Fixed point and equilibrium theorems in pseudoconvex and related spaces

Convexity spaces defined in the paper are generated by families of continuous functions. Without imposing any explicitly stated linear structure on the spaces, Browder's, Brouwer's and Kakutani's fixed point theorems are proved and used for deriving generalized Fan inequalities and two-function minimax theorems. The existence of Nash equilibria in noncooperative games is also established under more general conditions than known before. The convexity spaces studied in the paper allow for unusual (generalized) convex sets and functions.