Consider the differential equation $\dot x = f(x)$. The standard proofs are 

1. The Picard iteration based proof of existence/uniqueness for Lipschitz $f$. 

2. The Peano existence theorem for continuous $f$.

3. 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?