Exceptional Families of Elements for Variational Inequalities in Banach Spaces

  • M. Bianchi
  • N. Hadjisavvas
  • S. Schaible
Article

Abstract

In keeping with very recent efforts to establish a useful concept of an exceptional family of elements for variational inequality problems rather than complementarity problems as in the past, we propose such a concept. It generalizes previous ones to multivalued variational inequalities in general normed spaces and allows us to obtain several existence results for variational inequalities corresponding to earlier ones for complementarity problems. Compared with the existing literature, we consider problems in more general spaces and under considerably weaker assumptions on the defining map.

Keywords

Variational inequalities complementarity problems quasimonotone maps exceptional families of elements 

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References

  1. 1.
    SMITH, T. E., A Solution Condition for Complementarity Problems with an Application to Spatial Price Equilibrium, Applied Mathematics and Computation, Vol. 15, pp. 61–69, 1984.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    ISAC, G., and KALASHNIKOV, V. V., Exceptional Family of Elements, Leray-Schauder Alternative, Pseudomonotone Operators, and Complementarity, Journal of Optimization Theory and Applications, Vol. 109, pp. 69–83, 2001.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    ISAC, G., BULAVSKI, V., and KALASHNIKOV, V., Exceptional Families, Topological Degree, and Complementarity Problems, Journal of Global Optimization, Vol. 10, pp. 207–225, 1997.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    ZHAO, Y. B., and YUAN, J. Y., An Alternative Theorem for Generalized Variational Inequalities and Solvability of Nonlinear Quasi-P M_∗- Complementarity Problems, Applied Mathematics and Computation, Vol. 109, pp. 167–182, 2000.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    ZHAO, Y. B., and SUN, D., Alternative Theorems for Nonlinear Projection Equations, Nonlinear Analysis, Vol. 46, pp. 853–868, 2001.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    ZHAO, Y. B., and HAN, J., Exceptional Family of Elements for a Variational Inequality Problem and Its Applications, Journal of Global Optimization, Vol. 14, pp. 313–330, 1999.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    ISAC, G., and ZHAO, Y. B., Exceptional Family of Elements and the Solvability of Variational Inequalities for Unbounded Sets in Infinite Dimensional Hilbert Spaces, Journal of Mathematical Analysis and Applications, Vol. 246, pp. 544–556, 2000.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    ZHOU, S. Z., and BAI, M. R., A New Exceptional Family of Elements for a Variational Inequality Problem on Hilbert Space, Applied Mathematics Letters, Vol. 17, pp. 423–428, 2004.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    HAN, J., HUANG, Z. H., and FANG, S. C., Solvability of Variational Inequality Problems, Journal of Optimization Theory and Applications, Vol. 122, pp. 501–520, 2004.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    ISAC, G., and LI, J. L., Exceptional Family of Elements and the Solvability of Complementarity Problems in Uniformly Smooth and Uniformly Convex Banach Spaces, Journal of the Zhejiang University of Science, Vol. 6A, pp. 1–9, 2005.Google Scholar
  11. 11.
    CARBONE, A., and ZABREIKO, P. P., Some Remarks on Complementarity Problems in a Hilbert Space, Zeitschrift fuer Analysis und ihre Anwendungen, Vol. 21, pp. 1005–1014, 2002.MATHMathSciNetGoogle Scholar
  12. 12.
    BIANCHI, M., HADJISAVVAS, N., and SCHAIBLE, S., Minimal Coercivity Conditions and Exceptional Families of Elements in Quasimonotone Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 122, pp. 1–17, 2004.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    KARAMARDIAN, S., Generalized Complementarity Problems, Journal of Optimization Theory and Applications, Vol. 8, pp. 161–168, 1971.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    AUBIN, J. P., and EKELAND, I., Applied Nonlinear Analysis, John Wiley and Sons, New York, NY, 1984.Google Scholar
  15. 15.
    HADJISAVVAS, N., Continuity and Maximality Properties of Pseudomonotone Operators, Journal of Convex Analysis, Vol. 20, pp. 465–475, 2003.MathSciNetGoogle Scholar
  16. 16.
    AUSSEL, D., and HADJISAVVAS, N., On Quasimonotone Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 121, pp. 445–450, 2004.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • M. Bianchi
    • 1
  • N. Hadjisavvas
    • 2
  • S. Schaible
    • 3
  1. 1.Istituto di Econometria e Matematica per le Applicazioni Economiche, Finanziarie e AttuarialiUniversità Cattolica del Sacro CuoreMilanoItaly
  2. 2.Department of Product and Systems Design EngineeringUniversity of the AegeanHermoupolis, SyrosGreece
  3. 3.A.G. Anderson Graduate School of ManagementUniversity of CaliforniaRiversideCalifornia

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