Existence of solutions for a generalized vector quasivariational inequality

  • G. Y. Chen
  • S. J. Li
Contributed Papers


The paper deals with a generalization of a vector quasivariational inequality. An existence theorem for its solutions is established; it is based on a kind of minimax inequality, which is here established for continuous affine mappings and differs from previous results. Fan's section theorem for set-valued mappings is extended. An application for an equilibrium problem of a network with vector-valued cost functions is given.

Key Words

Minimax inequalities generalized vector quasivariational inequalities minimal points maximal points section theorem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Giannessi, F.,Theorems of the Alternative, Quadratic Problems, and Complementarity Problems, Variational Inequalities and Complementarity Problems, Edited By R. W. Cottle, F. Giannessi, and J. L. Lions, John Wiley, Chichester, England, pp. 151–186, 1980.Google Scholar
  2. 2.
    Chen, G. Y.,Existence of Solutions for a Vector Variational Inequality: An Extension of the Hartmann-Stampacchia Theorem, Journal of Optimization Theory and Applications, Vol. 74, pp. 445–456, 1992.Google Scholar
  3. 3.
    Chen, G. Y., andYang, X. Q.,The Vector Complementarity Problem and Its Equivalence with the Weak Minimal Element in Ordered Spaces, Journal of Mathematical Analysis and Applications, Vol. 153, pp. 136–158, 1990.Google Scholar
  4. 4.
    Chen, G. Y.,A Generalized Section Theorem and a Minimax Inequality for a Vector-Valued Mapping, Optimization, Vol. 22, pp. 745–754, 1991.Google Scholar
  5. 5.
    Nieuwenhuis, J. W.,Some Minimax Theorems in Vector-Valued Functions, Journal of Optimization Theory and Applications, Vol. 40, pp. 463–475, 1983.Google Scholar
  6. 6.
    Ferro, F.,A Minimax Theorem for Vector-Valued Functions, Part 1, Journal of Optimization Theory and Applications, Vol. 60, pp. 19–31, 1989.Google Scholar
  7. 7.
    Ferro, F.,A Minimax Theorem for Vector-Valued Functions, Part 2, Journal of Optimization Theory and Applications, Vol. 68, pp. 35–48, 1991.Google Scholar
  8. 8.
    Tanaka, T. Existence Theorems for Cone Saddle Points of Vector-Valued Functions in Infinite-Dimensional Spaces, Journal of Optimization Theory and Applications, Vol. 62, pp. 127–138, 1989.Google Scholar
  9. 9.
    Chan, D., andPang, J. S.,The Generalized Quasivariational Inequality Problem, Mathematics of Operations Research, Vol. 7, pp. 211–222, 1982.Google Scholar
  10. 10.
    Aubin, J. P.,Applied Abstract Analysis, John Wiley and Sons, New York, New York, 1977.Google Scholar
  11. 11.
    Knesev, H.,Sur un Theorème Fundamental de la Theorie des Jeux, Comptes Rendus de l'Academie des Sciences, Paris, Vol. 234, pp. 2418–2420, 1952.Google Scholar
  12. 12.
    Jahn, J. Scalarization in Vector Optimization, Mathematical Programming, Vol. 29, pp. 203–218, 1984.Google Scholar
  13. 13.
    Fan, F. A Generalization of Tychonoff's Fixed-Point Theorem, Mathematische Annalen, Vol. 142, pp. 305–310, 1961.Google Scholar
  14. 14.
    Penot, J. P., andSterna-Karwat, A.,Parametrized Multicriteria Optimization: Continuity and Closedness of Optimal Multifunctions, Journal of Mathematical Analysis and Applications Vol. 120, pp. 150–168, 1986.Google Scholar
  15. 15.
    Maugeri, A.,Convex Programming, Variational Inequalities, and Applications to the Traffic Equilibrium Problem, Applied Mathematics and Optimization, Vol. 16, pp. 169–185, 1987.Google Scholar
  16. 16.
    De Luca, M., andMaugeri, A.,Quasivariational Inequalities and Applications to Equilibrium Problems with Elastic Demand, Nonsmooth Optimization and Related Topics, Edited by F. H. Clarke, V. F. Demyanov, and F. Giannessi, Plenum Press, New York, New York, 1989.Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • G. Y. Chen
    • 1
  • S. J. Li
    • 2
  1. 1.Institute of Systems ScienceAcademia SinicaBeijingChina
  2. 2.Chongqing Architecture and Engineering InstituteChongqingChina

Personalised recommendations