Journal of Global Optimization

, Volume 34, Issue 1, pp 15–32 | Cite as

System of Generalized Vector Quasi-Equilibrium Problems with Applications to Fixed Point Theorems for a Family of Nonexpansive Multivalued Mappings

  • Lai-jiu Lin


In this paper, we establish the existence theorems of the generalized vector quasi-equilibrium problems. From some existence theorem, we establish fixed point theorems for a family of lower semicontinuous or nonexpansive multivalued mappings. We also obtain the existence theorems of system of mixed generalized vector variational-like inequalities and existence theorems of the Debreu vector equilibrium problems and the Nash vector equilibrium problems.


generalized vector quasi-equilibrium problem nonexpansive multivalued map quasiconvex quasiconvex-like upper(lower) semicontinuous multivalued map 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ansari, Q.H., Schaible, S., Yao, J.C. 2000The system of vector equilibrium problems and its applicationsJournal of Optim Theory and Applications.107547557Google Scholar
  2. 2.
    Ansari, Q.H., Schaible, S., Yao, J.C. 2000The system of generalized vector equilibrium problems with applicationsJournal of Global Optim.22316Google Scholar
  3. 3.
    Ansari, Q.H., Chan, W.K., Yang, X.Q. 2004The system of vector quasi-equilibrium problems with applicationsJournal of Global Optim.294557Google Scholar
  4. 4.
    Aubin, J.P., Cellina, A. 1994Differential InclusionsSpringer VerlagBerlin, Heidelberg, GermanyGoogle Scholar
  5. 5.
    Blum, E., Oettli, W. 1994From optimization and variational inequalities to equilibrium problemThe Mathematical Students.63123146Google Scholar
  6. 6.
    Browder, F.E. 1965Nonexpansive nonlinear operators in a Banach spaceProceeding National Academic Science USA.5410411044Google Scholar
  7. 7.
    Browder, F.E. 1968The fixed point theorem of multivalued mappings in topological linear spacesMathamatical Annal.177283301Google Scholar
  8. 8.
    Chen, M.P, Lin, L.J., Park, S. 2003Remarks on generalized quasi-equilibrium problemsNonlinear Analysis.52433444CrossRefGoogle Scholar
  9. 9.
    Debreu, G. 1952A social equilibrium existence theoremProceeding National Academic Science USA.38886893Google Scholar
  10. 10.
    Deguire, P., Tan, K.K., Yuan, G.X.Z. 1999The study of maximal elements fixed point for Ls-majorized mappings and the quasi-variational inequalities in product spacesNonlinear Analysis.37933951CrossRefGoogle Scholar
  11. 11.
    Ding, X.P., Tarafdar, E. 2000Generalized vector variational-like inequalities without monotonicityGiannessi, F eds. Vector Variational Inequalities and Vector Equilibria: Mathematical Theorem.Kluwer Academic PublishersDordrecht, Holland113124Google Scholar
  12. 12.
    Fu, J.Y., Wan, A.H. 2002Generalized vector equilibria problems with set-valued mappingsMathematical Methods Operativen Research.56259268Google Scholar
  13. 13.
    Gohde, D. 1965Zum Prinzip der kontraktuen AbbildungMathematical Nachs.30251258Google Scholar
  14. 14.
    Goebel, K., Kirk, W.A. 1990Topics in Metric Fixed Point TheoremCambridge University PressCambridgeGoogle Scholar
  15. 15.
    Hadjisavvas, N., Schaible, S. 1998From scalar to vector equilibrium problem in the quasimonotone caseJournal of Optim Theory and Applications.96297307Google Scholar
  16. 16.
    Kirk, W.A. 1965A fixed point theorem for mappings which do not increase distanceAmerican Mathematical Monthly7210041006Google Scholar
  17. 17.
    Kirk, W.A. (1986), Nonexpansive mapping on product spaces, set-valued mappings, k-uniform rotundity, nonlinear functional analysis and its applications. In:Browder, F.E. (ed.), American Mathematical Society Symposium Pure Mathematical 45, 51–64.Google Scholar
  18. 18.
    Khanh, P.Q. and Luu, L.M. Some existence results for vector quasi-variational inequalities involving multifunctions and applications to traffic equilibrium problems, Journal of Global Optim (to appear).Google Scholar
  19. 19.
    Lin, L.J., Park, S. 1998On some generalized quasi-equilibrium problemsJournal of Mathematical Analysis Applications.224167181Google Scholar
  20. 20.
    Lin, L.J., Yu, Z.T. 2001Fixed point theorems and equilibrium problemsNonlinear Analysis.43987999CrossRefGoogle Scholar
  21. 21.
    Lin, L.J., Yu, Z.T., Kassay, G. 2002Existence of equilibrium for multivalued mappings and its applications to vectorial equilibiraJournal of Optim Theory and Applications.114189208Google Scholar
  22. 22.
    Lin, L.J., Ansari, Q.H., Wu, J.Y. 2003Geometric properties and coincidence theorems with applications to generalized vector equilibrium problemJournal of Optim Theory and Applications.117121137Google Scholar
  23. 23.
    Lin, L.J. and Ansari, Q.H., Some existence results for solutions of generalized vector quasi-equilibrium problems (to appear).Google Scholar
  24. 24.
    Lin, L.J., Yu, Z.T. 2001On some equilibrium problems for multivalued mapsJournal of Computational Applied Mathematics.129171183Google Scholar
  25. 25.
    Lin, L.J., Chen, L.F. and Ansari, Q.H., Generalized abstract economy and solution of generalized vector quasi problems, Journal of Optimization Theory and Applications (to appear).Google Scholar
  26. 26.
    Lin, L.J. and Liu, Y.H. Existence theorems of system of generalized vector quasi- equilibrium problems and optimization problems, (preprint).Google Scholar
  27. 27.
    Nash, J. 1951Non-cooperative gamesAnnals of Mathematics.54286295Google Scholar
  28. 28.
    Takahashi, W. 2000Nonlinear Functional AnalysisYokohama PublishersYokohamaGoogle Scholar
  29. 29.
    Tan, N.X. 1985Quasi-variational inequalities in topological linear locally convex Hausdorff spacesMathematicsche Nachrichen.122231245Google Scholar
  30. 30.
    Wu, X. 1997A new fixed point theorem and its applicationsProceeding American Mathematical Society.12511791183Google Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • Lai-jiu Lin
    • 1
  1. 1.Department of MathematicsNational Changhua University of EducationChanghuaTaiwan

Personalised recommendations