Table of Contents

1. Attractor

  • Multiple application of the same function f will likely ends up on the fixed-point of f.
  • f is on the fixed-points.

1.1. Babylonian Method

  • Method of finding the square roots.
  • f(x)=x+a/x2 admits a fixed point a.
  • Start with a guess x0, then fn(x0)na.

2. Brouwer Fixed Point Theorem

  • The theorem is for finite dimensional spaces.

2.1. Statement

  • For a convex compact subset of X, any continuous function φ:XX admits a fixed-point.

3. Lipschitz Contraction

  • A function f:MM such that L<1,d(f(x),f(y))Ld(x,y)
  • Lipschitz continuous function with K<1.
    • Intuitively, the function is under the line with the slope K at every point.

4. Banach Fixed Point Theorem

4.1. Statement

  • For a complete metric space X, the Lipschitz contraction Φ:XX has a unique fixed-point x.
  • 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.

7. See Also

8. Reference

Created: 2025-05-06 Tue 23:34