Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 234706, 23 pages
doi:10.1155/2010/234706
Research Article

Generalizations of the Nash Equilibrium Theorem in the KKM Theory

Sehie Park1,2

1The National Academy of Sciences, Seoul 137-044, Republic of Korea
2Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea

Received 5 December 2009; Accepted 2 February 2010

Academic Editor: Anthony To Ming Lau

Copyright © 2010 Sehie Park. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The 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 G-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.