On vector variational inequalities

  • S. J. Yu
  • J. C. Yao
Contributed Papers


In this paper, we study vector variational inequalities. The concept of weaklyC-pseudomonotone operator is introduced. By employing the Fan lemma, we establish several existence results. The new results extend and unify existence results of vector variational inequalities for monotone operators under a Banach space setting. In particular, existence results for the generalized vector complementarity problem with weaklyC-pseudomonotone operators in Banach space are obtained.

Key Words

Vector variational inequalities generalized vector complementarity problems weaklyC-pseudomonotone operators weakly (C+)-pseudomonotone operators weaklyv-coercive conditions generalizedL-condition 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Giannessi, F.,Theorem of the Alternative, Quadratic Programs, and Complementarity Problems, Variational Inequalities and Complementarity Problems, Edited by R. W. Cottle, F. Giannessi, and J. L. Lions, Wiley, New York, New York, pp. 151–186, 1980.Google Scholar
  2. 2.
    Chen, G. Y., andYang, X. Q.,Vector Complementarity Problem and Its Equivalence with Weak Minimal Element in Ordered Spaces, Journal of Mathematical Analysis and Applications, Vol. 153, pp. 136–158, 1990.CrossRefGoogle Scholar
  3. 3.
    Chen, G. Y., andCheng, G. M.,Vector Variational Inequalities and Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Heidelberg, Germany, Vol. 258, 1987.Google Scholar
  4. 4.
    Chen, G. Y., andCraven, B. D.,A Vector Variational Inequality and Optimization over an Efficient Set, Zeitschrift für Operations Research, Vol. 3, pp. 1–12, 1990.Google Scholar
  5. 5.
    Chen, G. Y.,Existence of Solution for a Vector Variational Inequality: An Extension of the Hartmann-Stampacchia Theorem, Journal of Opimization Theory and Applications, Vol. 74, pp. 445–456, 1992.CrossRefGoogle Scholar
  6. 6.
    Yang, X. Q.,Vector Variational Inequality and Its Duality, Nonlinear Analysis: Theory, Methods, and Analysis, Vol. 21, pp. 869–877, 1993.Google Scholar
  7. 7.
    Hartmann, G. J., andStampacchia, G.,On Some Nonlinear Elliptic Differential Functional Equations, Acta Mathematica, Vol. 115, pp. 271–310, 1966.Google Scholar
  8. 8.
    Fan, K.,A Generalization of Tychonoff's Fixed-Point Theorem Mathematische Annalen, Vol. 142, pp. 305–310, 1961.CrossRefGoogle Scholar
  9. 9.
    Knaster, B., Kurotowski, C., andMazukiewicz, S.,Ein Beweis des Fixpunktsatzes für N-Dimensionale Simplexe, Fundamental Mathematica, Vol. 14, pp. 132–137, 1929.Google Scholar
  10. 10.
    Karmardian, S.,Complementarity over Cones with Monotone and Pseudomonotone Maps, Journal of Optimization Theory and Applications, Vol. 18, pp. 445–454, 1976.CrossRefGoogle Scholar
  11. 11.
    Lee, G. M., Kim, D. S., Lee, B. S., andCho, S. J.,Generalized Vector Variational Inequality and Fuzzy Extension, Applied Mathematics Letters, Vol. 6, pp. 47–51, 1993.CrossRefGoogle Scholar
  12. 12.
    Minty, G.,Monotone Nonlinear Operators in Hilbert Space, Duke Mathematical Journal, Vol. 29, pp. 341–346, 1962.CrossRefGoogle Scholar
  13. 13.
    Yao, J. C.,Variational Inequalities with Generalized Monotone Operators, Mathematics of Operations Research, Vol. 19, pp. 691–705, 1994.Google Scholar
  14. 14.
    Yu, L. P.,Multiple-Criteria Decision Making: Concepts, Techniques, and Extensions, Plenum Press, New York, New York, 1985.Google Scholar
  15. 15.
