(a) Carefully state the Picard-Lindelöf theorem on solutions to ordinary differential equations.

(b) Let $X=C\left([1, b], \mathbb{R}^{n}\right)$ be the set of continuous functions from a closed interval $[1, b]$ to $\mathbb{R}^{n}$, and let $\|\cdot\|$ be a norm on $\mathbb{R}^{n}$.

(i) Let $f \in X$. Show that for any $c \in[0, \infty)$ the norm

$$\|f\|_{c}=\sup _{t \in[1, b]}\left\|f(t) t^{-c}\right\|$$

is Lipschitz equivalent to the usual sup norm on $X$.

(ii) Assume that $F:[1, b] \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is continuous and Lipschitz in the second variable, i.e. there exists $M>0$ such that

$$\|F(t, x)-F(t, y)\| \leqslant M\|x-y\|$$

for all $t \in[1, b]$ and all $x, y \in \mathbb{R}^{n}$. Define $\varphi: X \rightarrow X$ by

$$\varphi(f)(t)=\int_{1}^{t} F(l, f(l)) d l$$

for $t \in[1, b]$.

Show that there is a choice of $c$ such that $\varphi$ is a contraction on $\left(X,\|\cdot\|_{c}\right)$. Deduce that for any $y_{0} \in \mathbb{R}^{n}$, the differential equation

$$D f(t)=F(t, f(t))$$

has a unique solution on $[1, b]$ with $f(1)=y_{0}$.