  We show that a continuous map or a continuous flow on $\R^{n}$ 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
$\R^{n}$ 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.
