Acta Applicandae Mathematica

, Volume 54, Issue 2, pp 135–166

The Study of Minimax Inequalities, Abstract Economics and Applications to Variational Inequalities and Nash Equilibria

  • George Xian-Zhi Yuan
  • George Isac
  • Kok-Keong Tan
  • Jian Yu


In this survey, a new minimax inequality and one equivalent geometricform are proved. Next, a theorem concerning the existence of maximalelements for an LC-majorized correspondence is obtained.By the maximal element theorem, existence theorems of equilibrium point fora noncompact one-person game and for a noncompact qualitative game withLC-majorized correspondences are given. Using the lastresult and employing 'approximation approach', we prove theexistence of equilibria for abstract economies in which the constraintcorrespondence is lower (upper) semicontinuous instead of having lower(upper) open sections or open graphs in infinite-dimensional topologicalspaces. Then, as the applications, the existence theorems of solutions forthe quasi-variational inequalities and generalized quasi-variationalinequalities for noncompact cases are also proven. Finally, with theapplications of quasi-variational inequalities, the existence theorems ofNash equilibrium of constrained games with noncompact are given. Our resultsinclude many results in the literature as special cases.

KKM principle minimax inequality lower (upper) open sections lower (upper) semicontinuous class LC LC-majorant LC-majorized qualitative game abstract economics equilibrium Nash equilibrium the property (K) quasi-variational inequality 


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Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • George Xian-Zhi Yuan
    • 1
  • George Isac
    • 2
  • Kok-Keong Tan
    • 3
  • Jian Yu
    • 3
  1. 1.Department of MathematicsThe University of QueenslandBrisbaneAustralia
  2. 2.Department of Mathematics and Computer ScienceRoyal Military College of CanadaKingstonCanada
  3. 3.Department of Mathematics and StatisticsDalhousie UniversityHalifaxCanada

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