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.
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
as a solution of the differential equation. Strictly speaking though, it does not satisfy the differential equation at , 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 with initial condition 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
with defined on the rectangular domain . If the function 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 is said to satisfy the Carathéodory conditions on if it fulfills the condition of the theorem.[5]
Uniqueness of a solution
Assume that the mapping satisfies the Carathéodory conditions on and there is a Lebesgue-integrable function , such that
for all Then, there exists a unique solution to the initial value problem
Moreover, if the mapping is defined on the whole space and if for any initial condition , there exists a compact rectangular domain such that the mapping satisfies all conditions from above on . Then, the domain of definition of the function is open and is continuous on .[6]
Example
Consider a linear initial value problem of the form
Here, the components of the matrix-valued mapping and of the inhomogeneity 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]
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In mathematics Caratheodory 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 Caratheodory s theorem shows existence of solutions in a more general sense for some discontinuous equations The theorem is named after Constantin Caratheodory Contents 1 Introduction 2 Statement of the theorem 3 Uniqueness of a solution 4 Example 5 See also 6 Notes 7 ReferencesIntroduction EditConsider the differential equation y t f t y t displaystyle y t f t y t with initial condition y t 0 y 0 displaystyle 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 displaystyle 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 y 0 0 displaystyle y t H t quad y 0 0 where H denotes the Heaviside function defined by H t 0 if t 0 1 if t gt 0 displaystyle H t begin cases 0 amp text if t leq 0 1 amp text if t gt 0 end cases It makes sense to consider the ramp function y t 0 t H s d s 0 if t 0 t if t gt 0 displaystyle y t int 0 t H s mathrm d s begin cases 0 amp text if t leq 0 t amp text if t gt 0 end cases 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 Statement of the theorem EditConsider the differential equation y t f t y t y t 0 y 0 displaystyle y t f t y t quad y t 0 y 0 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 f t y displaystyle f t y is continuous in y displaystyle y for each fixed t displaystyle t f t y displaystyle f t y is measurable in t displaystyle t for each fixed y displaystyle y there is a Lebesgue integrable function m t 0 a t 0 a 0 displaystyle m t 0 a t 0 a to 0 infty such that f t y m t displaystyle f t y leq m t for all t y R displaystyle 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 R R n displaystyle f colon R to mathbf R n is said to satisfy the Caratheodory conditions on R displaystyle R if it fulfills the condition of the theorem 5 Uniqueness of a solution EditAssume that the mapping f displaystyle f satisfies the Caratheodory 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 f t y 1 f t y 2 k t y 1 y 2 displaystyle f t y 1 f t y 2 leq k t y 1 y 2 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 y t f t y t y t 0 y 0 displaystyle y t f t y t quad y t 0 y 0 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 Example EditConsider a linear initial value problem of the form y t A t y t b t y t 0 y 0 displaystyle y t A t y t b t quad y t 0 y 0 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 Caratheodory conditions and there exists a unique solution to the initial value problem 7 See also Edit Mathematics portalPicard Lindelof theorem Cauchy Kowalevski theoremNotes Edit Coddington amp Levinson 1955 Theorem 1 2 of Chapter 1 Coddington amp Levinson 1955 page 42 Rudin 1987 Theorem 7 18 Coddington amp Levinson 1955 Theorem 1 1 of Chapter 2 Hale 1980 p 28 Hale 1980 Theorem 5 3 of Chapter 1 Hale 1980 p 30References EditCoddington 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 Retrieved from https en wikipedia org w index php title Caratheodory 27s existence theorem amp oldid 993657825, wikipedia, wiki, book, books, library,