A Schauder-type fixed point theorem

In this brief note we study Schauder's second fixed point theorem in the space (BC,‖⋅‖)(BC,‖⋅‖) of bounded continuous functions ϕ:[0,∞)→ℜnϕ:[0,∞)→ℜn with a view to reducing the requirement that there is a compact map to the requirement that the map is locally equicontinuous. Several examples are given, both motivating and applying the theory.