The Fixed Point Method Versus the KKM Method

In this survey, we compare the fixed point method and the KKM method in nonlinear analysis. Especially, we consider two methods in the proofs of the following important theorems in the chronological order: (1) The von Neumann minimax theorem, (2) The von Neumann intersection lemma, (3) The Nash equilibrium theorem, (4) The social equilibrium existence theorem of Debreu, (5) The Gale-Nikaido-Debreu theorem, (6) The Fan-Browder fixed point theorem, (7) Generalized Fan minimax inequality, and (8) The Himmelberg fixed point theorem.