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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 mapsends 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,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.

The growth bound is $|F(t,x)|\le K(1+|x|)$. A characteristic curve through $(s,a)$ solvesThe $C^1$ field is locally Lipschitz in space, giving local existence and uniqueness. On either direction of a bounded time interval the Gronwall inequality gives $1+|X(t;s,a)|\le (1+|a|)e^{K|t-s|}$. Thus no trajectory can escape to infinity at finite time. On the resulting compact space-time region, local ordinary differential equation existence continues it, proving global definition for all $t\in \mathbb{R}$.

Uniqueness gives the flow composition law and inverse $X(s;t,\cdot )$. Differentiation with respect to $a$ solves the variational equation, making each map a $C^1$ diffeomorphism. Its positive Jacobian determinant isThese facts constitute the global characteristic flow under linear growth. A uniform global bound on $D_{x}F$ is not required; its bounds on each compact characteristic tube suffice.