How does Lipschitz condition relate to Osgood's (https://www.encyclopediaofmath.org/index.php/Osgood_criterion) ?
In connection with ODEs, more specifically the IVP (for both scalar and vector case) $$y' = f(x,y), \ y(x_0) = y_0$$
the Osgood condition is more general/weaker requirement. I don't really see how? I think I should compare a constant (from Lipschitz) to the function $w$ (from Osgood, following the notation from the link).
However, I don't see how a constant is a stronger requirement, since a constant function has bigger domain (not just $[0, \infty]$) and it doesn't satisfy $w(0) = 0$. I also have no clue where does the integral $\int_{0}^{1} \frac{d\xi}{w(\xi)} = \infty $ come from?
Thank you in advance for an answer.
Notice that the criterion has $\omega(|x_1-x_2|)$ as a function of $|x_1-x_2|$. So for the Lipchitz case, let $\omega(x)=Mx$. Hence $\omega(|x_1-x_2|)=M|x_1-x_2|$. With this definition, $\omega(0)=0$ and $$\int_0^1\frac{1}{\omega(x)}dx=\int^1_0\frac{1}{Mx}dx=\infty.$$