ODE's Hale's Book pg. 27 Lemma 4.1 Exercises 4.1 I am studying with Hale's book on ODE's.
There it does not have the demonstration of the following motto:
LEMMA 4.1 If $f$ is either independent of $t$ or periodie in $t$, then the solution $x=0$ of $\dot{x}(t)=f(t,x(t))$ being stable (asymptotically stable) implies the solution $x=0$ of $\dot{x}(t)=f(t,x(t))$ is uniformly stable (uniformly asymptotically stable).
could you help me with the proof ? I couldn't develop anything. Any script or hints would be helpful
 A: Denote $g_{t_0}^t(x_0)$ the solution of the equation at the moment $t$ which satisfies $g_{t_0}^{t_0}(x_0) = x_0$.
Case 1: $f$ does not depend on $t$ $\Leftrightarrow$ $\forall \tau \in \mathbb{R}$ $g_{t_0 + \tau}^{t}(x_0) = g_{t_0}^{t-\tau}(x_0)$ (time-translation invariance).
Let $x^* = 0$ be stable at $t_0$. Then it is uniformly stable, because solutions are invariant under time-translations.
Case 2: $f$ is $T$-periodic function of $t$ $\Leftrightarrow$ $\forall n \in \mathbb{Z}$ $g_{t_0 + nT}^{t}(x_0) = g_{t_0}^{t-nT}(x_0)$ (the solutions are invariant under $T$-translations).
Suppose $x^* = 0$ is stable at $t_0$. Hence $\forall \varepsilon > 0$ $\exists \delta_\varepsilon > 0$ s.t. $||x_0|| \leq \delta_\varepsilon$ $\Rightarrow$ $\forall t \in \mathbb{R}_+$ $||g_{t_0}^{t_0 + t}(x_0)|| \leq \varepsilon$. But then $\forall \tau \in [t_0, t_0 + T]$ $\forall \varepsilon > 0$ $\exists \delta_\varepsilon(\tau) = \min_{||x|| = \delta_\varepsilon}||g_{t_0}^{\tau}(x)||  > 0$ (by the uniqueness theorem) s.t. $||x_0|| \leq \delta_\varepsilon(\tau)$ $\Rightarrow$ $\forall t \in \mathbb{R}_+$ $||g_{\tau}^{\tau + t}(x_0)|| \leq \varepsilon$. But $[t_0, t_0 + T]$ is compact, and $\delta_\varepsilon(\tau)$ is a continuous function of $\tau$ (follows from continuous dependence of the solutions on initial conditions and $t$). So its minimum $\overline\delta_\varepsilon$ exists and is non-zero by the uniqueness theorem. Thus $\forall \varepsilon > 0$ $\forall \tau \in \mathbb R$ $\exists \overline\delta_\varepsilon > 0$ s.t. $||x_0|| \leq \overline\delta_\varepsilon$ $\Rightarrow$ $\forall t \in \mathbb{R}_+$ $||g_{\tau}^{\tau + t}(x_0)|| \leq \varepsilon$ because the solutions are invariant under $T$-translations.
The case of asymptotic stability is similar.
