I would like to show that the null solution $ψ(t) ≡ 0$ of the equation $\frac{dx}{dt} = − \frac{x}{t}$, $(x,t)∈R×[a,+∞)$, $(a >0)$, is uniformly stable, asymptotically stable, but not uniformly asymptotically stable.
I've achieved to show the first two (uniformly stable and asymptotically stable), but I'm stuck on the last one. Indeed The definiton of uniformly asymptotically stability that I have is the following:
A solution $ψ(t)$ of $\frac{dx}{dt} = f(t,x)$ is uniformly asymptotically stable $\iff \exists ζ > 0$, $\forall η > 0$, $\exists T(η)$ such that $\forall t_0$, $||ξ - \psi(t_0)|| < ζ$, $t > t_0 + T \implies ||φ(t;t0,ξ) - \psi(t)|| < \eta$.
I've tried showing this property isn't true directly, however I always ended with an inequality where the right-hand side depended on $T(\eta)$, and since it should be true $\forall T(\eta)$, I couldn't conclude. Thus I tried showing it wasn't true by contradiction, i.e showing that:
$\forall ζ > 0$, $\exists η > 0$, $\forall T(η)$ such that $\exists t_0$, $||ξ - \psi(t_0)|| \geq ζ$, $t \leq t_0 + T$ and $||φ(t;t0,ξ) - \psi(t)|| \geq \eta$.
But again I didn't achieve the desired result.
Thus if anyone has an idea how to show this property, I'd love some help on that!
Thanks in advance