What is the necessary condition for ODE to have unique solution? For the ODE:
\begin{align}
\dot{x}(t)&=f(x,t) \\
x(t_{0})&=x_{0}
\end{align}
If $\;\;f:\mathbb{R}^{n}\rightarrow{}\mathbb{R}^{n}$ is Lipschitz continuous on $\mathbb{R}^{n}$, then there exists a unique solution for the ODE.
The reverse is not always true, right? and what is the necessary condition for the uniqueness for the ODE?
I have read a book about numerical computing for ODE. The author only assumed there exists a unique solution for the ODE. In proving some properties in the numerical scheme, the author used the Lipschitz continuous property of the ODE. I wonder it is not a correct proof.
 A: The Lipstcitz condition is not necessary for uniqueness of solutions. There are other sufficient conditions, like the Osgood criterion, or monotony conditions like
$$
(f(t,x)-f(t,y)\cdot(x-y)\le0.
$$
As far as I know, there are not known necessary and sufficient conditions.
A: The first question is answered by Julián Aguirre. The second is not, justifying this addition:
Yosie's theorem of 1926, as presented on p.82 of [1] at least, provides a characterisation of such ODEs' uniqueness, and so a necessary condition as well as sufficient. This uniqueness theorem, however, does not look like many (actually any, to my knowledge) of the classical ones (also presented in [1]). This is because the classical ones (by definition, if you like) boil down to a spatial regularity condition like Lipschitz's or Osgood's. Yosie's theorem, on the other hand, necessarily covers functions which have no meaningful spatial regularity. There are indeed examples of uniqueness for such `spatially irregular' ODEs, as demonstrated quite generally by the elegant idea of [2] to invert the spatial and temporal components in any classical result. 
[1] R P Agarwal & V Lakshmikantham, Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations, DOI: https://doi.org/10.1142/1988
[2] J A Cid & R L Pouso, Does Lipschitz with Respect to x Imply Uniqueness for the Differential Equation y' = f(x,y)?, URL: https://www.jstor.org/stable/27642664
