existence and uniqueness of solution to Cauchy problem



be a Cauchy problemMathworldPlanetmath, where F:U→ℝ is

Then there exists a unique solution 𝐱:I→ℝn of the Cauchy problem, defined in a neighborhood IβŠ†β„ of t0.


Solving the Cauchy problem is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to solving the following integral equation


Let X be the set of continuous functions 𝐟:[t0-Ξ΄,t0+Ξ΄]β†’B⁒(𝐱0,Ο΅). We’ll assume Ο΅ to be chosen such that the B⁒(𝐱0,Ο΅)βŠ†U 11B⁒(𝐱0,Ο΅) denotes the closed ballPlanetmathPlanetmath {𝐱:βˆ₯𝐱0-𝐱βˆ₯≀ϡ}. In this ball, therefore, F is Lipschitz continuous with respect to the first n variable, in other words, there exists a real number L such that


for all points 𝐱,𝐲 sufficiently near to 𝐱0.

Now let’s define the mapping T:Xβ†’X as follows


We make the following observations about T.

  1. 1.

    Since F is continuous, βˆ₯Fβˆ₯ attains a maximum value M on the compact set B⁒(𝐱0,Ο΅)Γ—[t0Β±Ξ΄]. But by hypothesisMathworldPlanetmath, βˆ₯𝐱⁒(t)-𝐱0βˆ₯≀ϡ, hence


    for all t∈[t0±δ].

  2. 2.

    The Lipschitz continuity of F yields


If we choose δ<min⁑{1/L,ϡ/M} these conditions ensure that

  • β€’

    T⁒(X)βŠ†X, i.e. T doesn’t send us outside of X.

  • β€’

    T is a contraction mapping with respect to the uniform convergenceMathworldPlanetmath metric d∞ on X, i.e. there exists Ξ»βˆˆβ„ such that for all 𝐱,𝐲∈X,


In particular, the second point allows us to apply Banach’s theorem and define


to find the unique fixed pointPlanetmathPlanetmathPlanetmath of T in X, i.e. the unique function which solves

T⁒𝐱=𝐱⁒ in other words ⁒𝐱⁒(t)=𝐱0+∫t0tF⁒(𝐱⁒(Ο„),Ο„)⁒dΟ„

and which therefore locally solves the Cauchy problem.

Title existence and uniqueness of solution to Cauchy problem
Canonical name ExistenceAndUniquenessOfSolutionToCauchyProblem
Date of creation 2013-03-22 16:54:45
Last modified on 2013-03-22 16:54:45
Owner ehremo (15714)
Last modified by ehremo (15714)
Numerical id 20
Author ehremo (15714)
Entry type Theorem
Classification msc 34A12