Existence and uniqueness of the solution of an ordinary differential equation with parameters on Banach spaces 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 $(x, s) ∈ U_0\times V_0$
 A: There are counterexamples in the low regularity case where $q = 0$. 
For equations where there exists a unique solution for all initial data, we can still have lack of continuous dependence on initial data. They are originally constructed by Pasika for Banach spaces with countable Schauder basis, and later by Garay and Schäffer, and separately De Blasi and Pianigiani for general Banach spaces. But I haven't checked completely the regularity properties of their constructions to make sure those are what you want. 
Picard's existence and uniqueness theorem, however, holds in a very large class of locally convex topological spaces. With smoothness, things improve. Theorem 25 in the article of Lobanov and Smolyanov (see citation below) is similar to what you have in the question. If I am comparing notations correctly, in their case $f: J\times U\times V$ is assumed to be $C^r$, $r\geq 1$ in $J\times U$, and $C^0$ in $V$. They show that for every $(t_0,x_0) \in J\times U$ the existence of a unique solution map $\psi: J_0\times J_0 \times U_0\times V$ where $t_0 \in J_0 \subseteq J$ and $x_0 \in U_0\subseteq U$ are open neighborhoods such that $\psi$ is $C^r$ on $J_0\times J_0\times U_0$. 
References:


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*E.E. Pasika, An example of a first-order differential equation in a Hilbert space without continuous dependence of the solution on the initial condition, Ukrain. Mat. Zh. 35 (1983), 786-788. MR 85c:34070.

*F.S. De Blasi and G. Pianigiani, Uniqueness for differential equations implies continuous dependence only in finite dimension, Bull. London Math. Soc. 18 (1986), 379-382. MR 87f:34072.

*B.M. Garay and J.J. Schaffer, More on uniqueness without continuous dependence in infinite dimension, J. Differential Equations 64 (1986), 48-50. MR 88e:34110.


See also


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*K. Deimling, Ordinary differential equations in Banach spaces, Lecture Notes in Math. 596 (1977). MR 57 # 3546

*S.G. Lobanov and O.G. Smolyanov, Ordinary differential equations in locally convex spaces, Uspekhi Mat. Nauk, 1994, 49, 93-168

