Generalizations of the FKKM Theorem and Ky-Fan Minimax Inequality, with Applications to Maximal Elements, Price Equilibrium, and Complementarity

This paper generalizes the Fan-Knaster-Kuratowski-Mazurkiewicz (FKKM) theorem of Ky Fan (“Game Theory and Related Topics,” pp. 151–156, North-Holland, Amsterdam, 1979; and Math. Ann.266, 1984, 519–537) and the Ky Fan minimax inequality by introducing a class of the generalized closedness and continuity conditions, which are called the transfer closedness and transfer continuities. We then apply these results to prove the existence of maximal elements of binary relations under very weak assumptions. We also prove the existence of price equilibrium and the complementarity problem without the continuity assumptions. Thus our results generalize many of the existence theorems in the literature.