Contraction Mapping: Case Studies

TL;DR

This paper explains the principle of contraction mapping and Banach fixed point theorem through detailed case studies in ordinary differential equations, aiming to clarify their proofs for students with calculus background.

Contribution

It provides a clear, step-by-step narration of two major contraction mapping proofs in ODEs, enhancing understanding for learners.

Findings

Clarified the proofs of contraction mapping in ODEs

Enhanced understanding of fixed point theorems

Educational resource for students in analysis

Abstract

While exploring dynamical systems, we often come across the principle of contraction mapping, or better known as the Banach fixed point theorem. It is an essential concept based on successive approximation, whose utility comes from two main guarantees: establishing existence and uniqueness of a solution, and establishing constructive proof. The intent of this manuscript is to break down two major proofs incorporating this in ordinary differential equations (ODEs), and make them a little more understandable step-by-step to an audience that presumably has adequate knowledge of modern calculus and real analysis. These are not original proofs, only original narration.