PICARD THEOREM: MATHEMATICAL FOUNDATIONS, APPLICATION IN RL CIRCUITS AND POSSIBILITIES FOR HIGH SCHOOL

Abstract

This work aims to study Picard’s Theorem based on concepts from metric spaces, Banach spaces, and the Banach Fixed Point Theorem, and to present an application to a series RL circuit. Initially, the concepts required to understand these results are presented. Then, Picard’s Theorem is proved by relating the solution of an initial value problem to the fixed point of an operator. In the application, the physical principles of the RL circuit are introduced, and the mathematical model that describes the electric current in the circuit is developed. After verifying the continuity, Lipschitz, and boundedness conditions, Picard’s Theorem guarantees the existence and uniqueness of a solution at a given interval. In a particular case, Picard iterations are used to obtain approximations for the electric current. Finally, possibilities for exploring this application in High School through calculations, tables, and graphs are presented, connecting Mathematics and Physics.