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.
Figure 6: A pictoral representation of the Intermediate Value Theorem. When dealing with one dimension, any closed and convex subset of R is homeomorphic to [0;1]. We can then show that any one-dimensional case for the Brouwer Fixed Point Theorem is equivalent to the case in [0;1], and thus, the Theorem applies there. 6
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