= Solution
A local form of the <Picard-Lindelöf theorem> is as follows. Let $F(t,y)$ be continuous on a rectangle about $(t_0,y_0)$ and uniformly <Lipschitz continuity>[Lipschitz] in $y$ there. Then some interval about $t_0$ admits a unique continuously differentiable solution of
$$
y'(t)=F(t,y(t)),
\qquad y(t_0)=y_0.
$$
The solution is the unique fixed point of the associated integral operator. If $F$ is continuous on $[a,b]\times\mathbb R^n$ and globally Lipschitz in $y$ with one constant, the solution exists uniquely throughout $[a,b]$.
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