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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 $(x, s) ∈ U_0\times V_0$

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1 Answer 1

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:

  • 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

  • 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
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Thanks. Unfortunately I don't have an easy access to those papers. Under the additional condition that $f$ is continuous, do you think there are still counterexamples? I thought I proved the theorem except the analytic case under the additional condition $f$ is continuous. The proof is similar to that of finite dimensional case. Maybe I should open another thread asking if my proof is correct. –  Makoto Kato Oct 10 '12 at 20:02
    
The constructed examples always have $f$ continuous. What I am not sure about is whether the counterexamples work for $f$ smooth. See, e.g. the reviews at MathSciNet (I assume you at least have access to that). –  Willie Wong Oct 11 '12 at 7:03
    
Um, it looks like the counterexamples cannot hold when the RHS is smooth. See edit. –  Willie Wong Oct 11 '12 at 7:28
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