Proof of Nagumo's Theorem of invariance

Question

I didn't find the proof of the following invariance Theorem due to Nagumo (1942). The statement of the Theorem is as next:`

Given a continuous system $x'(t)= p(x(t))$, $t\geq 0$ on $\mathbb{R}^n$ with initial data $x(0) \in \mathbb{R}^n$ and assuming that solutions exist and are unique in $ \mathbb{R}^n$, let $S\subset \mathbb{R}^n$ be a closed set. Then, $S$ is positively invariant under the flow of the system ($x(t) \in S$ for all $t\geq 0$) if and only if $p(x) \in K_x (S)$ for $x\in \partial S$ (the boundary of $S$) where $K_x(S)$ is the set of all sub-tangential vectors to $S$ at $x$, i.e., $$ K_x (S):=\lbrace z\in \mathbb{R}^n: \lim_{h-->0}\dfrac{d(x+hp(x);S)=0}{h}\rbrace, $$ where $d(w;S)$ is the distance from the vector $w$ to the subset $S$.

If someone has an idea or a reference where I can find the proof. Thank you in advance!

Answer 1

Here I have a proof for ** the necessary condition** which is more easier. That is, let $x\in S$, if we assume that the system $x'(t)=p(x(t)), \; t\geq 0 $ with initial data $x(0)=x$ has a unique solution such that $$ x(t) \in S \text{ for all } t\geq 0 .$$ We prove that $p(x) \in K_x (S)$. So, if $x(0) \in int(S)$, it is clear that $K_x (S)=\mathbb{R}^n$ (to prove that) so it is trivial. But, if $x(0)\in \partial S$, then for $h>0$, we have $$ h^{-1} d(x(0)+hp(x(0));S) \leq h^{-1} \|(x(0)+hp(x(0))-x(h)\|.$$ The right hand side of the last inequality goes to zero as $h$ goes to zero, since $x'(0)=p(x(0))$ and $x(h) \in S$ by assumption.