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Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2020, Director: Francesc Xavier Massaneda Clares

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[en] In Mathematics, there exist several theorems (or family of theorems) that do not seem as powerful as they actually are. These are the type of theorems which, at first glance, we do not perceive the big potencial they have. Maybe because of the simplicity of the theorem or maybe because we could believe the theorem shows an obvious fact. The fixedpoint theorems (or the theory of the fixed-point theorems) could be an example of this
phenomenon.
In this line, the aim of this document is to demonstrate that the theory of fixed-points theorems is an indispensable tool in Mathematics and its applications in Science. In order to do so, three relevant fixed-point theorems and their respective applications are presented: the Banach’s Fixed Point Theorem and its application to the Cauchy-Lipschitz’s Theorem, the Brouwer’s Fixed Point Theorems and its application to the existence of
Nash’s Equilibrium and the Schauder’s Fixed Point Theorem and its application to the Lomonosov’s Theorem.
Apart from showing the mathematics hidden behind the theorems and their applications, the main objective is to exhibit the power of this theory in such different fields as Analysis or Topology, and in different domains such as Economy or Game Theory.

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cc-by-nc-nd (c) Daniel García Álvaro, 2020

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Treballs Finals de Grau (TFG) - Matemàtiques