Picard’s theorem

Theorem 1 (Picard’s theorem [KF]).

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

|f(x,y)-f(x,y_{1})|\\leq 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 $y(x)$ on some interval $|x-x_{0}|\\leq\\delta$ .

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 $E$ 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 $f$ satisfies the Lipschitz condition in the variable $y_{1},\\ldots,y_{n}$ in $E$ :

|f_{i}(x,y_{1},\\dots,y_{n})-f_{i}(x,y_{1}^{\\prime}\\ldots,y_{n}^{\\prime})|\\leq M%\\max_{1\\leq j\\leq n}|y_{j}-y_{j}^{\\prime}|

where $M$ 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

y_{1}(x_{0})=y_{10},\\ldots,y_{n}(x_{0})=y_{n0}

has a unique solution

y_{1}(x),\\ldots,y_{n}(x)

on some interval $|x-x_{0}|\\leq\\delta$ .

References

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