# Alternate proof of uniqueness of integral curves to vector fields

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).

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.