Picard’s theorem

Theorem 1 (Picard’s theorem [KF]).

Let $E$ be an open subset of ${\mathrm{R}}^{\mathrm{2}}$ and a continuous function $f\mathit{}\mathrm{(}x\mathrm{,}y\mathrm{)}$ defined as $f\mathrm{:}E\mathrm{\to }\mathrm{R}$. If $\mathrm{(}{x}_{\mathrm{0}}\mathrm{,}{y}_{\mathrm{0}}\mathrm{)}\mathrm{\in }E$ and $f$ satisfies the Lipschitz condition in the variable $y$ in $E$:

$$|f(x,y)-f(x,y_1)|\le M|y-y_1|$$

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

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

with the initial condition

$$y(x_0)=y_0$$

has a unique solution $y\mathit{}\mathrm{(}x\mathrm{)}$ on some interval $\mathrm{|}x\mathrm{-}{x}_{\mathrm{0}}\mathrm{|}\mathrm{\le }\delta$.

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 $E$ be an open subset of ${\mathrm{R}}^{n\mathrm{+}\mathrm{1}}$ and a continuous function $f\mathit{}\mathrm{(}x\mathrm{,}{y}_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{y}_n\mathrm{)}$ defined as $f\mathrm{=}\mathrm{(}{f}_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{f}_n\mathrm{)}\mathrm{:}E\mathrm{\to }{\mathrm{R}}^n$. If $\mathrm{(}{t}_{\mathrm{0}}\mathrm{,}{y}_{\mathrm{10}}\mathrm{,}\mathrm{\dots }\mathrm{,}{y}_{n\mathit{}\mathrm{0}}\mathrm{)}\mathrm{\in }E$ and $f$ satisfies the Lipschitz condition in the variable $y_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{y}_n$ in $E$:

$$|f_i(x,y_1,\mathrm{\dots },y_n)-f_i(x,y_1^{\prime }\mathrm{\dots },y_n^{\prime })|\le M\underset{1\le j\le n}{\max }|y_j-y_j^{\prime }|$$

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

${\displaystyle \frac{d{y}_1}{dx}}$ $=f_1(x,y_1,\mathrm{\dots },y_n)$

$\mathrm{⋮}$

${\displaystyle \frac{d{y}_n}{dx}}$ $=f_n(x,y_1,\mathrm{\dots },y_n)$

with the initial condition

$$y_1(x_0)=y_{10},\mathrm{\dots },y_n(x_0)=y_{n0}$$

has a unique solution

$$y_1(x),\mathrm{\dots },y_n(x)$$

on some interval $\mathrm{|}x\mathrm{-}{x}_{\mathrm{0}}\mathrm{|}\mathrm{\le }\delta$.

see also:

• the entry existence and uniqueness of solution of ordinary differential equations

Title	Picard’s theorem
Canonical name	PicardsTheorem
Date of creation	2013-03-22 14:59:57
Last modified on	2013-03-22 14:59:57
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	6
Author	Daume (40)
Entry type	Theorem
Classification	msc 34A12
Synonym	Picard-Lindelöf theorem
Related topic	ExistenceAndUniquenessOfSolutionOfOrdinaryDifferentialEquations