I looking at a time variant first order system. I am trying to prove that a sequence of functions $x^{[k]}(t)$ generated using Picard iterations converges uniformly on some interval $[0,T]$.
Given:
$\dot{x}(t)=A(t)x(t)$ where $x(0)=x_0$
$x(t)=x∈R^n$ where $A(t)$ is a $n×n$ matrix
Assume there exists $M$ such that $||A(t)||\le M$ for all $t\ge 0$ and that $A(t)$ is a piecewise continuous function. Show that, for any $T > 0$, a solution exists in the interval $[0,T]$.
I know that $x(t)=x_0+\int^t_0A(τ)x(τ) dτ$ must be satisfied if the solution does indeed exist.
I am supposed to use Picard iterations to prove this, which I am not familiar with yet. But I know that they are of the form: $x^{[k+1]}=x_0+\int^t_0A(τ)x^{[k]}(τ) dτ$. Where $x^{[0]}(t)=x_0$
Can I use the standard Picard proof to prove this or do I need to start somewhere else?
I was also asked to show that this method satisfies
$|x^{[k+1]}(t)-x^{[k]}(t)|\le |x_0| $$\frac{π^{k+1}(t)}{(k+1)!}$ where $π(t)=\int^t_0||A(τ)||dτ$.
I have no idea where to start with this, but hopefully it will become clearer once I finish the first part.
What you are to use is a generalized version of the Banach fixed point theorem. If $F$ is a fixed-point operator on some complete space $X$ and each iterated composition $F^m$ has a Lipschitz constant $L_m$ on the whole of $X$ and $\sum_m L_m<\infty$, then the fixed point iteration converges to the unique fixed point.
What you are to show is that for $X=C([0,T])$ and $F$ the Picard iteration, you get $L_m=\frac{\pi(T)^m}{m!}$ so that $\sum_{m=1}^\infty L_m=e^{\pi(T)}-1<\infty$. This is proved via induction. The induction claim is that $$ \|F^n(x)-F^n(y)\|_{C([0,t]}\le \frac{\pi(t)^m}{m!}\|x-y\|_{C([0,t]} $$ where $$ \|u\|_{C([0,t]}=\max_{s\in [0,t]}|u(s)| $$ is the usual supremum norm on the space of continuous functions restricted to the indicated domain $[0,t])$.
In the induction proof for the step from $n=k-1$ to $n=k$ you compute, using $\pi'(t)=\|A(t)\|$, \begin{align} \|F^k(y)(t)-F^k(x)(t)\| &=\left\|\int_0^tA(s)[F^{k-1}(y)(s)-F^{k-1}(x)(s)]\,ds\right\| \\ &\le\int_0^t\|A(s)\|\,\left\|F^{k-1}(y)(s)-F^{k-1}(x)(s)\right\|\,ds \\ &\le\int_0^t\pi'(s)\frac{\pi(s)^{k-1}}{(k-1)!}\|y-x\|_{C([0,s])}\,ds \\ &\le \frac{\pi(t)^k}{k!}\|y-x\|_{C([0,t])}\le \frac{\pi(T)^k}{k!}\|y-x\|_{C([0,T])} \end{align}