# Parameter dependent differential equation in a Lie group

It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference where this result is extended to the case of a differential equation in a finite-dimensional Lie group. In particular, I need the parameter space be merely a metrisable topological vector space.

(If this question and its answer are folklore and thus not appropriate for Mathoverflow, I apologize in advance)

In more detail, one finds the following result (slightly paraphrased) in Dieudonne's Treatise on Analysis Vol.1 (10.7.2):

Let $I \subseteq \mathbb R$ be an open interval and $P$ be a metric space (the parameter space). Consider the linear differential equation in the finite-dimensional vector space $E$ $$\dot{x}(t) = A(t, p) \cdot x(t) + b(t, p),$$
where $A: I \times P \to L(E, E)$ and $b: I \times P \to E$ are continuous. If $t \mapsto u(t,p )$ is a solution to the initial value problem $u(t_0, \cdot ) = x_0$, then $u$ is continuous in $I \times P$.

I'm interested in the case where $E$ is not a vector space but a finite-dimensional Lie group and the differential equation is adapted accordingly.

• I think you refer to Theorem B3, which however only handles the case of a scalar parameter $\epsilon \in \mathbb{R}$. Thanks nonetheless! Dec 7 '13 at 17:21
• Indeed, but this result extends straigthforwardly at least to normed vector spaces (not necessarily complete), see for example Theorem A6 in my book (or also PhD thesis). Note that there I prove a bit more abstractly that the flow depends smoothly on the vector field in a uniform context, but the same argument can be used to prove that it depends $C^k$ on parameters in a normed space. I don't know if this implies metrisable TVS's. Dec 8 '13 at 12:47