Picard-Lindelöf theorem variations

There are so many slight variations of this theorem around. I wrote the following in my own words from different sources online in a way that makes sense to me. Is what I have written accurate? I want to make sure I have understood the theorem correctly.

Let $U$ be an open subset of $\mathbb{R}\times \mathbb{R}^m$ containing $(t_0,y_0)$ and let $f:U\to \mathbb{R}^m$. If $f$ is continuous in $t$ and satisfies for some $K\geqslant 0$, $|f(t,x)-f(t,y)|\leqslant K\,|x-y|$ for every $x,y \in U$. Then for some $\tau >0$ there exists a unique solution $y(t)$ on the interval $[t_0 - \tau, t_0 +\tau]$ for the initial value problem $$y'(t)=f(t,y(t)), \,\,\,\,\, y(t_0)=y_0$$

Solution 1:

The most general statements are:

• Peano: $f$ is continuous on $U$, then a local solution to the IVP exists.
• Picard-Lindelöf - local variant: $f$ is continuous on $U$ and is locally Lipschitz in $y$ direction (for instance if $f$ is continuously differentiable in that direction), then a local solution to the IVP exists and is unique.

Where the idea of a local solution is as you wrote. Then there is

• Picard-Lindelöf - global variant: If $U=(a,b)\times \Bbb R^n$ and $f$ is continuous with a global Lipschitz condition in $y$ direction, then a global solution $y:(a,b)\to\Bbb R^n$ exists and is unique.

As you can see, I avoided the naming of constants in the theorem, as they are really only needed in the proof. This would change if the statement of the theorem is to serve at the same time as definition of the constants and properties. For instance to make the theorem pseudo-constructive.

Your formulation is in the middle of all of this,

• it is a local theorem with a global Lipschitz condition, and thus slightly more restrictive than the more general formulation, especially if $U$ is not bounded,
• and it is naming the constant in the Lipschitz condition, but not with the full construction of the interval radius $τ$.

Remember that the proof of the general local theorem first selects a cylindrical domain and 1.) determines the maximum of $|f|$ over it, then restricts this to a smaller cylinder where all solutions would have to exit through the sides, then 2.) on that determines the local $y$-Lipschitz constant and then 3.) further makes the cylinder narrower so that the Picard iteration is contractive. In your formulation the first two steps are exchanged, resp. the second is left out as there is a global $y$-Lipschitz constant.