Transcription of Brouwer Fixed-Point Theorem
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Brouwer Fixed-Point Theorem
Brouwer Fixed-Point TheoremColin BuxtonMentor: Kathy PorterMay 17, 201611 Introduction to Fixed PointsFixed points have many applications. One of their prime applications is in the math-ematical field of game theory; here, they are involved in finding equilibria. The existenceand location of the fixed point(s) is important in determining the location of any are then applied to some economics, and used to justify the existence of economicequilibriums in the market, as well as equilibria in dynamical systemsDefinition Fixed Point:For a functionf:X X, a fixed pointc Xis a pointwheref(c) = a function has a fixed point,c, the point (c,c) is on its graph. The functionf(x) =xis composed entirely of fixed points , but it is largely unique in this respect.
point as a retraction that violates the above theorem. Because so much of the proof of the Brouwer Fixed-Point Theorem rests on the No-Retraction theorem, we also present its proof here for D ˆR2. [3] Proof: Let r: D !Cbe a retraction from the unit disk D to its boundary, C. Consider
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