Fixed Points, Maximal Elements and Equilibria in Generalized Convex Spaces

In this paper, a generalized version of the Fan-Knaster-Kuratowski-Mazurkiewicz lemma is obtained and used to prove the existence of fixed points for correspondenced defined on a generalized convex spaces. A result on the existence of maximal elements is deduced and is applied to prove the existence of equilibrium for qualitative games with an infinite number of agents. The last result is used to establish the existence of equilibrium in generalized games (or abstract economies) with a generalized convex choice sets.