A fixed point theorem for bounded dynamical systems

Abstract

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