A fixed-point theorem and applications to problems on sets with convex sections and to Nash equilibria

A new fixed-point theorem for a family of maps defined on product spaces is obtained. The new result requires the functions involved to satisfy the local intersection properties. Previous results required the functions to have the open lower sections which are more restrictive conditions. New properties of multivalued maps are provided and applied to prove the new fixed-point theorem. Applications to problems on sets with convex sections and to the existence of Nash equilibria for a family of continuous functions are given.