existence and uniqueness of solution to Cauchy problem

existence and uniqueness of solution to Cauchy problem

Let

{

𝐱

˙

=

F

⁢

(

𝐱

,

t

)

𝐱

⁢

(

t

0

)

=

𝐱

0

be a

Cauchy problem

, where

F

:

U

→

ℝ

is

•

a

continuous function

of

n

+

1

variables defined in a

neighborhood

U

⊆

ℝ

n

+

1

of

(

𝐱

0

,

t

0

)

•

Lipschitz continuous

with respect to the first

n

variables (i.e. with respect to

𝐱

).

Then there exists a unique solution

𝐱

:

I

→

ℝ

n

of the Cauchy problem, defined in a neighborhood

I

⊆

ℝ

of

t

0

.

Proof

Solving the Cauchy problem is

equivalent

to solving the following integral equation

x

⁢

(

t

)

=

x

⁢

(

t

0

)

+

∫

t

0

t

F

⁢

(

𝐱

⁢

(

τ

)

,

τ

)

⁢

d

τ

Let

X

be the set of continuous functions

𝐟

:

[

t

0

-

δ

,

t

0

+

δ

]

→

B

⁢

(

𝐱

0

,

ϵ

)

. We’ll assume

ϵ

to be chosen such that the

B

⁢

(

𝐱

0

,

ϵ

)

⊆

U

1

1

B

⁢

(

𝐱

0

,

ϵ

)

denotes the

closed ball

{

𝐱

:

∥

𝐱

0

-

𝐱

∥

≤

ϵ

}

. 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

)

≤

L

⁢

∥

𝐱

-

𝐲

∥

for all points

𝐱

,

𝐲

sufficiently near to

𝐱

0

.

Now let’s define the

mapping

T

:

X

→

X

as follows

T

⁢

𝐱

:

t

↦

𝐱

0

+

∫

t

0

t

F

⁢

(

𝐱

⁢

(

τ

)

,

τ

)

⁢

d

τ

We make the following observations about

T

.

1.

Since

F

is

continuous

,

∥

F

∥

attains a

maximum

value

M

on the

compact set

B

⁢

(

𝐱

0

,

ϵ

)

×

[

t

0

±

δ

]

. But by

hypothesis

,

∥

𝐱

⁢

(

t

)

-

𝐱

0

∥

≤

ϵ

, hence

∥

𝐱

⁢

(

t

)

-

𝐱

0

∥

≤

∫

t

0

t

∥

F

⁢

(

𝐱

⁢

(

τ

)

,

τ

)

∥

⁢

d

τ

≤

M

⁢

(

t

-

t

0

)

≤

M

⁢

δ

for all

t

∈

[

t

0

±

δ

]

.

2.

The Lipschitz continuity of

F

yields

∥

T

⁢

𝐱

⁢

(

t

)

-

T

⁢

𝐲

⁢

(

t

)

∥

≤

∫

t

0

t

∥

F

⁢

(

𝐱

⁢

(

τ

)

,

τ

)

-

F

⁢

(

𝐲

⁢

(

τ

)

,

τ

)

∥

⁢

d

τ

≤

∫

t

0

t

L

⁢

∥

𝐱

⁢

(

τ

)

-

𝐲

⁢

(

τ

)

∥

⁢

d

τ

≤

L

⁢

δ

⁢

d

∞

⁢

(

𝐱

,

𝐲

)

If we choose

δ

<

min

⁡

{

1

/

L

,

ϵ

/

M

}

these conditions ensure that

•

T

⁢

(

X

)

⊆

X

, i.e.

T

doesn’t send us outside of

X

.

•

T

is a

contraction mapping

with respect to the

uniform convergence

metric

d

∞

on

X

, i.e. there exists

λ

∈

ℝ

such that for all

𝐱

,

𝐲

∈

X

,

d

∞

⁢

(

T

⁢

𝐱

,

T

⁢

𝐲

)

≤

λ

⁢

d

∞

⁢

(

𝐱

,

𝐱

)

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

theorem

and define

𝐱

⋆

=

lim

k

→

∞

⁡

T

k

⁢

𝐱

0

to find the unique

fixed point

of

T

in

X

, i.e. the unique

function

which solves

T

⁢

𝐱

=

𝐱

⁢

in other words

⁢

𝐱

⁢

(

t

)

=

𝐱

0

+

∫

t

0

t

F

⁢

(

𝐱

⁢

(

τ

)

,

τ

)

⁢

d

τ

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

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