Abstract
AbstractThe partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. In this paper, we derive generalized forms of the Ky Fan minimax inequality, the von Neumann-Sion minimax theorem, the von Neumann-Fan intersection theorem, the Fan-type analytic alternative, and the Nash equilibrium theorem for abstract convex spaces satisfying the partial KKM principle. These results are compared with previously known cases for "Equation missing"-convex spaces. Consequently, our results unify and generalize most of previously known particular cases of the same nature. Finally, we add some detailed historical remarks on related topics.
Highlights
In 1928, John von Neumann found his celebrated minimax theorem 1 and, in 1937, his intersection lemma 2, which was intended to establish his minimax theorem and his theorem on optimal balanced growth paths
In our recent works 24–26, we studied the foundations of the KKM theory on abstract convex spaces
In our recent works 24–26, we studied the foundations of the KKM theory on abstract convex spaces and noticed that many important results therein are related to the partial KKM principle
Summary
In 1928, John von Neumann found his celebrated minimax theorem 1 and, in 1937, his intersection lemma 2 , which was intended to establish his minimax theorem and his theorem on optimal balanced growth paths. In 1961, Ky Fan 9 obtained his KKM lemma and, in 1964 , applied it to another intersection theorem for a finite family of sets having convex sections This was applied in 1966 to a proof of the Nash equilibrium theorem. Our aim in this paper is to obtain generalized forms of the KKM space versions of known results due to von Neumann, Sion, Nash, Fan, Ma, and many followers. These results are mainly obtained by 1 fixed point method, 2 continuous selection method, or 3 the KKM method. This paper is a revised and extended version of 22, 27 and a supplement to 24–26 , where some other topics on abstract convex spaces can be found
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