existence and uniqueness of solution to Cauchy problem
Let
{Λπ±=F(π±,t)π±(t0)=π±0 |
be a Cauchy problem, where F:Uββ is
-
β’
a continuous function
of n+1 variables defined in a neighborhood Uββn+1 of (π±0,t0)
-
β’
Lipschitz continuous with respect to the first n variables (i.e. with respect to π±).
Then there exists a unique solution π±:Iββn of the Cauchy problem, defined in a neighborhood Iββ of t0.
Proof
Solving the Cauchy problem is equivalent to solving the following integral equation
x(t)=x(t0)+β«tt0F(π±(Ο),Ο)dΟ |
Let X be the set of continuous functions π:[t0-Ξ΄,t0+Ξ΄]βB(π±0,Ο΅). Weβll assume Ο΅ to be chosen such that the B(π±0,Ο΅)βU 11B(π±0,Ο΅) denotes the closed ball {π±:β₯π±0-π±β₯β€Ο΅}. 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(π±,t)-F(π²,t)β€Lβ₯π±-π²β₯ |
for all points π±,π² sufficiently near to π±0.
Now letβs define the mapping T:XβX as follows
Tπ±:tβ¦π±0+β«tt0F(π±(Ο),Ο)dΟ |
We make the following observations about T.
-
1.
Since F is continuous, β₯Fβ₯ attains a maximum value M on the compact set B(π±0,Ο΅)Γ[t0Β±Ξ΄]. But by hypothesis
, β₯π±(t)-π±0β₯β€Ο΅, hence
β₯π±(t)-π±0β₯β€β«tt0β₯F(π±(Ο),Ο)β₯dΟβ€M(t-t0)β€MΞ΄ for all tβ[t0Β±Ξ΄].
-
2.
The Lipschitz continuity of F yields
β₯Tπ±(t)-Tπ²(t)β₯β€β«tt0β₯F(π±(Ο),Ο)-F(π²(Ο),Ο)β₯dΟβ€β«tt0Lβ₯π±(Ο)-π²(Ο)β₯dΟβ€LΞ΄dβ(π±,π²)
If we choose Ξ΄<min{1/L,Ο΅/M} these conditions ensure that
-
β’
T(X)βX, i.e. T doesnβt send us outside of X.
-
β’
T is a contraction mapping with respect to the uniform convergence
metric dβ on X, i.e. there exists Ξ»ββ such that for all π±,π²βX,
dβ(Tπ±,Tπ²)β€Ξ»dβ(π±,π±)
In particular, the second point allows us to apply Banachβs theorem and define
π±β=limkββTkπ±0 |
to find the unique fixed point of T in X, i.e. the unique function which solves
Tπ±=π± in other words π±(t)=π±0+β«tt0F(π±(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 |