The Brouwer fixed point theorem and some of its applications
Date of Publication
1985
Document Type
Bachelor's Thesis
Degree Name
Bachelor of Science in Applied Mathematics
College
College of Science
Department/Unit
Mathematics and Statistics
Abstract/Summary
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.
Abstract Format
html
Language
English
Format
Accession Number
TU05757
Shelf Location
Archives, The Learning Commons, 12F Henry Sy Sr. Hall
Physical Description
73 leaves
Keywords
Brouwerian algebras; Mathematics--Formulae
Recommended Citation
Macalalag, E. B. (1985). The Brouwer fixed point theorem and some of its applications. Retrieved from https://animorepository.dlsu.edu.ph/etd_bachelors/15991
