Teorema da existência e unicidade de soluções de equações diferenciais ordinárias no contexto de espaços métricos

Abstract

The present work seeks to present the Existence and Uniqueness Theorem of solutions for an initial value problem (IPV) within the context of metric spaces. For this, we start by reviewing some real analysis contents that help in understanding the concepts of metric spaces. We present a bit of topology in metric spaces, among them, the Cauchy sequence, which is extremely important for understanding the definition of complete metric spaces. We discuss the method of successive approximations due to Émille Picard, which, from a continuous path, allows us to approximate the solution to a PVI and, furthermore, to verify whether this solution found is unique from convergence of the iterated ones. Then, we present the Uniform Contraction Theorem, also known as Banach’s Fixed Point Theorem, which formalizes the successive approximations. Finally, we discuss the Existence and Uniqueness Theorem, its proof, as well as applications to some ordinary di