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.

Introduction
Consider the differential equation
 * $$ y'(t) = f(t,y(t)) $$

with initial condition
 * $$ y(t_0) = y_0, $$

where the function ƒ is defined on a rectangular domain of the form
 * $$ R = \{ (t,y) \in \mathbf{R}\times\mathbf{R}^n \,:\, |t-t_0| \le a, |y-y_0| \le b \}. $$

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
 * $$ y'(t) = H(t), \quad y(0) = 0, $$

where H denotes the Heaviside function defined by
 * $$ H(t) = \begin{cases} 0, & \text{if } t \le 0; \\ 1, & \text{if } t > 0. \end{cases} $$

It makes sense to consider the ramp function
 * $$ y(t) = \int_0^t H(s) \,\mathrm{d}s = \begin{cases} 0, & \text{if } t \le 0; \\ t, & \text{if } t > 0 \end{cases} $$

as a solution of the differential equation. Strictly speaking though, it does not satisfy the differential equation at $$t=0$$, 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 $$y' = f(t,y)$$ with initial condition $$y(t_0)=y_0$$ 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.

Statement of the theorem
Consider the differential equation
 * $$ y'(t) = f(t,y(t)), \quad y(t_0) = y_0, $$

with $$f$$ defined on the rectangular domain $$ R=\{(t,y) \, | \, |t - t_0 | \leq a, |y - y_0| \leq b\} $$. If the function $$f$$ satisfies the following three conditions: then the differential equation has a solution in the extended sense in a neighborhood of the initial condition.
 * $$f(t,y)$$ is continuous in $$y$$ for each fixed $$t$$,
 * $$f(t,y)$$ is measurable in $$t$$ for each fixed $$y$$,
 * there is a Lebesgue-integrable function $$m : [t_0 - a, t_0 + a] \to [0, \infty)$$ such that $$|f(t,y)| \leq m(t)$$ for all $$(t, y) \in R $$,

A mapping $$f \colon R \to \mathbf{R}^n$$ is said to satisfy the Carathéodory conditions on $$R$$ if it fulfills the condition of the theorem.

Uniqueness of a solution
Assume that the mapping $$f$$ satisfies the Carathéodory conditions on $$R$$ and there is a Lebesgue-integrable function $$k : [t_0 - a, t_0 + a] \to [0, \infty)$$, such that
 * $$|f(t,y_1) - f(t,y_2)| \leq k(t) |y_1 - y_2|,$$

for all $$(t,y_1) \in R, (t,y_2) \in R.$$ Then, there exists a unique solution $$y(t) = y(t,t_0,y_0)$$ to the initial value problem
 * $$ y'(t) = f(t,y(t)), \quad y(t_0) = y_0.$$

Moreover, if the mapping $$f$$ is defined on the whole space $$\mathbf{R} \times \mathbf{R}^n$$ and if for any initial condition $$(t_0,y_0) \in \mathbf{R} \times \mathbf{R}^n$$, there exists a compact rectangular domain $$R_{(t_0,y_0)} \subset \mathbf{R} \times \mathbf{R}^n$$ such that the mapping $$f$$ satisfies all conditions from above on $$R_{(t_0,y_0)}$$. Then, the domain $$E \subset \mathbf{R}^{2+n}$$ of definition of the function $$y(t,t_0,y_0)$$ is open and $$y(t,t_0,y_0)$$ is continuous on $$E$$.

Example
Consider a linear initial value problem of the form
 * $$ y'(t) = A(t)y(t) + b(t), \quad y(t_0) = y_0.$$

Here, the components of the matrix-valued mapping $$A \colon \mathbf{R} \to \mathbf{R}^{n \times n}$$ and of the inhomogeneity $$b \colon \mathbf{R} \to \mathbf{R}^{n}$$ are assumed to be integrable on every finite interval. Then, the right hand side of the differential equation satisfies the Carathéodory conditions and there exists a unique solution to the initial value problem.