2016-1.md

course	course_year	question_number	tags	title	year
Analysis II	IB	1	IB 2016 Analysis II	Paper 4, Section I, G	2016

(a) What does it mean to say that a mapping $f: X \rightarrow X$ from a metric space to itself is a contraction?

(b) State carefully the contraction mapping theorem.

(c) Let $\left(a_{1}, a_{2}, a_{3}\right) \in \mathbb{R}^{3}$. By considering the metric space $\left(\mathbb{R}^{3}, d\right)$ with

$$d(x, y)=\sum_{i=1}^{3}\left|x_{i}-y_{i}\right|$$

or otherwise, show that there exists a unique solution $\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}$ of the system of equations

$$\begin{aligned} &x_{1}=a_{1}+\frac{1}{6}\left(\sin x_{2}+\sin x_{3}\right), \\ &x_{2}=a_{2}+\frac{1}{6}\left(\sin x_{1}+\sin x_{3}\right), \\ &x_{3}=a_{3}+\frac{1}{6}\left(\sin x_{1}+\sin x_{2}\right) . \end{aligned}$$