Let F:

**R**x

**R**^{n} ->

**R**^{n} be an

ordinary differential equation of the form dx = F(t,x).

If F is

Lipschitz continuous in x and

coninuous in t, then there exists a

unique solution f:

**R**->

**R**^{n} (here: df=F(t,f(t))) defined on an maximal

open set for a given starting value f(x

_{0})=y

_{0}.

This a very useful existence theorem: You can just guess a solution, and therefore know it's unique !

Note that such a strong theorem doesn't hold for partial differential equations.

Therefore working with ODEs is much easier, but don't think now getting a solution is always simple ! There are a lot of cases where you don't find one !

This theorem might work on Banach spaces, too, but I'm really not sure.