Advertisement

Journal of Optimization Theory and Applications

, Volume 74, Issue 3, pp 445–456 | Cite as

Existence of solutions for a vector variational inequality: An extension of the Hartmann-Stampacchia theorem

  • G. Y. Chen
Contributed Papers

Abstract

A vector variational inequality is studied. The paper deals with existence theorems for solutions under convexity assumptions and without convexity assumptions.

Key Words

Variational inequalities vector extremum problems existence theorems convexity coercivity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Giannessi, F.,Theorems of Alternative, Quadratic Programs, and Complementarity Problems, Variational Inequalities and Complementarity Problems, Edited by R. W. Cottle, F. Giannessi, and J. L. Lions, John Wiley and Sons, Chichester, England, pp. 151–186, 1980.Google Scholar
  2. 2.
    Chen, G. Y., andCheng, G. M.,Vector Variational Inequality and Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Heidelberg, Germany, Vol. 285, 1987.Google Scholar
  3. 3.
    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
  4. 4.
    Chen, G. Y., andYang, X. Q.,Vector Complementarity Problem and Its Equivalences with Weak Minimal Element in Ordered Space, Journal of Mathematical Analysis and Applications, Vol. 153, pp. 136–158, 1990.Google Scholar
  5. 5.
    Hartmann, G. J., andStampacchia, G.,On Some Nonlinear Elliptic Differential Functional Equations, Acta Mathematica, Vol. 115, pp. 271–310, 1966.Google Scholar
  6. 6.
    Fan, K.,A Generalization of Tychonoff's Fixed-Point Theorem, Mathematics Annals, Vol. 142, pp. 305–310, 1961.Google Scholar
  7. 7.
    Ferrero, O.,Theorems of the Alternative for Set-valued Functions in Infinite-Dimensional Spaces, Optimization, Vol. 20, pp. 167–175, 1989.Google Scholar
  8. 8.
    Dugudji, J., andGranas, A.,KKM Maps and Variational Inequalities, Annali dell Scuola Normale Superiore, Pisa, Italy, pp. 679–682, 1979.Google Scholar
  9. 9.
    Bardaro, C., andCappitelli, R.,Some Further Generalizations of the Knoster-Kuratowski-Mazukiewicz Theorem and Minimax Inequalities, Journal of Mathematical Analysis and Applications, Vol. 132, pp. 484–490, 1988.Google Scholar
  10. 10.
    De Luca, M., andMaugeri, A.,Quasi-Variational 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, New York, New York, pp. 61–77, 1989.Google Scholar
  11. 11.
    Holmes, R. B.,Geometric Functional Analysis and Its Applications, Springer-Verlag, Heidelberg, Germany, 1975.Google Scholar
  12. 12.
    Martein, L.,Stationary Points and Necessary Conditions in Vector Extremum Problems, Research Report No. 133, Optimization and Operations Research Group, Department of Mathematics, University of Pisa, Pisa, Italy, 1986.Google Scholar
  13. 13.
    Jameson, G.,Ordered Linear Spaces, Springer-Verlag, Heidelberg, Germany, 1970.Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • G. Y. Chen
    • 1
  1. 1.Institute of Systems ScienceAcademia SinicaBeijingChina

Personalised recommendations