Kakutani fixed-point theorem

Theorem

Let $S$ be a (nonempty), compact and convex subset of $\mathbb{R}^n$, and let $f$ be an Upper hemicontinuous function (u.h.c.) which assigns to each $x \in S$ a closed and convex subset of $S$. Then there exists some $x \in S$ such that $x \in f(x)$.

Notes

There is an alternative definition with upper semicontinuous functions (slight difference from u.h.c.)

See also:

• Brouwer fixed-point theorem
• Weierstrass extreme value theorem

References

1. https://en.wikipedia.org/wiki/Kakutani_fixed-point_theorem
2. T. Başar and G.J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, Classics in Applied Mathematics, SIAM, Philadelphia, 1999.
   • Appendix C, theorem C.2
3. Kakutani, 1941