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.
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 ) ∈ R × R n : | t − t 0 | ≤ a , | y − y 0 | ≤ 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 ) , y ( 0 ) = 0 , where H denotes the Heaviside function defined by
H ( t ) = { 0 , if t ≤ 0 ; 1 , if t > 0. It makes sense to consider the ramp function
y ( t ) = ∫ 0 t H ( s ) d s = { 0 , if t ≤ 0 ; t , if t > 0 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.
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 | ≤ a , | y − y 0 | ≤ 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 ) , | t − t 0 | ≤ a , such that | f ( t , y ) | ≤ m ( t ) for all ( t , y ) ∈ R ,then the differential equation has a solution in the extended sense in a neighborhood of the initial condition.
A mapping f : R → R n is said to satisfy the Carathéodory conditions on R if it fulfills the condition of the theorem.
Assume that the mapping f satisfies the Carathéodory conditions on R and there is a Lebesgue-integrable function k ( t ) , | t − t 0 | ≤ a , such that
| f ( t , y 1 ) − f ( t , y 2 ) | ≤ k ( t ) | y 1 − y 2 | , for all ( t , y 1 ) ∈ R , ( t , y 2 ) ∈ 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 ) ) , y ( t 0 ) = y 0 . Moreover, if the mapping f is defined on the whole space R × R n and if for any initial condition ( t 0 , y 0 ) ∈ R × R n , there exists a compact rectangular domain R ( t 0 , y 0 ) ⊂ R × R n such that the mapping f satisfies all conditions from above on R ( t 0 , y 0 ) . Then, the domain E ⊂ 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 .
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 : R → R n × n and of the inhomogeneity b : R → 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.
