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
\( {\displaystyle y'(t)=f(t,y(t))} \)
with initial condition
\( {\displaystyle y(t_{0})=y_{0},} \)
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.[1]
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.[2] The absolute continuity of y implies that its derivative exists almost everywhere.[3]
Statement of the theorem
Consider the differential equation
\( {\displaystyle y'(t)=f(t,y(t)),\quad y(t_{0})=y_{0},} \)
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 \( {\displaystyle m:[t_{0}-a,t_{0}+a]\to [0,\infty )} \) 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.[4]
A mapping \( {\displaystyle f\colon R\to \mathbf {R} ^{n}}\) is said to satisfy the Carathéodory conditions on R if it fulfills the condition of the theorem.[5]
Uniqueness of a solution
Assume that the mapping f satisfies the Carathéodory conditions on R and there is a Lebesgue-integrable function \( {\displaystyle k:[t_{0}-a,t_{0}+a]\to [0,\infty )} \), such that
\( {\displaystyle |f(t,y_{1})-f(t,y_{2})|\leq k(t)|y_{1}-y_{2}|,} \)
for all \( {\displaystyle (t,y_{1})\in R,(t,y_{2})\in R.} \) Then, there exists a unique solution \( {\displaystyle y(t)=y(t,t_{0},y_{0})} \) to the initial value problem
\( {\displaystyle y'(t)=f(t,y(t)),\quad y(t_{0})=y_{0}.} \)
Moreover, if the mapping f is defined on the whole space \( {\displaystyle \mathbf {R} \times \mathbf {R} ^{n}} \) and if for any initial condition\( {\displaystyle (t_{0},y_{0})\in \mathbf {R} \times \mathbf {R} ^{n}} \), there exists a compact rectangular domain \( {\displaystyle R_{(t_{0},y_{0})}\subset \mathbf {R} \times \mathbf {R} ^{n}} \) such that the mapping f satisfies all conditions from above on \( {\displaystyle R_{(t_{0},y_{0})}} \). Then, the domain \( {\displaystyle E\subset \mathbf {R} ^{2+n}} \) of definition of the function\( {\displaystyle y(t,t_{0},y_{0})} \) is open and \( {\displaystyle y(t,t_{0},y_{0})} \) is continuous on E.[6]
Example
Consider a linear initial value problem of the form
\( } {\displaystyle y'(t)=A(t)y(t)+b(t),\quad y(t_{0})=y_{0}.} \)
Here, the components of the matrix-valued mapping \( {\displaystyle A\colon \mathbf {R} \to \mathbf {R} ^{n\times n}} \) and of the inhomogeneity \( {\displaystyle b\colon \mathbf {R} \to \mathbf {R} ^{n}} \) 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.[7]
Notes
Coddington & Levinson (1955), Theorem 1.2 of Chapter 1
Coddington & Levinson (1955), page 42
Rudin (1987), Theorem 7.18
Coddington & Levinson (1955), Theorem 1.1 of Chapter 2
Hale (1980), p.28
Hale (1980), Theorem 5.3 of Chapter 1
Hale (1980), p.30
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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