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.

Contenuti 1 introduzione 2 Enunciato del teorema 3 Uniqueness of a solution 4 Esempio 5 Guarda anche 6 Appunti 7 References Introduction Consider the differential equation {displaystyle y'(t)=f(t,y(t))} con condizione iniziale {stile di visualizzazione y(t_{0})=y_{0},} where the function ƒ is defined on a rectangular domain of the form {stile di visualizzazione R={(t,y)in matematica {R} volte 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] Tuttavia, 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 {stile di visualizzazione H(t)={inizio{casi}0,&{testo{Se }}tleq 0;\1,&{testo{Se }}t>0.end{casi}}} It makes sense to consider the ramp function {stile di visualizzazione y(t)=int _{0}^{t}H(S),matematica {d} s={inizio{casi}0,&{testo{Se }}tleq 0;\t,&{testo{Se }}t>0end{casi}}} as a solution of the differential equation. Strictly speaking though, it does not satisfy the differential equation at {stile di visualizzazione 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)} con condizione iniziale {stile di visualizzazione 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},} insieme a {stile di visualizzazione f} defined on the rectangular domain {stile di visualizzazione R={(t,y),|,|t-t_{0}|leq a,|y-y_{0}|leq b}} . If the function {stile di visualizzazione f} satisfies the following three conditions: {stile di visualizzazione f(t,y)} is continuous in {stile di visualizzazione y} for each fixed {stile di visualizzazione t} , {stile di visualizzazione f(t,y)} is measurable in {stile di visualizzazione t} for each fixed {stile di visualizzazione y} , there is a Lebesgue-integrable function {stile di visualizzazione m:[t_{0}-un,t_{0}+un]a [0,infty )} tale che {stile di visualizzazione |f(t,y)|leq m(t)} per tutti {stile di visualizzazione (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 {displaystyle fcolon Rto mathbf {R} ^{n}} is said to satisfy the Carathéodory conditions on {stile di visualizzazione R} if it fulfills the condition of the theorem.[5] Uniqueness of a solution Assume that the mapping {stile di visualizzazione f} satisfies the Carathéodory conditions on {stile di visualizzazione R} and there is a Lebesgue-integrable function {stile di visualizzazione k:[t_{0}-un,t_{0}+un]a [0,infty )} , tale che {stile di visualizzazione |f(t,si_{1})-f(t,si_{2})|leq k(t)|si_{1}-si_{2}|,} per tutti {stile di visualizzazione (t,si_{1})in r,(t,si_{2})in R.} Quindi, there exists a unique solution {stile di visualizzazione y(t)=y(t,t_{0},si_{0})} to the initial value problem {displaystyle y'(t)=f(t,y(t)),quad y(t_{0})=y_{0}.} Inoltre, if the mapping {stile di visualizzazione f} is defined on the whole space {displaystyle mathbf {R} volte mathbf {R} ^{n}} and if for any initial condition {stile di visualizzazione (t_{0},si_{0})in matematica {R} volte mathbf {R} ^{n}} , there exists a compact rectangular domain {stile di visualizzazione R_{(t_{0},si_{0})}subset mathbf {R} volte mathbf {R} ^{n}} such that the mapping {stile di visualizzazione f} satisfies all conditions from above on {stile di visualizzazione R_{(t_{0},si_{0})}} . Quindi, the domain {displaystyle Esubset mathbf {R} ^{2+n}} of definition of the function {stile di visualizzazione y(t,t_{0},si_{0})} is open and {stile di visualizzazione y(t,t_{0},si_{0})} is continuous on {stile di visualizzazione 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}.} Qui, the components of the matrix-valued mapping {displaystyle Acolon mathbf {R} to mathbf {R} ^{volte n}} and of the inhomogeneity {displaystyle bcolon mathbf {R} to mathbf {R} ^{n}} are assumed to be integrable on every finite interval. Quindi, 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), pagina 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, New York: McGraw-Hill. Hale, Jack K. (1980), Ordinary Differential Equations (2nd ed.), Malabar: Robert E. Krieger Publishing Company, ISBN 0-89874-011-8. Rudino, Walter (1987), Analisi reale e complessa (33a ed.), New York: McGraw-Hill, ISBN 978-0-07-054234-1, SIG 0924157. Categorie: Ordinary differential equationsTheorems in analysis

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