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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.

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What books have you looked in? Have you checked the standard references? (P.Hartman's ODE book, or Coddington and Levinson) If so, what's unsatisfactory about the presentation therein? –  Willie Wong Aug 23 '10 at 10:11
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I do not know any recent book on ODE's satisfying yours criterium. The classical books below do a fairly good job though. The first being more comprehensive and the second more elementary(no measure theory needed).

  1. Coddington, Earl A.; Levinson, Norman. Theory of ordinary differential equations.
  2. Petrovski, I. G. Ordinary differential equations.
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Thanks jvp and Willie Wong. I shall dig around in Coddington and Levinson. As jvp mentions I was hoping for something that also contained the advances that had occurred in the last 30 years. Nevertheless it's nice to have confirmation of what to look at. –  Ben Whale Aug 23 '10 at 23:30
    
An update for those looking for an inexpensive reference. Hale's book "Ordinary Differential Equations" provides the same level of rigour as Coddington and Levinson (at least for this specific question). Importantly, it's also published by Dover. You can pick up a copy from Amazon for less than $15... –  Ben Whale Nov 20 '10 at 6:15
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