Continuous map in $\mathbb{R}^2$ has a (scaled) fixed point

Question

Let $\phi:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be a continuous map.

How do I prove that there exist $a>0$ and $x\in\mathbb{R}^2$ such that $\phi(x)=ax$?

What I know:
I thought maybe this can be proved by contradiction. So suppose that there is no $x$ such that $\phi(x)=ax$. How would I continue?
Also this question makes me think of Brouwer's fixed point theorem, can I use the strategy of his proof?

Answer 1

Yes, you can reuse Brouwer's fixed point theorem. Let $\bar{\phi} : D^2 \to \mathbb{R}^2$ be the restriction of $\phi$ to $D^2 = \{ x \in \mathbb{R}^2 \mid \|x\| \le 1 \}$ (for whichever norm you want, usually the $L^2$ norm). Then since $D^2$ is compact, $\|\bar{\phi}\|$ reaches a maximum.

• Either $\max \|\bar{\phi}\| = 0$, in which case $\bar{\phi}(0) = 0 = 1 \cdot 0$;
• Or $a = \max \|\bar{\phi}\| > 0$, in which case the map $$x \mapsto \frac{\bar{\phi}(x)}{a}$$ is a map $D^2 \to D^2$ (because $\|\frac{\bar{\phi}(x)}{a}\| \le 1$). Hence by Brouwer's fixed point theorem, it has a fixed point $x \in D^2$, i.e. $\bar{\phi}(x)/a = x$. It follows that $\phi(x) = ax$.