A fixed point theorem for bounded dynamical systems

We show that a continuous map or a continuous flow on Rⁿ with a certain recurrence relation must have a fixed point. Specifically, if there is a compact set W with the property that the forward orbit of every point in Rⁿ intersects W, then there is a fixed point in W. Consequently, if the omega limit set of every point is nonempty and uniformly bounded, then there is a fixed point.