A Fixed Point Theorem for Bounded Dynamical Systems
Illinois Journal of Mathematics
We show that a continuous map or a continuous flow 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 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.
Richeson, David, and Jim Wiseman. "A Fixed Point Theorem for Bounded Dynamical Systems." Illinois Journal of Mathematics 46, no. 2 (2002): 491-495. https://www.projecteuclid.org/euclid.ijm/1258136205