Advertisement

Journal of Global Optimization

, Volume 49, Issue 2, pp 343–357 | Cite as

Degree theory for a generalized set-valued variational inequality with an application in Banach spaces

  • Zhong Bao Wang
  • Nan Jing Huang
Article

Abstract

In this paper, a degree theory for a generalized set-valued variational inequality is built in a Banach space. As an application, an existence result of solutions for the generalized set-valued variational inequality is given under some suitable conditions.

Keywords

Generalized set-valued variational inequality Topological degree Generalized f-projection operator Normalized duality mapping Set-valued mapping 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alber, Ya.: Generalized projection operators in Banach spaces: properties and applications. In: Proceedings of the Israel Seminar Ariel, Israel. Funct. Differ. Equ. 1:1–21 (1994)Google Scholar
  2. 2.
    Alber, Ya.: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos, A. Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 15–50. Dekker, New York (1996)Google Scholar
  3. 3.
    Aubin J.P., Ekland I.: Applied Nonlinear Analysis. Wiley, New York (1984)Google Scholar
  4. 4.
    Browder F.E.: Fixed point theory and nonlinear problems. Bull. Am. Math. Soc. 9, 1–39 (1983)CrossRefGoogle Scholar
  5. 5.
    Chen Y., Cho Y.J.: Topological degree theory for muti-valued mapping of class (S +)L. Arch. Math. 84, 325–333 (2005)CrossRefGoogle Scholar
  6. 6.
    Cioranescu I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer, Dordrecht (1990)Google Scholar
  7. 7.
    Chinchuluun, A., Migdalas, A., Pardalos, P.M., Pitsoulis, L. (eds): Pareto Optimality, Game Theory and Equilibria. Springer, Berlin (2008)Google Scholar
  8. 8.
    Cohen G.: Nash equilibria: gradient and decomposition algorithms. Large Scale Syst. 12, 173–184 (1987)Google Scholar
  9. 9.
    Dugundji J.: An extension of Tietze’s theorem. Pac. J. Math. 1, 353–367 (1951)Google Scholar
  10. 10.
    Facchinei F., Pang J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)Google Scholar
  11. 11.
    Fan J.H., Liu X., Li J.L.: Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces. Nonlinear Anal. TMA 70, 3997–4007 (2009)CrossRefGoogle Scholar
  12. 12.
    Fonseca, I., Gangbo, W.: Degree Theory in Analysis and Applications. Oxford (1995)Google Scholar
  13. 13.
    Giannessi, F., Maugeri, A., Pardalos, P.M. (eds): Equilibrium Problems and Variational Models. Kluwer, Boston (2001)Google Scholar
  14. 14.
    Guo D.J.: Nonlinear Functional Analysis. Shandong Science and Technology Publishing Press, Shandong (1985)Google Scholar
  15. 15.
    Hu S., Parageorgiou N.S.: Generalization of Browder’s degree theory. Trans. Am. Math. Soc. 347, 233–259 (1995)CrossRefGoogle Scholar
  16. 16.
    Ibrahimou B., Kartsatos A.G.: The Leray-Schauder approach to the degree theory for (S +)-perturbations of maximal monotone operators in separable reflexive Banach spaces. Nonlinear Anal. TMA 70, 4350–4368 (2009)CrossRefGoogle Scholar
  17. 17.
    Kartsatos A.G., Skrypnik I.V.: Topological degree theories for densely defined mapping involving operators of type (S +). Adv. Differ. Equ. 4, 413–456 (1999)Google Scholar
  18. 18.
    Kien B.T., Yao J.C., Yen N.D.: On the solution existence of pseudomonotone variational inequalities. J. Glob. Optim. 41, 135–145 (2008)CrossRefGoogle Scholar
  19. 19.
    Kien B.T., Wong M.-M., Wong N.C., Yao J.C.: Degree theory for generalized variational inequalities and applications. Eur. J. Oper. Res. 192, 730–736 (2009)CrossRefGoogle Scholar
  20. 20.
    Konnov I.: A combined relaxation method for a class of nonlinear variational inequalities. Optimization 51, 127–143 (2002)CrossRefGoogle Scholar
  21. 21.
    Li J.L.: The generalized projection operator on reflexive Banach spaces and its applications. J. Math. Anal. Appl. 306, 55–71 (2005)CrossRefGoogle Scholar
  22. 22.
    Li J.L., Whitaker J.: Exceptional family of elements and solvability of variational inequalities for mappings defined only on closed convex cones in Banach spaces. J. Math. Anal. Appl. 310, 254–261 (2005)CrossRefGoogle Scholar
  23. 23.
    Lloyd N.G.: Degree Theory. Cambridge University Press, Cambridge (1978)Google Scholar
  24. 24.
    Mazur, S., Schauder, S.J.: Über ein prinzip in der variationscechnung. Proc. Int. Congress Math. Oslo. 65 (1936)Google Scholar
  25. 25.
    Panagiotopoulos P., Stavroulakis G.: New types of variational principles based on the notion of quasidifferentiability. Acta Mech. 94, 171–194 (1994)CrossRefGoogle Scholar
  26. 26.
    Pardalos, P.M., Rassias, T.M., Khan, A.A. (eds): Nonlinear Analysis and Variational Problems. Springer, Berlin (2010)Google Scholar
  27. 27.
    Takahashi W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)Google Scholar
  28. 28.
    Vainberg M.M.: Variational Methods and Method of Monotone Operators. Wiley, New York (1973)Google Scholar
  29. 29.
    Wu K.Q., Huang N.J.: The generalized f-projection operator with an application. Bull. Aust. Math. Soc. 73, 307–317 (2006)CrossRefGoogle Scholar
  30. 30.
    Xia F.Q., Huang N.J., Liu Z.B.: A projected subgradient method for solving generalized mixed variational inequalities. Oper. Res. Lett. 36, 637–642 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Department of MathematicsSichuan UniversityChengduPeople’s Republic of China

Personalised recommendations