The book I am using for this course is Differential Equations and Dynamical Systems by Lawrence Perko.
The question is as follow:
Use the method of successive approximations to show that if $\vec{f}(\vec{x},t)$ is continuous in $t$ for all $t$ in some interval containing $t=0$ and continuously differentiable in $\vec{x}$ for all $\vec{x}$ in some open set $E \subset \mathbb{R}^n$ containing $\vec{x}_0$, then there exists an $a>0$ such that the initial value problem \begin{eqnarray} \vec{x}^{'} &=&\vec{f}(\vec{x},t)\\ \vec{x}(0) & = & \vec{x}_0 \end{eqnarray} has a unique solution $\vec{x}(t)$ on the interval $[-a,a]$.
Now this problem reminds me of the Fundamental Existence-Uniqueness Theorem; however, the Fundamental Existence-Uniqueness Theorem dealt with the autonomous system ($\vec{x}^{'} =\vec{f}(\vec{x})$, $\vec{x}(0) = \vec{x}_0$) as suppose to the nonautonomous case we have in this problem.
In the autonomous case, since $\vec{f}\in C^1(E)$, $\vec{f}$ was said to be locally Lipschitz. From there we used method of successive approximation, the fact $\vec{f}$ was locally Lipschitz, and the fact we had a complete normed linear space $C([-a,a])$ to prove existence and uniqueness.
I want to prove this case in a similar fashion. However, since $\vec{f}(\vec{x},t)$ depends on the independent variable $t$; I decided to rewrite our system into an autonomous system where $\vec{x}\in \mathbb{R}^{n+1}$ simply by letting $x_{n+1}=t$ and $x^{'}_{n+1}=1$.
The hypothesis states $\vec{f}$ is continuous in $t$ for all $t$ in some intervals containing $t=0$; therefore, I want to assume that our newly defined autonomous system (let call it $\vec{g}(\vec{x}$) is continuously differentiable on our open $D\subset \mathbb{R}^{n+1}$. Then from here I would continue the problem similar to the proof for the autonomous case.
If my idea of approaching the proof on the right track? If so, how would I properly go about continuing the proof? Or is there an alternative way to go about it that is more efficient?
Sorry for the rather long read, but I humbly Thank you for taking the time to read this post. Thank you in advance for any suggestions or feedback.
Take care and have a wonderful day.
your method is not quite correct to apply to time-varying systems. The time component should have a treatment different from the state. To that end, you should take care of the Lipschitz bound of the function $f$. For a reference of proof, you may check the Chapter 2 from Teschl, Ordinary DIfferential Equations and Dynamical Systems.
I hope this helps you.