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I'm studying ODE's and I found this problem in the notes our teacher gave us. The thing is that I haven't been able to prove it. So, any help would be awesome.

Problem:

Suppose f(t,x) is continuous in (t,x) \in \mathbb{R}^{2} and log-Lipschitz only in x, i.e, $|f(t,x_{1})-f(t,x_{2})| \leq L|x_{1}-x_{2}||log|x_{1}-x_{2}||$ for $L>0$ and $x_{1}\neq x_{2} $.

Assuming that the IVP $\dot x = f(t,x), x(t_{0})=x_{0}$ has a solution, prove that solution is unique.

Thanks so much for your help.

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1 Answer 1

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The idea is to use the Osgood criterion (see Osgood criterion. Encyclopedia of Mathematics.), which states that if $f$ is continuous and satisifies $|f(t,x_1) - f(t,x_2)| \leq \omega(|x_1-x_2|)$ for some $\omega$ with $\int_0^\infty \frac{ds}{\omega(s)} = \infty$, then there exists a unique solution to the IVP $\dot x = f(t,x), x(t_0) = x_0$. It remains to check that $\int_0^\infty \frac{ds}{s|\log(s)|} = \infty$ (this is quite well known).

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