3 Update

## Background

At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is usually functional analytic and imposes strong conditions on the map $f:X \to X$ whereas the second class is usually algebraic topological and imposes strong conditions on the space $X$ itself.

A typical example of the first class of theorems is the fixed point theorem of Banach. While the spaces it applies to are fairly general (complete metric spaces), the function must have a Lipschitz constant strictly less than $1$. On the other hand, Brouwer's theorem falls into the second class. Any continuous map works, but the domain must be a compact and convex subset of Euclidean space (originally a disk). Of course, both these theorems have been vastly generalized from the versions that I am stating here.

## Question

One fundamental advantage of the Banach theorem is that it actually provides a recipe for converging to the fixed point as part of the standard proof: just start at an initial point and iterate. The proofs of the Brouwer theorem that I have seen do no such thing. The best known proof (I think) is the one by contradiction: assuming the domain is a disk, if $f(x)$ and $x$ are always distinct then the ray from $f(x)$ through $x$ to the boundary of said disk provides a deformation-retraction from the disk to its boundary, aha!

Here is my question:

Is there any way to actually find a fixed point when using Brouwer's theorem?

## A Possible Idea

One scheme that unfortunately fails is as follows. Consider the sequence of iterates $f^n(x)$ for $n \in \mathbb{N}$ and any initial $x$ in the domain. We have an infinite sequence in a compact set, and hence a convergent subsequence, so the limit point is a candidate. This won't work since a) we haven't used convexity at all, and b) one may just be converging to a periodic orbit of $f$.

Sorry if this is too half-baked or elementary, but I have reduced an annoying problem to finding (any!!) fixed point of a map on the unit disk in $\mathbb{R}^n$. But this infernal map is absolutely hideous and in no way satisfies the hypotheses for the Banach fixed point theorem, so I have to use Brouwer's theorem. There is also no Earthly hope of discretizing the domain and approximating this monstrosity by a simplicial map. If the question sounds desperate, that's because it is... All help is greatly appreciated.

## Update

Thanks to all the answerers and commenters for various helpful and constructive suggestions. If either of the articles referenced by Aaron or Willie turn out to contain directly useful information, I will write a brief summary of the relevant content here.

2 added tag, spelled deformation properly and fixed latex

## Background

At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is usually functional analytic and imposes strong conditions on the map $f:X \to X$ whereas the second class is usually algebraic topological and imposes strong conditions on the space X $X$ itself.

A typical example of the first class of theorems is the fixed point theorem of Banach. While the spaces it applies to are fairly general (complete metric spaces), the function must have a Lipschitz constant strictly less than $1$. On the other hand, Brouwer's theorem falls into the second class. Any continuous map works, but the domain must be a compact and convex subset of Euclidean space (originally a disk). Of course, both these theorems have been vastly generalized from the versions that I am stating here.

## Question

One fundamental advantage of the Banach theorem is that it actually provides a recipe for converging to the fixed point as part of the standard proof: just start at an initial point and iterate. The proofs of the Brouwer theorem that I have seen do no such thing. The best known proof (I think) is the one by contradiction: assuming the domain is a disk, if $f(x)$ and $x$ are always distinct then the ray from $f(x)$ through $x$ to the boundary of said disk provides a defermation-retraction deformation-retraction from the disk to its boundary, aha!

Here is my question:

Is there any way to actually find a fixed point when using Brouwer's theorem?

## A Possible Idea

One scheme that unfortunately fails is as follows. Consider the sequence of iterates $f^n(x)$ for $n \in \mathbb{N}$ and any initial $x$ in the domain. We have an infinite sequence in a compact set, and hence a convergent subsequence, so the limit point is a candidate. This won't work since a) we haven't used convexity at all, and b) one may just be converging to a periodic orbit of $f$.

Sorry if this is too half-baked or elementary, but I have reduced an annoying problem to finding (any!!) fixed point of a map on the unit disk in $\mathbb{R}^n$. But this infernal map is absolutely hideous and in no way satisfies the hypotheses for the Banach fixed point theorem, so I have to use Brouwer's theorem. There is also no Earthly hope of discretizing the domain and approximating this monstrosity by a simplicial map. If the question sounds desperate, that's because it is... All help is greatly appreciated.

1

# Can we actually find any fixed points with Brouwer's theorem?

## Background

At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is usually functional analytic and imposes strong conditions on the map $f:X \to X$ whereas the second class is usually algebraic topological and imposes strong conditions on the space X itself.

A typical example of the first class of theorems is the fixed point theorem of Banach. While the spaces it applies to are fairly general (complete metric spaces), the function must have a Lipschitz constant strictly less than $1$. On the other hand, Brouwer's theorem falls into the second class. Any continuous map works, but the domain must be a compact and convex subset of Euclidean space (originally a disk). Of course, both these theorems have been vastly generalized from the versions that I am stating here.

## Question

One fundamental advantage of the Banach theorem is that it actually provides a recipe for converging to the fixed point as part of the standard proof: just start at an initial point and iterate. The proofs of the Brouwer theorem that I have seen do no such thing. The best known proof (I think) is the one by contradiction: assuming the domain is a disk, if $f(x)$ and $x$ are always distinct then the ray from $f(x)$ through $x$ to the boundary of said disk provides a defermation-retraction from the disk to its boundary, aha!

Here is my question:

Is there any way to actually find a fixed point when using Brouwer's theorem?

## A Possible Idea

One scheme that unfortunately fails is as follows. Consider the sequence of iterates $f^n(x)$ for $n \in \mathbb{N}$ and any initial $x$ in the domain. We have an infinite sequence in a compact set, and hence a convergent subsequence, so the limit point is a candidate. This won't work since a) we haven't used convexity at all, and b) one may just be converging to a periodic orbit of $f$.

Sorry if this is too half-baked or elementary, but I have reduced an annoying problem to finding (any!!) fixed point of a map on the unit disk in $\mathbb{R}^n$. But this infernal map is absolutely hideous and in no way satisfies the hypotheses for the Banach fixed point theorem, so I have to use Brouwer's theorem. There is also no Earthly hope of discretizing the domain and approximating this monstrosity by a simplicial map. If the question sounds desperate, that's because it is... All help is greatly appreciated.