Abstract
In this paper, we give a uniform approach for generalized convexity by using the concept of L-convexity defined by Ben El-Mechaiekh et al. (J Math Anal Appl 222:138-150, 1998). We prove that the generalized notion of L-space contains well-known generalized convex spaces defined in the literature in topological vector spaces as well as several generalized convexity structures defined on metric spaces. In this context, we give a generalized version of the Fan-Knaster-Kuratowski-Mazurkiewicz Principle (FKKM Principle) in L-spaces and a Browder-Fan type theorem about the existence of fixed points for open lower section set-valued maps defined in an L-space. As an application, we prove the existence of equilibria for an abstract economy with an infinite number of agents.