Journal of Global Optimization

, Volume 46, Issue 3, pp 465–473 | Cite as

Variational inequalities on weakly compact sets

  • Y. Chiang


In this paper, we derive an existence result for generalized variational inequalities associated with multivalued mappings on weakly compact sets under a continuity assumption which is much weaker than the regular complete continuity. As an application, we prove the existence of exceptional families of elements for such mappings on closed convex cones in reflexive Banach spaces when the corresponding complementarity problems have no solutions.


Regular complete continuity Generalized variational inequalities Exceptional families of elements 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bianchi M., Hadjisavvas N., Schaible S.: Minimal coercivity conditions and exceptional families of elements in Quasimonotone variational inequalities. J. Optim. Theory Appl. 122, 1–17 (2004)CrossRefGoogle Scholar
  2. 2.
    Bianchi M., Hadjisavvas N., Schaible S.: Exceptional families of elements for variational inequalities in Banach spaces. J. Optim. Theory Appl. 129, 23–31 (2006)CrossRefGoogle Scholar
  3. 3.
    Isac G., Bulavski V., Kalashnikov V.: Exceptional families, topological degree and complementarity problems. J. Glob. Optim. 10, 207–225 (1997)CrossRefGoogle Scholar
  4. 4.
    Isac G., Zhao Y.B.: Exceptional family of elements and the solvability of variational inequalities for unbounded sets in infinite dimensional Hilbert spaces. J. Math. Anal. Appl. 246, 544–556 (2000)CrossRefGoogle Scholar
  5. 5.
    Isac G., Németh S.Z.: Duality in multivalued complementarity theory by using inversions and scalar derivatives. J. Glob. Optim. 33, 197–213 (2005)CrossRefGoogle Scholar
  6. 6.
    Isac G., Li J.: Exceptional family of elements and the solvability of complementarity problems in uniformly smooth anduniformly convex Banach spaces. J. Zhejiang Uni. Sci. Ed. 6A(4), 289–295 (2005)CrossRefGoogle Scholar
  7. 7.
    Isac G.: Topological methods in complementarity theory. Kluwer, Dordrecht (2000)Google Scholar
  8. 8.
    Isac, G.: Leray-Schauder type alternatives, complementarity problems and variational inequalities. Nonconvex optimization and its applications, vol. 87. Springer, New York, (2006)Google Scholar
  9. 9.
    Lunsford L.: Generalized variational and quasi-variational inequalities with discontinuous operators. J. Math. Anal. Appl. 214, 245–263 (1997)CrossRefGoogle Scholar
  10. 10.
    Megginson R.E.: An introduction to Banach space theory. Springer, New York (1998)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  1. 1.Department of Applied MathematicsNational Sun Yat-sen UniversityKaohsiungTaiwan, ROC

Personalised recommendations