scholar 

  11. Lai T.C., Yao J.C.: Existence results for VVIP. Appl. Math. Lett. 9, 17–19 (1996)

    Article  Google Scholar 

  12. Lin K.L., Yang D.P., Yao J.C.: Generalized vector variational inequalities. J. Optim. Theory. Appl. 92, 117–125 (1997)

    Article  Google Scholar 

  13. Nadler S.B. Jr: Multi-valued contraction mappings. Pac. J. Math. 30, 475–488 (1969)

    Google Scholar 

  14. Siddiqi A.H., Ansari Q.H., Khaliq A.: On vector variational inequalities. J. Optim. Theory. Appl. 84, 171–180 (1995)

    Article  Google Scholar 

  15. Yang X.Q.: Vector variational inequality and vector pseudolinear optimization. J. Optim. Theory. Appl. 95, 729–734 (1997)

    Article  Google Scholar 

  16. Yang X.Q., Goh C.J.: On vector variational inequality application to vector traffic equilibria. J. Optim. Theory. Appl. 95, 431–443 (1997)

    Article  Google Scholar 

  17. Yu S.J., Yao J.C.: On vector variational inequalities. J. Optim. Theory. Appl. 89, 749–769 (1996)

    Article  Google Scholar 

  18. Zeng L.C., Yao J.C.: Existence of solutions of generalized vector variational inequalities in reflexive Banach spaces. J. Glob. Optim. 36, 483–497 (2006)

    Article  Google Scholar 

  19. Zheng F.: Vector variational inequalities with semi-monotone operators. J. Glob. Optim. 32, 633–642 (2005)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, 200234, Shanghai, China

    Lu-Chuan Ceng

  2. Department of Applied Mathematics, Dong Hwa University, Hualien, 97401, Taiwan

    Shuechin Huang

Authors
  1. Lu-Chuan Ceng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shuechin Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shuechin Huang.

Additional information

Dedicated to Professor Franco Giannessi for his 75th Birthday.

This research was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and the Dawn Program Foundation in Shanghai.

This research was partially supported by a grant from the National Science Council.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ceng, LC., Huang, S. Existence theorems for generalized vector variational inequalities with a variable ordering relation. J Glob Optim 46, 521–535 (2010). https://doi.org/10.1007/s10898-009-9436-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-009-9436-9

Keywords

Mathematics Subject Classification (2000)

Search

Navigation