There is also a quick proof, by Morris Hirsch (indeed by Elon Lages Lima), based on the impossibility of a differentiable retraction.
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The theorem was first studied in view of work on differential equations by the French mathematicians around Poincaré and Picard.
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In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Gérard Debreu and Kenneth Arrow.
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More than 20 years earlier Henri Poincaré had proved an equivalent result, and 5 years before Brouwer P.
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The French Encyclopædia Universalis defines it as the branch which "treats the properties of an object that are invariant if it is deformed in any continuous way, without tearing".
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Shizuo Kakutani (1911-2004), Japanese mathematician known for the Kakutani fixed-point theorem.
The sequence Fk used in this proof corresponds to the Kleene chain in the proof of the Kleene fixed-point theorem.
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This restriction is similar to the restriction to continuous operators in the Kleene fixed-point theorem of order theory.