A Schauder-type fixed point theorem

In this brief note we study Schauderʹs second fixed point theorem in the space ( BC , ‖ ⋅ ‖ ) of bounded continuous functions ϕ : [ 0 , ∞ ) → ℜ n 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.