Carathéodory's existence theorem

In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem. Peano's theorem requires that the right-hand side of the differential equation be continuous, while Carathéodory's theorem shows existence of solutions (in a more general sense) for some discontinuous equations. The theorem is named after Constantin Carathéodory. Consider the differential equation with initial condition where the function ƒ is defined on a rectangular domain of the form Peano's existence theorem states that if ƒ is continuous, then the differential equation has at least one solution in a neighbourhood of the initial condition. However, it is also possible to consider differential equations with a discontinuous right-hand side, like the equation where H denotes the Heaviside function defined by It makes sense to consider the ramp function as a solution of the differential equation. Strictly speaking though, it does not satisfy the differential equation at , because the function is not differentiable there. This suggests that the idea of a solution be extended to allow for solutions that are not everywhere differentiable, thus motivating the following definition. A function y is called a solution in the extended sense of the differential equation with initial condition if y is absolutely continuous, y satisfies the differential equation almost everywhere and y satisfies the initial condition. The absolute continuity of y implies that its derivative exists almost everywhere. Consider the differential equation with defined on the rectangular domain . If the function satisfies the following three conditions: is continuous in for each fixed , is measurable in for each fixed , there is a Lebesgue-integrable function such that for all , then the differential equation has a solution in the extended sense in a neighborhood of the initial condition. A mapping is said to satisfy the Carathéodory conditions on if it fulfills the condition of the theorem.