Carathéodory's existence theorem

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 ) ) {\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. 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 > 0. {\displaystyle 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 ) = ∫ 0 t H ( s ) d s = { 0 , if t ≤ 0 ; t , if t > 0 {\displaystyle 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 {\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. The absolute continuity of y implies that its derivative exists almost everywhere.

Statement of the theorem Consider 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. 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.

Uniqueness of a solution 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

| 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} .

Example Consider 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 Carathéodory conditions and there exists a unique solution to the initial value problem.

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