A condition on uniform spaces for the existence of maximal elements and fixed points

We introduce extensions to uniform spaces of some classical theorems of nonlinear analysis, whose original setting corresponded to complete metric spaces. Our results are based on a condition for a filter base, on uniform spaces, to have a nonempty intersection. Using this condition, we prove the existence of maximal elements for a given preordering, the existence of fixed points for multivalued functions, and related issues. Well-known results by Nadler and Saint-Raymond, in the setting of metric spaces, are also extended to the uniform space scenario.