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Kakutani fixed-point theorem - Wikipedia

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In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions. The theorem was developed by Shizuo Kakutani in 1941,[1] and was used by John Nash in his description of Nash equilibria.[2] It has subsequently found widespread application in game theory and economics.[3] Kakutani's theorem states:[4] The function: 𝜑 ( 𝑥 ) = [ 1 − 𝑥 / 2 , 1 − 𝑥 / 4 ] , shown on the figure at the right, satisfies all Kakutani's cond

Kakutani fixed-point theorem - Wikipedia Jump to content From Wikipedia, the free encyclopedia Fixed-point theorem for set-valued functions In mathematical analysis , the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions . It provides sufficient conditions for a set-valued function defined on a convex , compact subset of a Euclidean space to have a fixed point , i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem . The Brouwer fixed point theorem is a fundamental result in topo

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