Teoria del grau, teorema de Schauder i aplicacions

Abstract

[en] The aim of this project is to introduce degree theory and its applications. We analyse the degree for functions defined over finite-dimensional and infinite-dimensional normed spaces, together with its main properties. One of these properties allows us to connect degree theory with fixed-point theory. As a result, we demonstrate some fundamental fixed-point theorems, like Brouwer’s theorem or its generalisation to infinite-dimensional spaces, Schauder’s theorem. The latter is applied in two instances. Firstly, we prove the existence of one-dimensional invariant manifolds for a certain class of planar maps. Secondly, we demonstrate the existence of periodic solutions for the forced pendulum equation.