Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $(x, s) ∈ U_0×V_0$

share|cite|improve this question
up vote 4 down vote accepted

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.

share|cite|improve this answer
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. – Makoto Kato Jun 6 '12 at 10:49

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.