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
with initial condition
where the function ƒ is defined on a rectangular domain of the form
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
where H denotes the Heaviside function defined by
It makes sense to consider the ramp function
as a solution of the differential equation. Strictly speaking though, it does not satisfy the differential equation at t = 0 {\displaystyle 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 ) {\displaystyle y'=f(t,y)} with initial condition y ( t 0 ) = y 0 {\displaystyle 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]
with f {\displaystyle f} defined on the rectangular domain R = { ( t , y ) | | t − − --> t 0 | ≤ ≤ --> a , | y − − --> y 0 | ≤ ≤ --> b } {\displaystyle R=\{(t,y)\,|\,|t-t_{0}|\leq a,|y-y_{0}|\leq b\}} . If the function f {\displaystyle f} satisfies the following three conditions:
then the differential equation has a solution in the extended sense in a neighborhood of the initial condition.[4]
A mapping f : : --> R → → --> R n {\displaystyle f\colon R\to \mathbf {R} ^{n}} is said to satisfy the Carathéodory conditions on R {\displaystyle R} if it fulfills the condition of the theorem.[5]
Assume that the mapping f {\displaystyle f} satisfies the Carathéodory conditions on R {\displaystyle R} and there is a Lebesgue-integrable function k : [ t 0 − − --> a , t 0 + a ] → → --> [ 0 , ∞ ∞ --> ) {\displaystyle k:[t_{0}-a,t_{0}+a]\to [0,\infty )} , such that
for all ( t , y 1 ) ∈ ∈ --> R , ( t , y 2 ) ∈ ∈ --> R . {\displaystyle (t,y_{1})\in R,(t,y_{2})\in R.} Then, there exists a unique solution y ( t ) = y ( t , t 0 , y 0 ) {\displaystyle y(t)=y(t,t_{0},y_{0})} to the initial value problem
Moreover, if the mapping f {\displaystyle f} is defined on the whole space R × × --> R n {\displaystyle \mathbf {R} \times \mathbf {R} ^{n}} and if for any initial condition ( t 0 , y 0 ) ∈ ∈ --> R × × --> R n {\displaystyle (t_{0},y_{0})\in \mathbf {R} \times \mathbf {R} ^{n}} , there exists a compact rectangular domain R ( t 0 , y 0 ) ⊂ ⊂ --> R × × --> R n {\displaystyle R_{(t_{0},y_{0})}\subset \mathbf {R} \times \mathbf {R} ^{n}} such that the mapping f {\displaystyle f} satisfies all conditions from above on R ( t 0 , y 0 ) {\displaystyle R_{(t_{0},y_{0})}} . Then, the domain E ⊂ ⊂ --> R 2 + n {\displaystyle E\subset \mathbf {R} ^{2+n}} of definition of the function y ( t , t 0 , y 0 ) {\displaystyle y(t,t_{0},y_{0})} is open and y ( t , t 0 , y 0 ) {\displaystyle y(t,t_{0},y_{0})} is continuous on E {\displaystyle E} .[6]
Consider a linear initial value problem of the form
Here, the components of the matrix-valued mapping A : : --> R → → --> R n × × --> n {\displaystyle A\colon \mathbf {R} \to \mathbf {R} ^{n\times n}} and of the inhomogeneity b : : --> R → → --> R n {\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]
Municipality in Northeast, BrazilSítio do Mato Município de Sítio do MatoMunicipalityVegetation in Sítio do Mato, BahiaLocation of Sítio do Mato in BahiaSítio do MatoLocation of Sítio do Mato in BrazilCoordinates: 13°05′06″S 43°27′54″W / 13.08500°S 43.46500°W / -13.08500; -43.46500Country BrazilRegionNortheastState BahiaGovernment • MayorAlfredo de Oliveira Magalhaes JuniorArea • Total1,627.776 km2 (628.488 sq …
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