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 $f(x) = x$.

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

Any continuous function $G : \mathbb{B}^n \to \mathbb{B}^n$ has a fixed point, $$\mathbb{B} = \{\mathbf{x} \in \mathbb{R}^n : x_1^2 + ... + x_n^2 \leq 1\}$$ 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

References:

1. T. Başar and G.J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, Classics in Applied Mathematics, SIAM, Philadelphia, 1999.
   • 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