Consider the differential equation $\dot x = f(x)$. The standard proofs are
The Picard iteration based proof of existence/uniqueness for Lipschitz $f$.
The Peano existence theorem for continuous $f$.
The Caratheodory existence theorem for measurable $f$.
My question is as follows. Assuming a Lipschitz $f(x)$, are there any other proofs out there for existence of solutions (in some reasonable sense) for ODEs?