Teodor Alfson: Topics in Nonlinear Ordinary Differential Equations

Abstract: In this paper we describe different methods used in showing existence of solutions to boundary value problems for nonlinear ordinary differential equations. We describe the shooting method and give an example how it can be applied to the pendulum equation. We look at examples of how the shooting method can be used together with a priori bounds and Poincaré maps. We look at contraction maps and we prove Banach's contraction mapping principle and give an example of its use. Finally, we give a brief overview of the Mountain Pass Theorem and return to the pendulum equation.