I have the following differential equation : $\dot{x}(t,\epsilon) = a(t,\epsilon) x(t,\epsilon)$ where $a(t,\epsilon)$ is assumed to be continuously differentiable both in $\epsilon$ and $t$, $t$ is time, $\epsilon$ is a parameter in a compact interval.
My goal is to show that the limit $l(\epsilon) = \lim_{t\to \infty} x(t,\epsilon)$ is continuous and even continuously differentiable with regards to $\epsilon$.
From my problem, I have the following properties on $x$ :
- the solution $x(t,\epsilon)$ exists and is continuously differentiable both in $\epsilon$ and $t$.
- $x(t,\epsilon)$ is non-increasing in $t$, lower bounded by a constant $C$ which is uniform in $\epsilon$, thus it converges.
$x(t,\epsilon)$ converges exponentially fast uniformly in $\epsilon$ : there is $K$ and $\lambda$ such that $$ |x(t,\epsilon) - l(\epsilon)| \le K e^{-\lambda t} $$ where both $K$ and $\lambda$ are independent of $\epsilon$.
My hint is that if I can prove uniform convergence of $x(t,\epsilon)$ when $\epsilon \to \epsilon_0$ for any $t\in [0,\infty)$, I will be able to use theorems of calculus to interchange $\lim_{t\to \infty}$ and $\lim_{\epsilon \to \epsilon_0}$. Same if I can prove uniform convergence of $\frac{\partial x}{\partial \epsilon}(t,\epsilon)$.
To prove uniform convergence of $x(t,\epsilon)$ I was hoping to use that $x(t,\epsilon)$ is monotonous in $t$ and point-wise converging, and then apply Dini's Theorem, but this theorem already requires continuity of $l(\epsilon)$ : this circular argument leads nowhere.
Any help would be more than welcome :)