Let F:RxRⁿ -> Rⁿ 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-> Rⁿ (here: df=F(t,f(t))) defined on an maximal open set for a given starting value f(x₀)=y₀.

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.