Picard’s theorem

Theorem 1 (Picard’s theorem [KF]).

Let E be an open subset of R2 and a continuous functionMathworldPlanetmathPlanetmath f(x,y) defined as f:ER. If (x0,y0)E and f satisfies the Lipschitz conditionMathworldPlanetmath in the variable y in E:


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


with the initial conditionMathworldPlanetmath


has a unique solution y(x) on some interval |x-x0|δ.

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 Rn+1 and a continuous function f(x,y1,,yn) defined as f=(f1,,fn):ERn. If (t0,y10,,yn0)E and f satisfies the Lipschitz condition in the variable y1,,yn in E:


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

dy1dx =f1(x,y1,,yn)
dyndx =fn(x,y1,,yn)

with the initial condition


has a unique solution


on some interval |x-x0|δ.


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