Contraction Mapping: Case Studies

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.