Английская Википедия:Carathéodory's existence theorem

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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

<math> y'(t) = f(t,y(t)) </math>

with initial condition

<math> y(t_0) = y_0, </math>

where the function ƒ is defined on a rectangular domain of the form

<math> R = \{ (t,y) \in \mathbf{R}\times\mathbf{R}^n \,:\, |t-t_0| \le a, |y-y_0| \le b \}. </math>

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

<math> y'(t) = H(t), \quad y(0) = 0, </math>

where H denotes the Heaviside function defined by

<math> H(t) = \begin{cases} 0, & \text{if } t \le 0; \\ 1, & \text{if } t > 0. \end{cases} </math>

It makes sense to consider the ramp function

<math> 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} </math>

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

<math> y'(t) = f(t,y(t)), \quad y(t_0) = y_0, </math>

with <math>f</math> defined on the rectangular domain <math> R=\{(t,y) \, | \, |t - t_0 | \leq a, |y - y_0| \leq b\} </math>. If the function <math>f</math> satisfies the following three conditions:

• <math>f(t,y)</math> is continuous in <math>y</math> for each fixed <math>t</math>,
• <math>f(t,y)</math> is measurable in <math>t</math> for each fixed <math>y</math>,
• there is a Lebesgue-integrable function <math>m : [t_0 - a, t_0 + a] \to [0, \infty)</math> such that <math>|f(t,y)| \leq m(t)</math> for all <math>(t, y) \in R </math>,

then the differential equation has a solution in the extended sense in a neighborhood of the initial condition.

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

Uniqueness of a solution

Assume that the mapping <math>f</math> satisfies the Carathéodory conditions on <math>R</math> and there is a Lebesgue-integrable function <math>k : [t_0 - a, t_0 + a] \to [0, \infty)</math>, such that

<math>|f(t,y_1) - f(t,y_2)| \leq k(t) |y_1 - y_2|,</math>

for all <math>(t,y_1) \in R, (t,y_2) \in R.</math> Then, there exists a unique solution <math>y(t) = y(t,t_0,y_0)</math> to the initial value problem

<math> y'(t) = f(t,y(t)), \quad y(t_0) = y_0.</math>

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

Example

Consider a linear initial value problem of the form

<math> y'(t) = A(t)y(t) + b(t), \quad y(t_0) = y_0.</math>

Here, the components of the matrix-valued mapping <math>A \colon \mathbf{R} \to \mathbf{R}^{n \times n}</math> and of the inhomogeneity <math>b \colon \mathbf{R} \to \mathbf{R}^{n}</math> 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.

See also

• Picard–Lindelöf theorem
• Cauchy–Kowalevski theorem

References

• Шаблон:Citation.
• Шаблон:Citation.
• Шаблон:Citation.