Table of Contents
1. Attractor
- Multiple application of the same function
will likely ends up on the fixed-point of . is on the fixed-points.
1.1. Babylonian Method
- Method of finding the square roots.
admits a fixed point .- Start with a guess
, then .
2. Brouwer Fixed Point Theorem
- The theorem is for finite dimensional spaces.
2.1. Statement
- For a convex compact subset of
, any continuous function admits a fixed-point.
3. Lipschitz Contraction
- A function
such that - Lipschitz continuous function with
.- Intuitively, the function is under the line with the slope
at every point.
- Intuitively, the function is under the line with the slope
4. Banach Fixed Point Theorem
4.1. Statement
- For a complete metric space
, the Lipschitz contraction has a unique fixed-point . - This can be any finite or infinite dimensional space.
5. Theorem (DeBlasi, Myjak)
- The set of non-expansive functions defined on a bounded set that admits fixed-points are co-porous with respect to the ((66478ab5-ed8d-4525-a7e1-a41671c644f9)).
- That is, almost all such functions admits fixed-point.
- There are almost no Lipschitz contraction, but most of them are Rakotch contractions which are the functions that are below an arbitrary curve.
6. Theorem (Bargetz, Reich, Thimm)
- There are almost no Rakotch contraction on an unbounded set. But almost all of them admits fixed-point, since most of them are locally Rakotch contraction.