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existence and uniqueness of solution to Cauchy problem


Let

{˙𝐱=F(𝐱,t)𝐱(t0)=𝐱0

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

  • β€’

    a continuous functionMathworldPlanetmathPlanetmath of n+1 variables defined in a neighborhood UβŠ†β„n+1 of (𝐱0,t0)

  • β€’

    Lipschitz continuous with respect to the first n variables (i.e. with respect to 𝐱).

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

Proof

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

x(t)=x(t0)+∫tt0F(𝐱(Ο„),Ο„)dΟ„

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

F(𝐱,t)-F(𝐲,t)≀Lβˆ₯𝐱-𝐲βˆ₯

for all points 𝐱,𝐲 sufficiently near to 𝐱0.

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

T𝐱:t↦𝐱0+∫tt0F(𝐱(Ο„),Ο„)dΟ„

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

    βˆ₯𝐱(t)-𝐱0βˆ₯β‰€βˆ«tt0βˆ₯F(𝐱(Ο„),Ο„)βˆ₯dτ≀M(t-t0)≀MΞ΄

    for all t∈[t0±δ].

  2. 2.

    The Lipschitz continuity of F yields

    βˆ₯T𝐱(t)-T𝐲(t)βˆ₯β‰€βˆ«tt0βˆ₯F(𝐱(Ο„),Ο„)-F(𝐲(Ο„),Ο„)βˆ₯dΟ„β‰€βˆ«tt0Lβˆ₯𝐱(Ο„)-𝐲(Ο„)βˆ₯dτ≀LΞ΄d∞(𝐱,𝐲)

If we choose Ξ΄<min{1/L,Ο΅/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,

    d∞(T𝐱,T𝐲)≀λd∞(𝐱,𝐱)

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

𝐱⋆=limkβ†’βˆžTk𝐱0

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

T𝐱=𝐱 in other words π±(t)=𝐱0+∫tt0F(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