Journal of Optimization Theory and Applications

, Volume 134, Issue 3, pp 515–531 | Cite as

Existence of Solutions of Systems of Generalized Implicit Vector Variational Inequalities

  • S. Al-Homidan
  • Q. H. Ansari
  • S. Schaible


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.


Systems of generalized implicit vector variational inequalities Pseudomonotonicity Existence results for a solution Gap functions Lower semicontinuity Upper hemicontinuity ℋ-Hemicontinuity 


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  1. 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. 2.
    Ansari, Q.H., Yao, J.C.: On nondifferentiable and nonconvex vector optimization problems. J. Optim. Theory Appl. 106, 487–500 (2000) CrossRefMathSciNetGoogle Scholar
  3. 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) MATHCrossRefMathSciNetGoogle Scholar
  4. 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. 5.
    Lee, G.M., Kim, D.S.: Existence of solutions for vector optimization problems. J. Math. Anal. Appl. 220, 90–98 (1998) MATHCrossRefMathSciNetGoogle Scholar
  6. 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) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ansari, Q.H., Yao, J.C.: Systems of generalized variational inequalities and their applications. Appl. Anal. 76, 203–217 (2000) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Patriksson, M.: Nonlinear Programming and Variational Inequality Problems. Kluwer Academic, Dordrecht (1999) MATHGoogle Scholar
  9. 9.
    Crouzeix, J.-P.: Pseudomonotone variational inequality problems: existence of solutions. Math. Program. 78, 305–314 (1997) MathSciNetGoogle Scholar
  10. 10.
    Ansari, Q.H., Yao, J.C.: Generalised variational-like inequalities and a gap function. Bull. Aust. Math. Soc. 59, 33–44 (1999) MATHCrossRefMathSciNetGoogle Scholar
  11. 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) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Li, J., He, Z.-Q.: Gap functions and existence of solutions to generalized vector variational inequalities. Appl. Math. Lett. 18, 989–1000 (2005) MATHCrossRefMathSciNetGoogle Scholar
  13. 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) MATHCrossRefMathSciNetGoogle Scholar
  14. 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) Google Scholar
  15. 15.
    Ansari, Q.H.: A note on generalized vector variational-like inequalities. Optimization 41, 197–205 (1997) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lee, G.M., Kim, D.S., Lee, B.S.: Generalized vector variational inequality. Appl. Math. Lett. 9(1), 39–42 (1996) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kum, S., Lee, G.M.: Remarks on implicit vector variational inequalities. Taiwan. J. Math. 6, 369–382 (2002) MATHMathSciNetGoogle Scholar
  18. 18.
    Lee, G.M., Kum, S.: On implicit vector variational inequalities. J. Optim. Appl. 104, 409–425 (2000) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990) MATHGoogle Scholar
  20. 20.
    Nadler, S.B. Jr.: Multi-valued contraction mappings. Pac. J. Math. 30, 475–488 (1969) MATHMathSciNetGoogle Scholar
  21. 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) Google Scholar
  22. 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) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Mathematical SciencesKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia
  2. 2.Department of MathematicsAligarh Muslim UniversityAligarhIndia
  3. 3.A.G. Anderson Graduate School of ManagementUniversity of CaliforniaRiversideUSA

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