    Sawaragi, Y., Nakayama, H., andTanino, T.,Theory of Multiobjective Optimization, Academic Press, New York, New York, 1985.Google Scholar
  16. 16.
    Moré, J. J.,Coercivity Conditions in Nonlinear Complementarity Problems, SIAM Review, Vol. 16, pp. 1–16, 1974.CrossRefGoogle Scholar
  17. 17.
    Browder, F. E.,Nonlinear Monotone Operators and Convex Sets in Banach Space, Bulletin of the American Mathematical Society, Vol. 71, pp. 780–785, 1965.Google Scholar
  18. 18.
    Opial, Z.,Nonexpansive Monotone Mapping in Banach Spaces, Technical Report 67-1, Department of Mathematics, Brown University, Providence, Rhode Island, 1967.Google Scholar
  19. 19.
    Stampacchia, G.,Variational Inequalities Theory and Applications of Monotone Operators, Edited by A. Ghizzetti, Edizioni Oderisi, Gubbio, Italy, 1969.Google Scholar
  20. 20.
    Karamardian, S.,The Nonlinear Complementarity Problem, with Applications, Parts 1 and 2, Journal of Optimization Theory and Applications, Vol. 4, pp. 87–98, 1969 and Vol. 4, pp. 167–181, 1969.CrossRefGoogle Scholar
  21. 21.
    Karamardian, S.,The Complementarity Problem, Mathematical Programming, Vol. 2, pp. 107–129, 1972.CrossRefGoogle Scholar
  22. 22.
    Rheinboldt, W. C.,On M-Functions and Their Application to Nonlinear Gauss-Seidel Iterations and Network Flows, Journal of Mathematical Analysis and Applications, Vol. 32, pp. 274–307, 1971.CrossRefGoogle Scholar
  23. 23.
    Schaible, S., andYao, J. C.,On the Equivalence of Nonlinear Complementarity Problems, Mathematical Programming, 1995.Google Scholar
  24. 24.
    Yao, J. C.,Multi-Valued Variational Inequalities with K-Pseudomonotone Operators, Journal of Optimization Theory and Applications, Vol. 83, pp. 445–454, 1994.CrossRefGoogle Scholar
  25. 25.
    Conway, J. B.,A Course in Functional Analysis, 2nd Edition, Springer Verlag New York, New York, 1990.Google Scholar
  26. 26.
    Karamardian, S.,Generalized Complementarity Problem, Journal of Optimization Theory and Applications, Vol. 8, pp. 161–168, 1971.CrossRefGoogle Scholar
  27. 27.
    Théra, M.,Existence Results for the Nonlinear Complementarity Problem and Applications to Nonlinear Analysis, Journal of Mathematical Analysis and Applications, Vol. 154, pp. 572–584, 1991.CrossRefGoogle Scholar
  28. 28.
    Harker, P. T., andPang, J. S.,Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms, and Applications, Mathematical Programming, Vol. 48B, pp. 161–220, 1990.CrossRefGoogle Scholar
  29. 29.
    Nanda, S.,Nonlinear Complementarity Problem of Mathematical Programming in Banach Space, Indian Journal of Pure Applied Mathematics, Vol. 18, pp. 215–218, 1987.Google Scholar
  30. 30.
    Isac, G., andThéra, M.,Complementarity Problem and the Existence of the Post-Critical Equilibrium State of a Thin Elastic Plate, Journal of Optimization Theory and Applications, Vol. 58, pp. 241–257, 1988.CrossRefGoogle Scholar
  31. 31.
    Dash, A. T., andNanda, S.,A Complementarity Problem in Mathematical Programming in Banach Space, Journal of Mathematical Analysis and Applications, Vol. 98, pp. 328–331, 1984.CrossRefGoogle Scholar
  32. 32.
    Borwein, J. M.,Generalized Linear Complementarity Problems Treated without Fixed-Point Theory, Journal of Optimization Theory and Applications, Vol. 43, pp. 445–454, 1984.CrossRefGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • S. J. Yu
    • 1
  • J. C. Yao
    • 1
  1. 1.Department of Applied MathematicsNational Sun Yat-Sen UniversityKaohsiungTaiwan, ROC

Personalised recommendations