This is used in a proof of the Brouwer fixed point theorem.

Furthermore, there is a simple proof which uses Tarski's fixed point theorem.

The vanishing point theorem is the principle theorem in the science of perspective.

By the Banach fixed point theorem, these have fixed points and respectively.

A common thread in many proofs is the Brouwer fixed point theorem.

Among his further results, the Brouwer fixed point theorem is also well known.

This version is known as the Schauder-Tychonoff fixed point theorem.

A proof using the Brouwer fixed point theorem by .

This version is essentially a special case of the Banach fixed point theorem.

Therefore we may apply the Brouwer fixed point theorem to .