Abstract Nonsense

Crushing one theorem at a time

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)<d(\varphi(x_0),x_0) 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))<d(x_0,x_1)

The conclusion follows. \blacksquare

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June 9, 2010 - Posted by | Fun Problems, Topology | , , ,

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 […]

    Pingback by Banach Fixed Point Theorem « Abstract Nonsense | June 13, 2011 | Reply


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