Theorem Kakutani Fixed Point Theorem
theorem staged

Kakutani Fixed Point Theorem

Let $C$ be a nonempty convex compact subset of a normed vector space. Let $F:C\rightrightarrows C$ be a correspondence such that:

  1. for every $c\in C$, $F(c)$ is nonempty, compact, and convex;
  2. the graph $\{(c,d)\in C\times C:d\in F(c)\}$ is closed.

Then the fixed point set $$ \{c\in C:c\in F(c)\} $$ is nonempty and compact.

References

  • [MFoGT, Thm. 4.11.5] Laraki, Renault, and Sorin, Mathematical Foundations of Game Theory. Kakutani fixed point theorem for closed-graph convex compact valued correspondences.

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