Q. Let $F : [a,b] \times \mathbb{R}^n \to \mathbb{R}^n$ be a continuous mapping, with $[a,b]$ a compact interval, such that $F$ is Lipschitz continuous for its second variable, meaning for all $t \in [a,b]$, $x,y \in \mathbb{R}^n$, and $L > 0$ one has $$ \|F(t,x) - F(t,y)\| \leq L\|x-y\|.$$
Then show for each $y_0 \in \mathbb{R}^n$ the inital value problem, defined as $$y'(x) = F(x, y(x)), ~~y(a) = y_0$$ has only one solution $y : [a,b] \to \mathbb{R}^n$.
To this end we can use the Banach space $\mathcal{C}([a,b], \mathbb{R}^n)$ with the norm defined as $$\|f\|_w = \sup\left\{e^{-2Lx}|f(x)| : x \in [a,b]\right\},$$ to reformulate the intital value problem above into fixed points of a linear operator on $\mathcal{C}([a,b], \mathbb{R}^n)$, and then can apply Banach's fixed point theorem.
So far I have proved that $\mathcal{C}([a,b], \mathbb{R}^n)$ with norm $\|f\|_w$ is in fact a Banach space, and we can use Picard's existence theorem to reformulate the initial value problem to something similar to the integral $$y(t) = y_0 + \int_0^t f(x, y(x))dx, ~~~0 \leq t \leq a,$$ and then can show that $F$ is a contraction, for which we can then use Banach's fixed point theorem.
But I'm not sure that exact reformulation is correct with respect to my Banach space?
Let us equip $X := C([0,a],\mathbb{R}^n)$ with the norm $\|\cdot\|_w$, so that $(X, \|\cdot\|_w)$ is a Banach space.
You have to prove that the Picard operator $\mathcal{P}\colon X\to X$ defined by $$ \mathcal{P}[y] (t) := y_0 + \int_0^t F(s, y(s))\, ds, \qquad t\in [0,a] $$ is a contraction.
At this point, its unique fixed point is the unique solution of the Cauchy problem $$ y'(t) = F(t, y(t)), \qquad y(0) = y_0. $$