= Solution
Put $z=(X,V)$ and $b(t,z)=(V,F(t,X,V))$. On a small closed time interval and a closed ball around the initial point, let $M$ bound $b$ and let $K$ be its spatial Lipschitz constant. The map
$$
(\mathcal Tz)(t)=z_0+\int_{t_0}^t b(s,z(s))\,ds
$$
sends the corresponding closed ball of continuous paths into itself when the time length times $M$ is at most the ball radius. If the time length times $K$ is less than one, it is a contraction in the <supremum norm>. The <Banach fixed-point theorem> gives a local solution and uniqueness; differentiating its <integral> equation gives the <ordinary differential equation>. Overlapping local solutions agree by this uniqueness.
For continuation, the growth assumption gives, on any bounded time interval,
$$
1+|z(t)|\leq1+|z_0|+C\int_{t_0}^t(1+|z(s)|)\,ds
$$
for forward time, with the analogous reversed-time bound. The <Gronwall inequality> bounds the trajectory on that interval. A finite terminal time is impossible: within the resulting compact ball the field is bounded, so the path has a limit at the endpoint, and the local construction restarts there. This proves the <global characteristic flow under linear growth>. No differentiability of the flow with respect to its initial point is needed for this existence and uniqueness proof.
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