Lemma 2.1.2.label Let $E, F$ be Banach spaces, $U \subset E$, $V \subset F$ be open sets, $\eps > 0$, $g \in C^{k}((-\eps, \eps) \times U \times V; F)$ ($k \ge 2$), and $(x_{0}, y_{0}) \in U \times V$, then there exists $\delta > 0$, $U_{0} \in \cn_{E}(x_{0})$, $V_{0} \in \cn_{F}(x_{0})$ and a unique $C^{k}$-mapping
such that
- (1)
For all $(x, y) \in U_{0} \times V_{0}$, $\beta(0, x, y) = y$.
- (2)
For all $(t, x, y) \in (-\delta, \delta) \times U_{0} \times V_{0}$,
\[\frac{d}{dt}\beta(t, x, y) = g(t, x, \beta(t, x, y))\] - (3)
For each fixed $y \in V_{0}$, let $\beta(t, x) = \beta(t, x, y)$, then for each $(t, x) \in (-\delta, \delta) \times U_{0}$,
\[(\partial_{t}\partial_{x}\beta)(t, x) = (\partial_{x}g)(t, x, \beta(t, x)) + (\partial_{y}g)(t, x, \beta(t, x)) \circ (\partial_{x}\beta)(t, x)\]
Proof, [Proposition VI.2.1, Lan12].Let
then by the existence and uniqueness of ODEs, there exists $\delta > 0$, $U_{0} \in \cn_{E}(x_{0})$, $V_{0} \in \cn_{F}(y_{0})$, and a unique $C^{k}$-mapping
such that
- (a)
For each $(x, y) \in U_{0} \times V_{0}$, $B(0, x, y) = (x, y)$.
- (b)
For each $(t, x, y) \in (-\delta, \delta) \times U_{0} \times V_{0}$,
\[\frac{d}{dt}B(t, x, y) = (0, g(t, B(t, x, y)))\]
Let $\beta(t, x, y) = \pi_{2}(t, x, y)$, then
- (1)
For each $(x, y) \in U_{0} \times V_{0}$, $\beta(0, x, y) = \pi_{2}(x, y) = y$.
- (2)
For each $(t, x, y) \in (-\delta, \delta) \times U_{0} \times V_{0}$,
\[\frac{d}{dt}\beta(t, x, y) = \pi_{2}(g(t, B(t, x, y))) = g(t, x, \beta(t, x, y))\] - (3)
For each fixed $y \in V_{0}$, let $\beta(t, x) = \beta(t, x, y)$, then for each $(t, x) \in (-\delta, \delta) \times U_{0}$, by the chain rule,
\[(\partial_{t}\partial_{x}\beta)(t, x) = (\partial_{x}g)(t, x, \beta(t, x)) + (\partial_{y}g)(t, x, \beta(t, x)) \circ (\partial_{x}\beta)(t, x)\]