Kakutani's Fixed Point Theorem

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Theorem

Let $S \subset \R^n$ be nonempty, compact, and convex.

Let $\Phi : S \to 2^S$ be a correspondence.

Let the following conditions be satisfied:

$(1): \quad \map \Phi x$ is nonempty and convex for all $x$

$(2): \quad \map \Phi \cdot$ is upper hemi-continuous

Then $\Phi$ has a fixed point.

Proof

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Source of Name

This entry was named for Shizuo Kakutani.

Sources

• 1994: Martin J. Osborne and Ariel Rubinstein: A Course in Game Theory: $\text I$ Strategic Games: Chapter $2$ Nash Equilibrium: $2.4$ Existence of a Nash Equilibrium: Lemma $20.1$ (Kakutani's fixed point theorem)