Carathéodory's existence theorem

Carathéodory's existence theorem Differential equations Navier–Stokes differential equations used to simulate airflow around an obstruction Scope show Fields Classification show Types show Relation to processes Solution show Existence and uniqueness show General topics show Solution methods People show List vte 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.

Conteúdo 1 Introdução 2 Declaração do teorema 3 Uniqueness of a solution 4 Exemplo 5 Veja também 6 Notas 7 References Introduction Consider the differential equation {displaystyle y'(t)=f(t,y(t))} with initial condition {estilo de exibição y(t_{0})=y_{0},} where the function ƒ is defined on a rectangular domain of the form {estilo de exibição R={(t,y)em matemática {R} vezes 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] No entanto, it is also possible to consider differential equations with a discontinuous right-hand side, like the equation {displaystyle y'(t)=H(t),quad y(0)=0,} where H denotes the Heaviside function defined by {estilo de exibição H(t)={começar{casos}0,&{texto{E se }}tleq 0;\1,&{texto{E se }}t>0.end{casos}}} It makes sense to consider the ramp function {estilo de exibição y(t)=int_{0}^{t}H(s),matemática {d} s={começar{casos}0,&{texto{E se }}tleq 0;\t,&{texto{E se }}t>0end{casos}}} as a solution of the differential equation. Strictly speaking though, it does not satisfy the differential equation at {estilo de exibição 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 {displaystyle y'=f(t,y)} with initial condition {estilo de exibição 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 {displaystyle y'(t)=f(t,y(t)),quad y(t_{0})=y_{0},} com {estilo de exibição f} defined on the rectangular domain {estilo de exibição R={(t,y),|,|t-t_{0}|leq a,|y-y_{0}|leq b}} . If the function {estilo de exibição f} satisfies the following three conditions: {estilo de exibição f(t,y)} is continuous in {estilo de exibição y} for each fixed {estilo de exibição t} , {estilo de exibição f(t,y)} is measurable in {estilo de exibição t} for each fixed {estilo de exibição y} , there is a Lebesgue-integrable function {estilo de exibição m:[t_{0}-uma,t_{0}+uma]para [0,infty )} de tal modo que {estilo de exibição |f(t,y)|leq m(t)} para todos {estilo de exibição (t,y)em R} , then the differential equation has a solution in the extended sense in a neighborhood of the initial condition.[4] A mapping {displaystyle fcolon Rto mathbf {R} ^{n}} is said to satisfy the Carathéodory conditions on {estilo de exibição R} if it fulfills the condition of the theorem.[5] Uniqueness of a solution Assume that the mapping {estilo de exibição f} satisfies the Carathéodory conditions on {estilo de exibição R} and there is a Lebesgue-integrable function {estilo de exibição k:[t_{0}-uma,t_{0}+uma]para [0,infty )} , de tal modo que {estilo de exibição |f(t,s_{1})-f(t,s_{2})|leq k(t)|s_{1}-s_{2}|,} para todos {estilo de exibição (t,s_{1})em R,(t,s_{2})in R.} Então, there exists a unique solution {estilo de exibição y(t)=y(t,t_{0},s_{0})} to the initial value problem {displaystyle y'(t)=f(t,y(t)),quad y(t_{0})=y_{0}.} Além disso, if the mapping {estilo de exibição f} is defined on the whole space {estilo de exibição mathbf {R} vezes mathbf {R} ^{n}} and if for any initial condition {estilo de exibição (t_{0},s_{0})em matemática {R} vezes mathbf {R} ^{n}} , there exists a compact rectangular domain {estilo de exibição R_{(t_{0},s_{0})}subset mathbf {R} vezes mathbf {R} ^{n}} such that the mapping {estilo de exibição f} satisfies all conditions from above on {estilo de exibição R_{(t_{0},s_{0})}} . Então, the domain {displaystyle Esubset mathbf {R} ^{2+n}} of definition of the function {estilo de exibição y(t,t_{0},s_{0})} is open and {estilo de exibição y(t,t_{0},s_{0})} is continuous on {estilo de exibição E} .[6] Example Consider a linear initial value problem of the form {displaystyle y'(t)=A(t)y(t)+b(t),quad y(t_{0})=y_{0}.} Aqui, the components of the matrix-valued mapping {displaystyle Acolon mathbf {R} to mathbf {R} ^{vezes n}} and of the inhomogeneity {displaystyle bcolon mathbf {R} to mathbf {R} ^{n}} are assumed to be integrable on every finite interval. Então, 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 Mathematics portal Picard–Lindelöf theorem Cauchy–Kowalevski theorem Notes ^ Coddington & Levinson (1955), Teorema 1.2 of Chapter 1 ^ Coddington & Levinson (1955), página 42 ^ Rudin (1987), Teorema 7.18 ^ Coddington & Levinson (1955), Teorema 1.1 of Chapter 2 ^ Hale (1980), p.28 ^ Hale (1980), Teorema 5.3 of Chapter 1 ^ Hale (1980), p.30 References Coddington, Earl A.; Levinson, Norman (1955), Theory of Ordinary Differential Equations, Nova york: McGraw-Hill. Hale, Jack K. (1980), Ordinary Differential Equations (2ª edição), Malabar: Robert E. Krieger Publishing Company, ISBN 0-89874-011-8. Rudin, Walter (1987), Análise real e complexa (3rd ed.), Nova york: McGraw-Hill, ISBN 978-0-07-054234-1, MR 0924157. Categorias: Ordinary differential equationsTheorems in analysis