# Banach contraction principle and Global Picard Theorem

These two theorems are quite similar, they both show that an IVP has a unique point. Is there any difference between them.

To show an IVP has a solution, I usually show that the map is a contraction, thus IVP has a unique solution by Banach contraction principle. But if we apply global Picard theorem, there is no need to proving T is a contraction (i.e. as long as T is Lipschitz, it's fine.

The immediate use of the Banach fixed point theorem only works up to a finite time. Specifically, you define a map $A$ and seek its fixed point as the solution, and you find something like $\| A(f) - A(g) \| \leq LT \| f - g \|$ where $L$ is the Lipschitz constant from the equation and $T$ is the length of the interval you are using to define $A$. This will only be a contraction for $T<1/L$, so this argument only gives existence/uniqueness up to a finite time.
Global Picard amounts to performing the Banach fixed point theorem argument repeatedly to get $T$ to be larger than $1/L$.