PlanetMath: Picard's theorem

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Picard's theorem

(Theorem)

Theorem 1 (Picard's theorem [KF]) Let be an open subset of $\mathbb{R}^2$ and a continuous function defined as $f\colon E \to \mathbb{R}$ . If $(x_0,y_0)\in E$ and satisfies the Lipschitz condition in the variable in :

$\displaystyle \vert f(x,y)-f(x,y_1)\vert \leq M\vert y-y_1\vert$

where is a constant. Then the ordinary differential equation defined as

$\displaystyle \frac{dy}{dx} = f(x,y)$

with the initial condition

$\displaystyle y(x_0) = y_0$

has a unique solution on some interval $\vert x- x_0\vert \leq\delta$ .

The above theorem is also named the Picard-Lindelöf theorem and can be generalized to a system of first order ordinary differential equations

Theorem 2 (generalization of Picard's theorem [KF]) Let be an open subset of $\mathbb{R}^{n+1}$ and a continuous function $f(x,y_1,\ldots,y_n)$ defined as $f=(f_1,\ldots,f_n)\colon E \to \mathbb{R}^n$ . If $(t_0,y_{10},\ldots,y_{n0})\in E$ and satisfies the Lipschitz condition in the variable $y_1,\ldots,y_n$ in :

$\displaystyle \vert f_i(x,y_1,\dots,y_n)-f_i(x,y_1'\ldots,y_n')\vert \leq M \max_{1\leq j\leq n}\vert y_j-y_j'\vert$

where is a constant. Then the system of ordinary differential equation defined as

$\displaystyle \frac{dy_1}{dx}$	$\displaystyle = f_1(x,y_1,\ldots,y_n)$
	$\displaystyle \vdots$
$\displaystyle \frac{dy_n}{dx}$	$\displaystyle = f_n(x,y_1,\ldots,y_n)$

with the initial condition

$\displaystyle y_1(x_0) = y_{10},\ldots, y_n(x_0) = y_{n0}$

has a unique solution

$\displaystyle y_1(x) ,\ldots, y_n(x)$

on some interval $\vert x- x_0\vert \leq\delta$ .

References

KF: Kolmogorov, A.N. & Fomin, S.V.: Introductory Real Analysis, Translated & Edited by Richard A. Silverman. Dover Publications, Inc. New York, 1970.

"Picard's theorem" is owned by Daume.

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Other names:

Picard-Lindelöf theorem

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Cross-references: existence and uniqueness of solution of ordinary differential equations, first order, interval, solution, initial condition, ordinary differential equation, variable, Lipschitz condition, continuous function, open subset
There are 3 references to this entry.

This is version 3 of Picard's theorem, born on 2005-02-03, modified 2008-01-09.
Object id is 6706, canonical name is PicardsTheorem2.
Accessed 4185 times total.

Classification:

AMS MSC:

34A12 (Ordinary differential equations :: General theory :: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions)

Pending Errata and Addenda

None.[ View all 3 ]

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