# Locally Lipschitz $\Rightarrow$ uniqueness; globally Lipschitz $\Rightarrow$ existence and uniqueness?

After reading a differential equations text and a bunch of sites in a google search, I'm still not perfectly clear on the difference between local and global Lipschitz and what these imply for the solutions of differential equations. I'll state my current understanding, but if I'm wrong on any point I'd appreciate a correction.

1. This is the Lipschitz condition:

$$|f(t,u)-f(t,v)|<M|u-v|$$

1. A function is locally Lipschitz if, for any compact subset of the domain, there is an $M$ for which it satisfies the condition on that subset.

2. A function is globally Lipschitz, which we also just call Lipschitz, if it satisfies this condition for one choice of $M$ on all of $\mathbb{R}$.

3. If a differential equation $y'=f(t,y)$ is locally Lipschitz and has an initial condition, then its solution is unique but existence is not guaranteed. If we further ensure that $\frac{\partial f}{\partial y}$ is continuous then existence is guaranteed.

4. If $y'=f(t,y)$ is globally Lipschitz and has an initial condition then existence and uniqueness are guaranteed.

• 1. In that case you say $f$ is Lipschitz with respect to $y$ 4. A differential equation can't be Lipschitz continuous, $f$ is – cronos2 Sep 26 '17 at 23:44
• If its Locally Lipschitz, uniqueness and existence are guaranteed. Even more, if its Locally Lipschitz in the second variable (Your Lipschitz condition, i.e 1.) then uniqueness and existence are guaranteed. Check Picard–Lindelöf theorem : en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem – Bajo Fondo Sep 26 '17 at 23:46
• @BajoFondo Okay, I know I've read some writing which says that existence is not guaranteed under some conditions. I think maybe this is it: When a function is locally Lipschitz, a solution exists locally but not for all time? I think that would be consistent with what you've said. – Addem Sep 27 '17 at 0:03
• This statement is true: let $F: \Omega \subset \mathbb{R}\textrm{x}\mathbb{R}^n \to \mathbb{R}^n$ be a continuous and locally Lipschitz in the second variable (your condition 1 but locally), then there exist an unique maximal solution to the differential equation: $x'=F(t,x)$ with initial condition $(x_0,t_0)$. – Bajo Fondo Sep 27 '17 at 0:30
• A maximal solution is a solution that cannot be extended into another solution. – Bajo Fondo Sep 27 '17 at 0:32