The Brouwer fixed point theorem and some of its applications

• Main Author: Macalalag, Emmanuel B.
• Format: text
• Language: English
• Published: Animo Repository 1985
• Subjects: Brouwerian algebras; Mathematics > Formulae
• Online Access: https://animorepository.dlsu.edu.ph/etd_bachelors/15991
• Institution: De La Salle University

The Brouwer Fixed Point Theorem states that a continuous function from a unit ball in Rn into itself possesses at least one point whose image under the continuous function is itself. While the usual proofs of this theorem use higher mathematical concepts, this paper presents an analytic proof of the theorem using the concepts of winding numbers, vector fields, and the idea that the function (1 + t2)n/2 is not a polynomial when n is odd. The one, two, and n dimensional versions of the theorem are discussed. The theorem is also shown to hold for compact subsets of Rn. The applicability of the theorem to fields of interest such as linear algebra and stochastic processes has been investigated and has yielded positive results. It is shown that a linear transformation possesses a positive eigenvalue provided that the transformation matrix has all entries positive. The existence of a state which is equally uncertain as the previous state under a simple Markov process is investigated. In all these applications, each problem is formulated as a fixed point problem so as to allow a solution via the Brouwer Fixed Point Theorem. The computation of these fixed points however, is not within the scope of this paper. The applicability of the theorem is geared toward encouraging further research on this field as it seems to be interesting and promising.