Journal of Optimization Theory and Applications

, Volume 99, Issue 1, pp 165–181 | Cite as

On Quasimonotone Variational Inequalities

  • I. V. Konnov
Article

Abstract

In this paper, we study variational inequalities with multivalued mappings. By employing Fan's lemma, we establish the existence result for the dual formulation of variational inequalities with semistrictly quasimonotone mappings. We also show that similar results for quasimonotone variational inequalities do not hold.

Semistrictly quasimonotone mappings dual formulation of variational inequalities multivalued mappings 

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References

  1. 1.
    Kinderlerer, D., and Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, New York, 1980.Google Scholar
  2. 2.
    Panagiotopoulos, P. D., Inequality Problems in Mechanics and Their Applications, Birkhauser, Boston, Massachusetts, 1985.Google Scholar
  3. 3.
    Harker, P. T., and Pang, J. S., Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms, and Applications, Mathematical Programming, Vol. 48, pp. 161–220, 1990.Google Scholar
  4. 4.
    Minty, G., Monotone (Nonlinear) Operators in Hilbert Space, Duke Mathematical Journal, Vol. 29, pp. 341–346, 1962.Google Scholar
  5. 5.
    Shih, M. H., and Tan, K. K., Browder-Hartmann-Stampacchia Variational Inequalities for Multivalued Monotone Operators, Journal of Mathematical Analysis and Applications, Vol. 134, pp. 431–440, 1988.Google Scholar
  6. 6.
    Yao, J. C., Multivalued Variational Inequalities with K-Pseudomonotone Operators, Journal of Optimization Theory and Applications, Vol. 83, pp. 391–403, 1994.Google Scholar
  7. 7.
    Hadjisavvas, N., and Schaible, S., Quasimonotone Variational Inequalities in Banach Spaces, Journal of Optimization Theory and Applications, Vol. 90, pp. 95–111, 1996.Google Scholar
  8. 8.
    Konnov, I. V., Combined Relaxation Methods for Finding Equilibrium Points and Solving Related Problems, Russian Mathematics (Izvestiya VUZ. Matematika), Vol. 37, pp. 44–51, 1993.Google Scholar
  9. 9.
    Antipin, A. S., On Convergence of Proximal Methods to Fixed Points of Extremal Mappings and Estimates of Their Rate of Convergence, Computational Mathematics and Mathematical Physics, Vol. 35, pp. 539–551, 1995.Google Scholar
  10. 10.
    Konnov, I. V., A General Approach to Finding Stationary Points and the Solution of Related Problems, Computational Mathematics and Mathematical Physics, Vol. 36, pp. 585–593, 1996.Google Scholar
  11. 11.
    Daniilidis, A., and Hadjisavvas, N., Variational Inequalities with Quasimonotone Multivalued Operators, Working Paper, Department of Mathematics, University of the Aegean, Samos, Greece, 1995.Google Scholar
  12. 12.
    Aubin, J. P., and Cellina, A., Differential Inclusions, Springer Verlag, Berlin, Germany, 1984.Google Scholar
  13. 13.
    Fan, K., A Generalization of Tychonoff's Fixed-Point Theorem, Mathematische Annalen, Vol. 142, pp. 305–310, 1961.Google Scholar
  14. 14.
    Hadjisavvas, N., and Schaible, S., On Strong Pseudomonotonicity and (Semi) Strict Quasimonotonicity, Journal of Optimization Theory and Applications, Vol. 79, pp. 139–155, 1993.Google Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • I. V. Konnov
    • 1
  1. 1.Department of Applied MathematicsKazan UniversityKazanRussia

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