Brouwer fixed-point theorem

Definition (compact and convex subset in real space)

If $S$ is a (nonempty) compact and convex subset of $\mathbb{R}^n$ and $f$ is a continuous function mapping $S$ into itself, then there exists at least one $x \in S$ such that $f(x) = x$.

Such a point $x$ is called a fixed point of $f$.

Alternative formulation ($n$-ball in real space)

Any continuous function $G : \mathbb{B}^n \to \mathbb{B}^n$ has a fixed point, where

is the unit $n$-ball.

Notes

• an extension exists for topological vector spaces, Schauder fixed point theorem
• Compare Banach fixed-point theorem, which involves contraction mappings
• also see Kakutani fixed-point theorem

References:

1. T. Başar and G.J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, Classics in Applied Mathematics, SIAM, Philadelphia, 1999, pp. 166, 483.
   • Appendix C, theorem C.1
2. https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem
3. https://www.homepages.ucl.ac.uk/~ucahjde/tg/html/pi1-08.html
4. https://mathworld.wolfram.com/BrouwerFixedPointTheorem.html
5. Brouwer, 1910
6. Kuga, 1974
7. https://bpb-us-e1.wpmucdn.com/wp.nyu.edu/dist/5/2123/files/2019/12/Lecture-3-Scribe.pdf
8. https://math.uchicago.edu/~may/REU2017/REUPapers/Katz.pdf
9. A. Hatcher, Algebraic Topology, 2001, pp. 31-32. http://pi.math.cornell.edu/~hatcher/AT/AT.pdf
   • https://math.stackexchange.com/questions/4540516/proof-of-brouwers-fixed-point-theorem-in-hatcher
10. https://ncatlab.org/nlab/show/Brouwer's+fixed+point+theorem