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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.

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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.

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