O teorema do ponto fixo de Banach para existência e unicidade de equações diferenciais ordinárias de primeira ordem

Abstract

This work is a collection of results and examples in metric spaces, organized in eight chapters, whose ultimate goal is the proof of the Banach Fixed Point Theorem for the existence and uniqueness of solutions to initial value problems of differential equations. In the development of the work, basic concepts and important examples are addressed that allow the reader to reflect on the main definitions and results established in metric spaces and the Banach Fixed Point Theorem. The notions of Lipschitzian applications, continuity and their relations with open and closed sets, limits and sequences, Cauchy sequences and complete metric spaces are relevant in the development of the study of the themes because they are central points in the discipline of Final Course Work 2. This work is a study of the work Metric Spaces (Lima, 1977), with influence from some other works, such as A Course in Analysis - Volume 1 (Lima, 1976), A Course in Analysis - Volume 2 (Lima, 1981), Metric Spaces and Introduction to Topology (Domingues, 1982) and Lessons in Ordinary Differential Equations (Sotomayor, 1979). In addition, the chosen theme is intended to complement the training of the author, an undergraduate student in Mathematics.