# Abstract Nonsense

## Simple Fixed Point Problem

This is just a simple fixed point problem I did on mathhelpforum a while ago. It’s not particularly difficult.

Problem: Let $\mathcal{M}$ a compact metric space and $\varphi:\mathcal{M}\to\mathcal{M}$ a contractive mapping. Then, $\varphi$ has a unique fixed point.

Proof: Clearly the mapping $\text{dist}:\mathcal{M}\to\mathbb{R}:x\mapsto d(x,\varphi(x))$ is continuous since $\text{dist}=d\circ\left(\text{id}\oplus\varphi\right)$. Thus, by the compactness of $\mathcal{M}$ we have that $\text{dist}$ attains a minimum at some point $x_0\in\mathcal{M}$. Now, suppose that $\varphi(x_0)\ne x_0$, then $d\left(\varphi(\varphi(x_0)),\varphi(x_0)\right) which contradicts the minimality of $x_0$. It follows that $x_0$ is in fact a fixed point. Now, clearly it is unique for if we supposed that $\varphi(x_1)=x_1,\text{ }x_1\ne x_0$ then we’d have that

$d(x_0,x_1)=d(f(x_0),f(x_1))

The conclusion follows. $\blacksquare$

June 9, 2010 -

## 1 Comment »

1. […] point theorems (analytic feeling is pretty vague, but most often has to do with metric spaces e.g. this theorem) is the Banach fixed point theorem which first appeared in the Ph.D. thesis of Stefan […]

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