Applied Mathematics and Mechanics

, Volume 22, Issue 2, pp 160–172 | Cite as

Quasi-equilibrium problems and constrained multiobjective games in generalized convex space

  • Ding Xie-ping


A class of quasi-equilibrium problems and a class of constrained multiobjective games were introduced and studied in generalized convex spaces without linear structure. First, two existence theorems of solutions for quasi-equilibrium problems are proved in noncompact generalized convex spaces. Then, as applications of the quasi-equilibrium existence theorem, several existence theorems of weighted Nash-equilibria and Pareto equilibria for the constrained multiobjective games are established in noncompact generalized convex spaces. These theorems improve, unify and generalize the corresponding results of the multiobjective games in recent literatures.

Key words

quasi-equilibrium problem constrained multiobjective game weighted Nash-equilibria Pareto equilibria generalized convex space 

CLC number



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    DING Xie-ping. Quasi-equilibrium problems with applications to infinite optimization and constrained games in general topological spaces[J].Appl Math Lett, 2000,12(3): 21–26.Google Scholar
  2. [2]
    Noor M A, Oettli W. On general nonlinear complementarity problems and quasiequilibria[J].Le Mathematiche, 1994,49: 313–331.MATHMathSciNetGoogle Scholar
  3. [3]
    Cubiotti P. Existence of solutions for lower semicontinuous quasi-equilibrium problems[J].Compu Math Appl, 1995,30(12): 11–22.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    DING Xie-ping. Existence of solutions for equilibrium problems[J].J Sichuan Normal Univ, 1998,21(6): 603–608.Google Scholar
  5. [5]
    DING Xie-ping. Quasi-equilibrium problems in noncompact generalized convex spaces[J].Applied Mathematics and Mechanics (English Edition), 2000,21(6): 637–644.MathSciNetGoogle Scholar
  6. [6]
    Lin L J, Park S. On some generalized quasi-equilibrium problems[J].J Math Anal Appl, 1998,224(1): 167–191.MATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    Nash J F. Equilibrium point inn-person games[J].Proc Nat Acad Sci USA, 1950,36(1): 48–49.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    Nash J F. Noncooperative games[J].Ann Math, 1951,54: 286–295.MATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    Szidarovszky F, Gershon M E, Duckstein L.Techniques for Multiobjective Decision Marking in System Management[M]. Amsterdam Holland: Elsevier, 1986.Google Scholar
  10. [10]
    Zeleny M. Game with multiple payoffs[J].International J Game Theory, 1976,4(1): 179–191.MathSciNetGoogle Scholar
  11. [11]
    Bergstresser K, Yu P L. Domination structures and multicriteria problem inN-person games[J].Theory and Decision, 1977,8(1): 5–47.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    Borm P E M, Tijs S H, Van Den Aarssen J C M. Pareto equilibrium in multiobjective games[J].Methods of Operations Research, 1990,60(2): 303–312.MATHGoogle Scholar
  13. [13]
    Yu P L. Second-order game problems: Decision dynamics in gaming phenomena[J].J Optim Theory Appl, 1979,27(1): 147–166.MATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    Chose D, Prasad U R. Solution concepts in two-person multicriteria games[J].J Optim Theory Appl, 1989,63(1): 167–189.MathSciNetGoogle Scholar
  15. [15]
    Wang S Y. An existence theorem of a Pareto equilibrium[J].Appl Math Lett, 1991,4(3): 61–63.MATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    Wang S Y. Existence of a Pareto equilibrium[J].J Optim Theory Appl, 1993,79(2): 373–384.MATHMathSciNetCrossRefGoogle Scholar
  17. [17]
    DING Xie-ping. Pareto equilibria of multicriteria games without compactness, continuity and concavity[J].Applied Mathematics and Mechanics (English Edition), 1996,17(9): 847–854.MathSciNetGoogle Scholar
  18. [18]
    Yuan X Z, Tarafdar E. Non-compact Pareto equilibria for multiobjective games[J].J Math Anal Appl, 1996,204(1): 156–163.MATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    Yu J, Yuan X Z. The study of Pareto equilibria for multiobjective games by fixed point and Ky Fan minimax inequality methods[J].Compu Math Appl, 1998,35(9): 17–24.MATHMathSciNetCrossRefGoogle Scholar
  20. [20]
    DING Xie-ping. Existence of Pareto equilibria for constrained multiobjective games inH-spaces [J].Compu Math Appl, 2000,39(9): 115–123.CrossRefGoogle Scholar
  21. [21]
    Park S, Kim H. Coincidence theorems for admissible multifunctions on generalized convex spaces [J].J Math Anal Appl, 1996,197(1): 173–187.MATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    Park S, Kim H. Foundation of the KKM theory on generalized convex spaces[J].J Math Anal Appl, 1997,209(3): 551–571.MATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    Tan K K.G-KKM theorem, minimax inequalities and saddle points[J].Nonlinear Anal, 1997,30 (7): 4151–4160.MATHMathSciNetCrossRefGoogle Scholar
  24. [24]
    Aubin J P.Mathematical Methods of Game and Economic Theory [M]. Amsterdam: North-Holland, 1982.Google Scholar
  25. [25]
    Aubin J P, Ekeland I.Applied Nonlinear Analysis[M]. New York: Wiley, 1984.Google Scholar
  26. [26]
    DING Xie-ping. Quasi-variational inequalities and social equilibrium[J].Applied Mathematics and Mechanics (English Edition), 1991,12(7): 639–646.Google Scholar
  27. [27]
    DING Xie-ping. Generalized quasi-variational inequalities, optimization and equilibrium problems [J].J Sichuan Normal Univ, 1998,21(1): 22–27.Google Scholar
  28. [28]
    Tian G. Generalizations of the FKKM theorem and the Fan minimax inequality with applications to maximal elements, price equilibrium and complementarity[J].J Math Anal Appl, 1992,170(2): 457–471.MATHMathSciNetCrossRefGoogle Scholar
  29. [29]
    Yuan X Z, Isac G, Tan K K, et al. The study of minimax inequalities, abstract economics and applications to variational inequalities and Nash equilibria[J].Acta Appl Math, 1998,54(1): 135–166.MATHMathSciNetCrossRefGoogle Scholar
  30. [30]
    DING Xie-ping. Generalized variational inequalities and equilibrium problems in generalized convex spaces[J].Compu Math Appl, 1999,38(7–8): 189–197.Google Scholar
  31. [31]
    DING Xie-ping. Fixed points, minimax inequalities and equilibria of noncompact abstract economies[J].Taiwanese J Math, 1998,2(1): 25–55.MathSciNetGoogle Scholar
  32. [32]
    Zhou J X, Chen G. Diagonally convexity conditions for problems in convex analysis and quasivariational inequalities[J].J Math Anal Appl, 1988,132(2): 213–225.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics and Mechanics 1980

Authors and Affiliations

  • Ding Xie-ping
    • 1
  1. 1.Department of MathematicsSichuan Normal UniversityChengduP R China

Personalised recommendations