Let $n\in\mathbb{N}$, let $I\subset \mathbb{R}$ be an interval, let $U\subset \mathbb{R}^n$ be open, and let $F:I\times U\to \mathbb{R}^n$ be continuous. Let $a,b\in I$ satisfy $a<b$ and $[a,b]\subset I$. Let $x,y:[a,b]\to U$ be solutions of the [ordinary differential equation](/page/Ordinary%20Differential%20Equation) $\dot z(t)=F(t,z(t))$ on $[a,b]$, with the differential equation imposed at interior points and by one-sided derivatives at the endpoints. Suppose there is a set $C\subset U$ such that $x([a,b])\subset C$ and $y([a,b])\subset C$, and suppose that $L\ge 0$ satisfies

\begin{align*} |F(t,p)-F(t,q)|\le L|p-q| \end{align*}

for every $t\in [a,b]$ and every $p,q\in C$. Then, for every $t\in [a,b]$,

\begin{align*} |x(t)-y(t)|\le e^{L(t-a)}|x(a)-y(a)|. \end{align*}