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.
Citation
David Richeson. Jim Wiseman. "A fixed point theorem for bounded dynamical systems." Illinois J. Math. 46 (2) 491 - 495, Summer 2002. https://doi.org/10.1215/ijm/1258136205
Information