Schauder fixed point theorem

Definition

#incomplete

Special case

Let $S$ be a bounded subset of $\mathbb{R}^n$, and let $C(S)$ be the space of real-valued bounded continuous functions on $S$, endowing with the $\sup$ norm. Let $F \subset C(S)$ be nonempty, closed, bounded, and convex. Then if the mapping $T: F \to F$ is continuous and the family $T(F)$ is equicontinuous, $T$ has a fixed point in $F$.

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.3