Skip to main content
SpringerLink
Log in

Existence of Solutions of Systems of Generalized Implicit Vector Variational Inequalities

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

Abstract

We consider five different types of systems of generalized vector variational inequalities and derive relationships among them. We introduce the concept of pseudomonotonicity for a family of multivalued maps and prove the existence of weak solutions of these problems under these pseudomonotonicity assumptions in the setting of Hausdorff topological vector spaces as well as real Banach spaces. We also establish the existence of a strong solution of our problems under lower semicontinuity for a family of multivalued maps involved in the formulation of the problems. By using a nonlinear scalar function, we introduce gap functions for our problems by which we can solve systems of generalized vector variational inequalities using optimization techniques.

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. Ansari, Q.H., Siddiqi, A.H.: A generalized vector variational-like inequality and optimization over an efficient set. In: Brokate, M., Siddiqi, A.H. (eds.) Functional Analysis with Current Applications in Science, Technology and Industry. Pitman Research Notes in Mathematics, vol. 377, pp. 177–191. Addison-Wesley, Essex (1998)

    Google Scholar 

  2. Ansari, Q.H., Yao, J.C.: On nondifferentiable and nonconvex vector optimization problems. J. Optim. Theory Appl. 106, 487–500 (2000)

    Article  MathSciNet  Google Scholar 

  3. Chen, G.Y., Craven, B.D.: A vector variational inequality and optimization over an efficient set. ZOR: Math. Methods Oper. Res. 34, 1–12 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  4. Lee, G.M.: On relations between vector variational inequality and vector optimization problem. In: Yang, X.Q. et al. (eds.) Progress in Optimization, pp. 167–179. Kluwer Academic, Dordrecht (2000)

    Google Scholar 

  5. Lee, G.M., Kim, D.S.: Existence of solutions for vector optimization problems. J. Math. Anal. Appl. 220, 90–98 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ansari, Q.H., Schaible, S., Yao, J.C.: The system of generalized vector equilibrium problems with applications. J. Glob. Optim. 22, 3–16 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ansari, Q.H., Yao, J.C.: Systems of generalized variational inequalities and their applications. Appl. Anal. 76, 203–217 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Patriksson, M.: Nonlinear Programming and Variational Inequality Problems. Kluwer Academic, Dordrecht (1999)

    MATH  Google Scholar 

  9. Crouzeix, J.-P.: Pseudomonotone variational inequality problems: existence of solutions. Math. Program. 78, 305–314 (1997)

    MathSciNet  Google Scholar 

  10. Ansari, Q.H., Yao, J.C.: Generalised variational-like inequalities and a gap function. Bull. Aust. Math. Soc. 59, 33–44 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Yang, X.Q., Yao, J.C.: Gap functions and existence of solutions to set-valued vector variational inequalities. J. Optim. Theory Appl. 115, 407–417 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Li, J., He, Z.-Q.: Gap functions and existence of solutions to generalized vector variational inequalities. Appl. Math. Lett. 18, 989–1000 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chen, G.Y., Yang, X.Q., Yu, H.: A nonlinear scalarization function and generalized vector quasi-equilibrium problems. J. Glob. Optim. 32, 451–466 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Huang, N.-J., Li, J., Yao, J.C.: Gap function and existence of solutions for a system of vector equilibrium problems. Preprint, National Sun Yat-sen University, Kaohsiung, Taiwan (2006)

  15. Ansari, Q.H.: A note on generalized vector variational-like inequalities. Optimization 41, 197–205 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lee, G.M., Kim, D.S., Lee, B.S.: Generalized vector variational inequality. Appl. Math. Lett. 9(1), 39–42 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kum, S., Lee, G.M.: Remarks on implicit vector variational inequalities. Taiwan. J. Math. 6, 369–382 (2002)

    MATH  MathSciNet  Google Scholar 

  18. Lee, G.M., Kum, S.: On implicit vector variational inequalities. J. Optim. Appl. 104, 409–425 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  19. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  21. Zeng, L.C., Yao, J.C.: Existence of solutions of generalized vector variational inequalities in reflexive Banach spaces. Preprint, National Sun Yat-sen University, Kaohsiung, Taiwan (2006)

  22. Lin, L.J., Ansari, Q.H.: Collective fixed points and maximal elements with applications to abstract economies. J. Math. Anal. Appl. 296, 455–472 (2004)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

    S. Al-Homidan & Q. H. Ansari

  2. Department of Mathematics, Aligarh Muslim University, Aligarh, India

    Q. H. Ansari

  3. A.G. Anderson Graduate School of Management, University of California, Riverside, CA, USA

    S. Schaible

Authors
  1. S. Al-Homidan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Q. H. Ansari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Schaible
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Schaible.

Additional information

The first two authors were supported by SABIC and Fast Track Research Grants SAB-2006-05. They are grateful to the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Al-Homidan, S., Ansari, Q.H. & Schaible, S. Existence of Solutions of Systems of Generalized Implicit Vector Variational Inequalities. J Optim Theory Appl 134, 515–531 (2007). https://doi.org/10.1007/s10957-007-9236-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-007-9236-7

Keywords

Search

Navigation