I am working on the following problem from Gerald Teschl's book on ODE's and am at a loss of how to proceed.
Suppose $f:\mathbb{R}^n→\mathbb{R}^n$ local Lipschitz. Show that,
if $\lim sup_{|x|\rightarrow \infty}\frac{<x,f(x)>}{|x|^2}<\infty$, then all solution of $x′=f(x)$ are global defined.
Any help would be appreciated. Thanks!
Confirm that $\dfrac{\langle x,f(x)\rangle}{|x|^2}$ is not only continuous but also bounded outside some disk of radius $R$, that is for $|x|\ge R$. Additionally, $\langle x,f(x)\rangle$ is continuous and thus bounded on $|x|\le R$. Let $M$ be a common bound for both cases.
Consider $V(x)=|x|^2$. Then $$ \frac{d}{dt}V(x(t))=2\langle x,f(x)\rangle\le 2M\,(1+V(x(t))) $$ which gives an exponential bound on $|x(t)|$ preventing divergence to infinity in finite time.