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.
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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".
Shizuo Kakutani (1911-2004), Japanese mathematician known for the Kakutani fixed-point theorem.