I have this problem from Perko(Page 85, Q-3) which says that :
Consider the initial value problem: $$\dot{x}=f(t,x,\mu)$$ $$x(0)=x_0$$
Given that $E$ is open subset of $\mathbb{R^{n+m+1}}$ containing the point $(0,x_0,\mu_0)$ where $x_0 \in \mathbb{R^n}$ and $\mu_{0} \in \mathbb{R^m}$ and that $f$ and $\frac{\partial f}{\partial x}$ are continuous on $E$, use Gronwall's Lemma and the method of successive approximations to show that there is an $a \gt 0$ and $\delta \gt 0$ such that the ivp has a unique solution $u(t,\mu)$ continuous on $[-a,a] \times N_{\delta}(\mu_{0})$
So I define: $$\mu_{0}(t,\mu)=\mu_{0}$$ $$\mu_{k+1}(t,\mu)=\mu_{0}+\int_{0}^{t} f(\mu_{k}(s,\mu)) ds$$
I guess that the interval $[-a,a]$ comes from the part of $t$ owing to existence and uniqueness theorem. I am not sure about the $N_{\delta}(\mu_{0})$ part.
Thanks for the help!!