Journal of Zhejiang University-SCIENCE A

, Volume 6, Issue 4, pp 289–295 | Cite as

Exceptional family of elements and the solvability of complementarity problems in uniformly smooth and uniformly convex Banach spaces

  • G. Isac
  • Li Jin-lu
Applied Mathematics and Engineering Management Science


The notion of “exceptional family of elments (EFE)” plays a very important role in solving complementarity problems. It has been applied in finite dimensional spaces and Hilbert spaces by many authors. In this paper, by using the generalized projection defined by Alber, we extend this notion from Hilbert spaces to uniformly smooth and uniformly convex Banach spaces, and apply this extension to the study of nonlinear complementarity problems in Banach spaces.

Key words

Exceptional family of elements (EFE) Banach spaces and complementarity 

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  1. Alber, Y., 1996. Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications.In: Kartsatos, A. (Ed.) Theory and Applications of Nonlinear Operators of Monatonic and Accretive Type. Marcel Dekker, New York, p. 15–50.Google Scholar
  2. Bulavski, V. A., Isac, G., Kalashnikov, V. V., 1998. Application of Topoligical Degree to Complementarity Problems.In: Migdalas, A., Pardalos, P.M., Värbrand, P. (Eds.), Multilevel Optimization: Algorithms and Applications. Kluwer Academic Publishers, p. 333–358.Google Scholar
  3. Cioranescu, I., 1990. Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic Publsiers.Google Scholar
  4. Cottle, R.W., Pang, J.S., Stone, R.E., 1992. The Linear Complementarity Problems. Academic Press, New York.MATHGoogle Scholar
  5. Isac, G., 1992. Complementarity Problems. Lecture Notes in Math., Vol. 1528. Springer-VerlagGoogle Scholar
  6. Isac, G., 1998. Exceptional Families of Elements fork-fields in Kilbert Spaces and Complementarity Theory. Proc. International Conf. Opt. Techniques. Appl. (ICOTA’98), Perth, Australia, p. 1135–1143.Google Scholar
  7. Isac, G., 1999a. A generalization of Karamardian’s condition in complementarity theory.Nonlinear Analysis Forum,4:49–63.MathSciNetMATHGoogle Scholar
  8. Isac, G., 1999b. On the Solvability, of Multi-values Complementarity Problem: A Topological Method. Fourth European Workshop on Fuzzy Decision Analysis and Recognition Technology (EFDAN’99), Dortmund, Germany, p. 51–66.Google Scholar
  9. Isac, G., 2000a. Topological Methods in Complementarity Theory. Kluwer Academic Publishers.Google Scholar
  10. Isac, G., 2000b. Exceptional family of elements, feasibility and complementarity.J. Opt. Theory Appl.,104:577–588.MathSciNetCrossRefMATHGoogle Scholar
  11. Isac, G., 2000c. Exceptional Family of Elements, Feasibility, Solvability and Continuous Paths of ε-solutions for Nonlinear Complementarity Problems.In: Pardalos, P.((Ed.), Approximation and Complexity in Numerical Optimization: Continuous and Discrete Problems. Kluwer Academic Publishers, p. 323–337.Google Scholar
  12. Isac, G., 2001. Leray-Schauder type alternatives and the solvability of complementarity problems.Topol. Methods Nonlinear Analysis,18:191–204MathSciNetMATHGoogle Scholar
  13. Isac, G., Obuchowska, V.T., 1998. Functions without exceptional families of elements and complementarity problems.J. Optim. Theory Appl.,99:147–163.MathSciNetCrossRefMATHGoogle Scholar
  14. Isac, G., Carbone, A., 1999. Exceptional families of elements for continuous functions: some applications to complementarity theory.J. Global Optim.,15:181–196.MathSciNetCrossRefMATHGoogle Scholar
  15. Isac, G., Zhao, Y.B., 2000. Exceptional family and the solvability of variational inequalities for unbounded sets in infinite dimensional Hilbert spaces.J. Math. Anal. Appl.,246:544–556.MathSciNetCrossRefMATHGoogle Scholar
  16. Isac, G., Li, J.L., 2001. Complementarity problems, Karamardian’s condition and a generalization of Harker-Pang condition.Nonlinear Anal. Forum,6(2):383–390.MathSciNetMATHGoogle Scholar
  17. Isac, G., Bulavski, V.A., Kalashnikov, V.V., 1997, Exceptional families, topological degree and complementarity problems.J. Global Optim.,10:207–225.MathSciNetCrossRefMATHGoogle Scholar
  18. Kalashnikov, V.V., 1995. Complementarity Problem and the Generalized Oligopoly Model, Habilitation Thesis. CEMI, Moscow.Google Scholar
  19. Kalashnikov, V.V., Isac, G., 2002. Solvability of implicit complementarity problems.Annals of Oper. Research,116:199–221.MathSciNetCrossRefMATHGoogle Scholar
  20. Takahashi, W., 2000. Nonlinear Functional Analysis (Fixed Point Theory and Its Applications). Yokohama Publishers, Inc.Google Scholar
  21. Zhao, Y.B., 1997. Exceptional family and finite-dimensional variational inequalities over polyhedral convex sets.Applied Math. Comput.,87:111–126.MATHGoogle Scholar
  22. Zhao, Y.B., 1998. Existence Theory and Algorithms for Finite-Dimensional Variational Inequalities and Complementarity Problems. Ph. D. Thesis, Institute of Applied Mathematics, Academia Sinica, Beijing China (in Chinese).Google Scholar
  23. Zhao, Y.B., Isac, G., 2000a. Quasi-P andP(τ, α, β)-maps, exceptional family of elements and complementarity problems.J. Opt. Theory Appl.,105:213–231.MathSciNetCrossRefGoogle Scholar
  24. Zhao, Y.B., Isac, G., 2000b. Properties of a multivalued mapping associated with some nonmonotone complementarity problems.SIAM J. Control Optim. 39:571–593.MathSciNetCrossRefMATHGoogle Scholar

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© Zhejiang University Press 2005

Authors and Affiliations

  1. 1.Department of MathematicsRoyal Military College of CanadaKingstonCanada
  2. 2.Department of MathematicsShawnee State UniversityPortsmouthUSA

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