For existence and uniqueness of first order initial value problems on $\mathbb{R}^n$, theorems like Picard's theorem gives only local information.
I wonder when we can say an initial value problem $y'=f(x,y),y\in\mathbb{R}^n$ has unique solution on $(a,b)\subset\mathbb{R}$, for any given initial condition $y(a)=y_1\in\mathbb{R}$.
(The condition on $f$ can be specified as much as needed and inequalities can also be given, but let's not assume to be linear.)