Carathéodory's existence theorem explained

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 conditiony(t _{0)} = y _{0,} where the function ƒ is defined on a rectangular domain of the formR = \{ (t,y) \in R x 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.^[1]

However, it is also possible to consider differential equations with a discontinuous right-hand side, like the equation

y'(t) = H(t), y( 0 ) = 0 , where H denotes the Heaviside function defined byH(t) = \begin{cases} 0 , & if t \le 0 ; \ 1 , & if t > 0 . \end{cases} It makes sense to consider the ramp functiony(t) = \int ^{t} _{0} H(s) d s = \begin{cases} 0 , & if t \le 0 ; \ t, & 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

y'(t) = f(t,y(t)), 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 , inf ty) 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

f \colon R \to 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 k : [t _{0} - a, t _{0} + a] \to [ 0 , inf ty) , 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 problemy'(t) = f(t,y(t)), y(t _{0)} = y _{0.}

Moreover, if the mapping

f is defined on the whole space R x R ^{n} and if for any initial condition (t _{0,y} _{0)} \in R x R ^{n} , there exists a compact rectangular domain R _{(t _{0,y} _{0)}} \subset R x R ^{n} such that the mapping f satisfies all conditions from above on R _{(t _{0,y} _{0)}} . Then, the domain E \subset 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 .^[6]

Example

Consider a linear initial value problem of the form

y'(t) = A(t)y(t) + b(t), y(t _{0)} = y _{0.} Here, the components of the matrix-valued mapping A \colon R \to R ^{n} and of the inhomogeneity b \colon R \to 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]

See also

• Picard–Lindelöf theorem
• Cauchy–Kowalevski theorem

Notes

1. , Theorem 1.2 of Chapter 1
2. , page 42
3. , Theorem 7.18
4. , Theorem 1.1 of Chapter 2
5. , p.28
6. , Theorem 5.3 of Chapter 1
7. , p.30

References

• .
• .
• .