Carathéodory's existence theorem

For other uses, see Carathéodory's theorem (disambiguation).

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

$R=\{(t,y)\in \mathbf {R} \times \mathbf {R} ^{n}\,:\,|t-t_{0}|\leq a,|y-y_{0}|\leq b\}.$

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

$H(t)={\begin{cases}0,&{\text{if }}t\leq 0;\\1,&{\text{if }}t>0.\end{cases}}$

It makes sense to consider the ramp function

$y(t)=\int _{0}^{t}H(s)\,\mathrm {d} s={\begin{cases}0,&{\text{if }}t\leq 0;\\t,&{\text{if }}t>0\end{cases}}$

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:

• $f(t,y)$ 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 $(t,y)\in R$,

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

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.

See also

• Picard–Lindelöf theorem
• Cauchy–Kowalevski theorem

References

• Coddington, Earl A.; Levinson, Norman (1955), Theory of Ordinary Differential Equations, New York: McGraw-Hill.
• Hale, Jack K. (1980), Ordinary Differential Equations (2nd ed.), Malabar: Robert E. Krieger Publishing Company, ISBN 0-89874-011-8.
• Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-054234-1, MR 0924157.