Consider $$x'(t)=A\cdot x(t)+f(t,x(t)),\,\,x(0)=a$$ for $t\geq0$. Assume that $A$ is real $N\times N$ matrix with negative real parts of eigenvalues, $f\colon \mathbb{R}\times \mathbb{R}^N\to\mathbb{R}^N$ is continuous and locally Lipschitz with respect to the second argument. Moreover, assume that $f(t,0)=0$ for all $t\geq 0$ and $$\frac{\|f(t,x)\|}{\|x\|}\to 0$$ uniformly as $x\to 0$ for all $t\geq 0$. Then, $x=0$ is asymptotically stable. How shall I understand the last hypothesis? Is it $$\forall \varepsilon>0\, \exists\,\delta>0\,\forall\, x\in\mathbb{R}^N\, \forall t\geq0\, \|x\|<\delta\implies \frac{\|f(t,x)\|}{\|x\|}<\varepsilon$$ or $$\forall \varepsilon>0\, \exists\,\delta>0\,\forall\, t\geq0\,\forall x(t)\in\mathbb{R}^N\, \|x(t)\|<\delta\implies \frac{\|f(t,x(t))\|}{\|x(t)\|}<\varepsilon\,\,?$$
Also, does it follow from the hypothesis that if $\|x(0)\|$ is sufficiently small, then for all $t\geq 0$ we have that $\frac{\|f(t,x(t))\|}{\|x(t)\|}<\varepsilon$? -$\delta$ is chosen independently from time.
Edit. Suppose for now, that $t\in [0,T]$. Let $x(\cdot)$ be a solution of our problem. We know that $$x(t)=ae^{At}+\int_{0}^{t}e^{(t-s)A}f(s,x(s))\,ds$$ and we can estimate the last term to get $$\|x(t)\|\le \|a\| c_1e^{-c_2t}+\int_{0}^{t}c_1e^{-c_2(t-s)}\|f(s,x(s))\|\,ds.$$ In order to use Gronwall's lemma we need to estimate $\|f(s,x(s))\|$. It follows from $$\forall \varepsilon>0\, \exists\,\delta>0\,\forall\, x\in\mathbb{R}^N\, \forall t\geq0\, \|x\|<\delta\implies \frac{\|f(t,x)\|}{\|x\|}<\varepsilon$$ that $$\forall \varepsilon>0\, \exists\,\delta>0\,\forall\, x\in\mathbb{R}^N\, \forall t\geq0\, \|x\|<\delta\implies \|f(t,x)\|<\varepsilon \|x\|.$$ In order to use the last estimate in our situation we need to assume however that for all $t\in [0,T]$, $\|x(t)\|< \delta$ which is an absurd. So in my opinion we are not allowed to use Gronwall lemma in here, at least directly. Another problem is, that Gronwall's lemma is valid if $t\in [0,T]$ (at least this version I can prove), so what happens if $t\in [0,\infty)$? I believe we can use Gronwall's lemma in the way I wanted, but first we have to introduce a Lyapunov funcion $L(x)=Q(x,x)$, where $$Q(x,y)=\int_{0}^{\infty}\langle e^{\xi A}x, e^{\xi A}y\rangle_{\mathbb{R}^N}\,d\xi$$ and secondly prove that if $x(0)\in M=\{x\in \mathbb{R}^N\,|\, L(x)\le r\}$, then $x(t)\in M$ for all $t\ge 0$. Then the assumption of $\|x(t)\|<\delta$ will be justified for $t\ge 0$ and applying Gronwall's lemma will get asymptotic stability.