| course |
course_year |
question_number |
tags |
title |
year |
Analysis II |
IB |
5 |
|
Paper 3, Section II, E |
2019 |
(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 $f(1)=y_{0}$.