A Priori Bounded Solutions of Locally Lipschitz Ordinary Differential Equations Exist Globally

Let $a<b$ be real numbers, let $k\ge1$ be a natural number, and for $x\in\mathbb{R}^{k}$ (Euclidean space) write $|x|=d(x,0)$ with the Euclidean distance $d$. Let $\xi\in\mathbb{R}^{k}$ and let $F:[a,b]\times\mathbb{R}^{k}\to\mathbb{R}^{k}$ satisfy the following, continuity of a map defined on a subinterval of $[a,b]$ being understood as continuity of a map of metric spaces, with that interval regarded as a subset of the real line with the absolute value metric and $\mathbb{R}$ carrying the same metric:

(i) (composition continuity) for every function $h:[a,b]\to\mathbb{R}^{k}$ with continuous components, $t\mapsto F(t,h(t))$ has continuous components;

(ii) (local Lipschitz condition) for every real $R>0$ there is a real $L_R\ge0$ with $d(F(t,x),F(t,y))\le L_R\,d(x,y)$ for all $t\in[a,b]$ and all $x,y\in\mathbb{R}^{k}$ with $|x|\le R$ and $|y|\le R$.

Call a function $h:[a,c]\to\mathbb{R}^{k}$ with continuous components, where $a<c\le b$, a solution on $[a,c]$ if, componentwise with the Riemann integral,

(The integrand $r\mapsto F(r,h(r))$ is continuous on $[a,c]$: extending $h$ to $\bar h:[a,b]\to\mathbb{R}^{k}$ by $\bar h(t)=h(\min(t,c))$ yields a function with continuous components, hypothesis (i) makes $t\mapsto F(t,\bar h(t))$ continuous on $[a,b]$, and its restriction to $[a,c]$ equals the integrand; the integrals then exist by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, with degenerate intervals following the convention of Mean-Square Riemann Integral of a Family of Random Variables.)

Suppose finally:

(iii) (a priori bound) there is a real $\rho\ge|\xi|$ such that every solution $h$ on every interval $[a,c]$ with $a<c\le b$ satisfies $|h(t)|\le\rho$ for all $t\in[a,c]$.

Then there is exactly one solution $x$ on $[a,b]$, and it satisfies $|x(t)|\le\rho$ for all $t\in[a,b]$.