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Let $f:\mathbb{R}\times\mathbb{R}^n \to \mathbb{R}^n$ be a continuous, locally Lipschitz function that satisfies the following condition:

$$|f(t,x)|\leq C(1+|x|)\; , \forall (t,x)\in \mathbb{R}\times\mathbb{R}^n$$

Show that the IVP $x' = f(t,x),\; x(t_0)=x_0$ admits a unique solution.

We can get existence of solutions from Peano's existence theorem. It's also easy to see that $C(1+|x|)$ is globally Lipschitz. Though, I don't know how to connect the dots. It feels like we would need $f$ to be globally Lipschitz to be able to say anything about uniqueness of solutions...

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  • $\begingroup$ The case where $f$ doesn't depend on $t$ is proved as Proposition IV.3 in Zehnder's book Lectures on Dynamical Systems, using Grönwall's lemma. Probably something similar works in the general case. $\endgroup$ Oct 10 '19 at 19:32
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With the equation locally Lipschitz you also get that the solutions are locally unique. As any branching of IVP solutions is also a local event, local uniqueness is sufficient for global uniqueness, that is, for as long as the solution is defined.

What is usually associated with a linear bound of $f$ of the given form is that the domain of any solution is $\Bbb R$. But your task description does not even mention that?

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  • $\begingroup$ Your question is correct, that's exactly what I'm supposed to prove. Seems like I forgot to write the most important part of the problem. $\endgroup$ Oct 10 '19 at 23:04

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