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Let $V$ be a continuous vector field on an open set $U \subset \mathbb{R}^n$ and let $p_0 \in U$. There are many ways to construct local integral curves of $V$ through $p_0$, i.e. differentiable maps $\gamma\colon (-\epsilon,\epsilon) \rightarrow U$ with $\gamma(0) = p_0$ whose derivatives equal $V$. This is another way of stating the existence of solutions to a system of first order ODE's given an initial value.

If $V$ is Lipshitz, then such integral curves are unique. Is there a direct way of seeing this, perhaps with a stronger smoothness assumption (e.g. $V$ being $C^{\infty}$)? The usual proof gives both existence and uniqueness by recasting this as an integral equation and showing that the solution to the integral equation is a fixed point for a contracting map on some appropriate space of functions. This has always seemed a little magical to me, in contrast to some of the more geometric arguments for existence of integral curves (but in contrast to existence, I don't know any other proofs of uniqueness).

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Osgood's criterion is known to be a necessary (and sufficient) condition for uniqueness, since there are non-Osgood vector-fields for which uniqueness does not hold (e.g. $V(x)=|x|^{\alpha}$ for any $\alpha\in(0,1)$ in dimension 1). Of course existence requires basically nothing, except continuity, by the Cauchy-Peano theorem.

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