Theorem. Consider a continuous map $f : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ and a nonempty subset $K \subseteq {\mathbb{R}}^{n}$. If $K$ is forward-invariant for the autonomous dynamical system $\dot{x} = f(x)$, and if $K$ is compact and convex, then $K$ contains an equilibrium of the dynamical system $\dot{x} = f(x)$, i.e. a zero of the map $f$.

According to a reliable source, the above theorem is a standard result everyone uses in dynamical systems without proof. I propose a proof in "Equilibria Exist in Compact Convex Forward-Invariant Sets" at http://math.GillesGnacadja.info/files/EquilExists.html. I am interested in comments on this proof, in references to this or other proofs in the literature, and in new/better proofs.

Theorem. Consider a continuous map $f : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ and suppose that the autonomous dynamical system $\dot{x} = f(x)$ has a semiflow $\varphi : {\mathbb{R}}_{\geq{0}} \times {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$. Let $K \subseteq {\mathbb{R}}^{n}$. If $K$ is nonempty, compact, convex and forward-invariant, then $K$ contains an equilibrium of the dynamical system, i.e. a zero of the map $f$.

According to a reliable source, the above theorem is a standard result everyone uses in dynamical systems without proof. I propose a proof in "Equilibria Exist in Compact Convex Forward-Invariant Sets" at http://math.GillesGnacadja.info/files/EquilExists.html. I am interested in comments on this proof, in references to this or other proofs in the literature, and in new/better proofs.