Teorema de puntos fijos con aplicaciones

Abstract

The main aim of this work is to develop several remarkable Fixed Point Theorems and apply them to different fields in Mathematical Analysis. In this context, we first consider complete metric spaces and we study the classical Banach Contraction Principle as well as several of its extensions. We apply some of these results to differential equations, fractal theory, Banach algebras and finally to convex metric spaces. In the next chapter, we switch to the Banach space framework and more generally to locally convex vectors spaces. We prove Schauder-Tychovoff Theorem, where the main features to obtain a fixed point are now the compactness and convexity of the domain together with the continuity of the mapping. This will lead us to the study of invariant measures for a continuous function defined on a compact Hausdorff space and to approach one of the most famous open problems inside Operator Theory: the Invariant Subspace Problem. In the third chapter, we study Markov-Kakutani Theorem to achieve a common fixed point for a commutative family of affine continuous functions. As applications, we prove the existence of a Haar measure on a compact topological group as well as the existence of invariant means on every commutative semigroup.