Pontos fixos em espaços métricos completos e o Teorema de Picard

Abstract

This work aims to deepen the study of complete metric spaces, focusing especially on the analysis of fixed points. Our intention is to demonstrate Banach’s Fixed Point Theorem and, subsequently, apply this theory to ordinary differential equations through Picard’s Theorem. To achieve this objective, we will begin by addressing the fundamental concepts of metric spaces, with an emphasis on understanding the basic elements , offering examples and introducing topological concepts, as well as the notion of continuity. We will conduct the study until we reach the definition of complete metric spaces, and then analyze the notion of fixed point. Finally, we will demonstrate the main theorem, which establishes the existence and uniqueness of solutions to initial value problems in ordinary differential equations.