My specific situation is that I have a non-spacelike continuous future directed curve $\gamma:[0,a)\to M$ in a Lorentzian manifold. The curve must necessarily satisfy a local Lipschitz condition and therefore the derivative of the curve exists almost everywhere. In these circumstances, as I understand it, one can show (using Carathéodory's existence theorem?) that given $v\in T_{\gamma(0)\}M$ there exists an absolutely continuous vector field $V$ along $\gamma$ satisfying a local Lipschitz condition with $V(0) = v$ to the equation $\nabla_{\gamma'}V=0$.

I am specifically interested in the details of the proof of this and other ODE existence and uniqueness theorems in order to gain deeper insight into the relation of the differentiability conditions to conditions on the solutions (existence, uniqueness, differentiability, etc...).

Thus I am looking for a reference that treats such theorems in full generality with regards to the assumptions used. I'm not so concerned about the generality of the domain of the equations, $\mathbb{R}^n$ is ok for me. Though a reference that covered function spaces would also be interesting. I'm interested in a reference that provide full (or near to full) proofs for each result. Think "Real Analysis" by Haaser and Sullivan or "Introduction To Commutative Algebra" by Atiyah and MacDonald.