Abstract
Many problems in physics, engineering, biology, economics, etc., can be modeled as relations between observables or states and their derivatives, hence as differential equations. When only derivatives with respect to one variable play a role, the differential equation is called an ordinary differential equation. The field of differential equations has a long history, starting with Newton and Leibniz in the seventeenth century. In the beginning of the study of differential equations, the focus is on finding explicit solutions as the emphasis is on solving the underlying physical problems. But soon one starts to wonder: If a starting point for a solution of a differential equation is given, does the solution always exist? And if such a solution exists, how long does it exist and is there only one such solution? These are the questions of existence and uniqueness of solutions of initial value problems. The first existence result is given in the middle of the nineteenth century by Cauchy. At the end of the nineteenth century and the beginning of the twentieth century, substantial progress is made on the existence and uniqueness of solutions of initial value problems and currently the heart of the topic is quite well understood. But there are many open questions as soon as one considers delay equations, functional differential equations, partial differential equations or stochastic differential equations. Another area of intensive current research, which uses the existence and uniqueness of differential equations, is the area of finite- and infinite-dimensional dynamical systems.