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The following theorem is stated in the appendix I of Foundation of differential geometry by Kobayashi-Nomizu. They say the proof will be found in various text books on differential equations. I checked several books, but could not find it.

Theorem Let $E$ and $F$ be finite dimensional normed spaces over $\mathbb R$. Let $U$ and $V$ be non-empty open subsets of $E$ and $F$ respectively. Let $J$ be an open interval of $\mathbb R$ containing $0$. Let $f:J×U×V → E$ be a map. Suppose $f$ is differentiable of class $C^p, 0 ≦ p ≦ ω$ in $J$ and of class $C^q, 1 ≦ q ≦ ω$ in $U$ and $V$.

Then there exist open subinterval $J_0$ of $J$ containing $0$, non-empty open connected subsets $U_0, V_0$ of $U, V$ respectively and a unique map $g:J_0×U_0×V_0 → U$ which satisfy the following properties.

(1) $g$ is differentiable of class $C^{p+1}$ in $J_0$ and of class $C^q$ in $U_0$ and $V_0$.

(2) $D_tg(t, x, s) = f(t, g(t, x, s)$, s) for all $(t, x, s) ∈ J_0×U_0×V_0$

(3) $g(0, x, s) = x$ for all $(x, s) ∈ U_0×V_0$

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This is a combination of the existence and uniqueness theorem and the smooth dependence on parameters. See e.g. Solomon Lefschetz, "Differential Equations: Geometric Theory", II.9.1.

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  • $\begingroup$ Thanks. I'll check the book. It may take a few days before I accept your answer. I'd like to see if there are any other books which prove the theorem. $\endgroup$ Jun 6, 2012 at 10:49

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