## Background ##

At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful [book][1]: *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][2]. 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][3] 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.


  [1]: http://www.amazon.com/Fixed-Point-Theory-Andrzej-Granas/dp/0387001735
  [2]: http://en.wikipedia.org/wiki/Banach_fixed-point_theorem
  [3]: http://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem