Application of Generalized Fixed Point Theorems to Ordinary Differential Equations

Abstract

Ordinary differential equations (ODEs) serve as a fundamental tool for modeling

various natural phenomena in science and engineering. The application of fixed point

theorems has proved to be a powerful technique in the study of the existence,

uniqueness, and stability of solutions to ODEs. However, traditional fixed point

theorems are limited to specific settings and may not be directly applicable to more

complex ODEs arising in practical scenarios.

In this research the Banach Contraction Fixed Point Theorems will be used, as well as

Schauder's Theorem and Picard Theorem. Said Theorems will be executed to ordinary

differential equations (ODE). The objective is to emphasize the utility of the different

fixed point theorems, when applying them to proofing the existence theorems and the

uniqueness theorems of the solutions of ODE in initial conditions or boundary

conditions.