I want to prove that, for any $t$, a solution exists in the interval $[0,T]$, when $T>0$.
$x'(t)=A(t)x(t)$
My question is quite similar to this one Picard iteration (general), but with one small difference. My $A(t)$ is a function of $t$, whereas in the other question $A$ seems to be constant.
Picard iteration: $x^{[k+1]}=x_0+\int^t_0A(τ)x^{[k]}(τ) dτ$. Where $x_0$ is arbitrary.
I want tor prove that $x^{[k+1]}$ converges on the given interval and that it satisfies the equation below.
$|x^{[k+1]}(t)-x^{[k]}(t)|\le |x_0| $$\frac{π^{k+1}(t)}{(k+1)!}$ where $π(t)=\int^t_0||A(τ)||dτ$.
I'm not sure how to account for the variant function $A(t)$ in my proof, so I was hoping someone could show me what it look like
