I asked this question. I wonder if it can be generalized to an infinite dimensional case. Namely, is the following theorem true? If yes, how do we prove it?
Theorem Let $E$ and $F$ be Banach spaces over $\mathbb{R}$. Let $U$ and $V$ be non-empty open subsets of $E$ and $F$ respectively. Let $J$ be a non-empty open interval of $\mathbb R$. Let $f\colon J\times U\times V → E$ be a map. Suppose $f$ is differentiable of class $C^p, 0 ≦ p ≦ \omega$ in $J$ and of class $C^q, 1 ≦ q ≦ \omega$ in $U$ and $V$. Let $(t_0, x_0, s_0) \in J\times U \times V$. Then there exist open subinterval $J_0$ of $J$, open subsets $U_0, V_0$ of $U, V$ respectively such that $(t_0, x_0, s_0) \in J_0\times U_0\times V_0$ and a unique map $g:J_0\times U_0 \times V_0 \rightarrow U$ which satisfy the following properties.
(1) $g$ is differentiable of class $C^{p+1}$ in $J$ and of class $C^q$ in $U_0$ and $V_0$.
(2) $D_t g(t, x, s) = f(t, g(t, x, s), s)$ for all $(t, x, s) \in J_0\times U_0\times V_0$
(3) $g(t_0, x, s) = x$ for all $(x, s) ∈ U_0\times V_0$