# Unique solution and log Lipschitz condition

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.

## 1 Answer

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).