Theory of Ordinary Differential Equations of Ordinary Differential Equations Basic Existence and Uniqueness John A. Burns C enter for O ptimal D esign A nd C ontrol I nterdisciplinary

Theory of Ordinary Differential Equations

Basic Existence and Uniqueness

John A. Burns

Center for Optimal Design And Control

Interdisciplinary Center for Applied Mathematics

Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061-0531

MATH 5245 - FALL 2012

1st Order Scalar Equations

( ) ( , ( ))x t f t x t

1st order ordinary differential equation

2( ) ( ) 5[ ( )]x t tx t x t

( ) 3 ( )d

p t p tdt

3( ) 5[ ( )] cos( )y t y t t

3( ) sin( ( ))y t y t t

( , ) 3f t p p

( , ) :f t x D R R R

2( , ) 5f t x t x x

3( , ) sin( )f t y y t

3( , ) 5 cos( )f t y y t

1st Order Autonomous DEs

2( ) ( ) 5[ ( )]x t tx t x t

( ) 3 ( )d

p t p tdt

3( ) 5[ ( )] cos( ( ))y t y t y t

3( ) sin( ( ))y t y t t

( , ) 3f t p p

2( , ) 5f t x t x x

3( , ) sin( )f t y y t

3( , ) 5 cos( )f t y y y

( ) ( , ( ))x t f t x t ( ) ( ( ))x t f x t

Auto

Non-auto

Non-auto

Auto

0( ) [1 (1/ ) ( )] ( )p t r k p t p t 0( ) [1 (1/ ) ]f p r k p p

Logistic Equation

Autonomous

Definition of Solution

A solution to the ordinary differential equation (Σ) is a

differentiable function

( ) , ( )x t f t x t(Σ)

( ) : ( , )x t a b R

defined on a connected interval (a,b) such that x(t)

satisfies (Σ) for all t (a,b).

2

( ) ( )d

x t x tdt

1( )

1x t

t

1( ) , 1

1lx t t

t

1( ) , 1

1rx t t

t

TWO SOLUTIONS

Solutions

-3 -2 -1 0 1 2 3 4-20

-15

-10

-5

0

5

10

15

20

1( ) , 1

1lx t t

t

1( ) , 1

1rx t t

t

Initial Condition

-3 -2 -1 0 1 2 3 4-20

-15

-10

-5

0

5

10

15

20

1( )0lx

1( ) , 1

1lx t t

t

1( ) , 1

1rx t t

t

1( )2rx

EXISTENCE AND UNIQUENESS

SCALAR PROBLEMS

Ordinary Differential Equations

1R

t

x0

t0

)(tx

1

0 0( )x t x R (IC)

( ) , ( )x t f t x t(Σ) {(IVP)

( , ) : Df t x R R R

D

D

Solutions to First Order Equations

A solution to the ordinary differential equation (Σ) is a

differentiable function

( ) , ( )x t f t x t(Σ)

1( ) : ( , )x t a b Rdefined on a connected interval (a, b) such that

1. (t, x(t)) D for all t (a, b),

2. x(t) satisfies (Σ) for all t (a, b).

1 ,1

1)(

t

ttxl

tt

txr 1 ,1

1)(

HAS TWO

GENERAL

SOLUTIONS

2

( ) ( )x t x t

( ) 3 ( )x t x t3( ) tx t e k ( , ) ( , )a b k

( , ) :f t x D R R R

Multiple Solutions

-3 -2 -1 0 1 2 3 4-20

-15

-10

-5

0

5

10

15

20

1 ,1

1)(

t

ttxl

tt

txr 1 ,1

1)(

( , ) ( ,1)l la b

( , ) (1, )r ra b

Initial Value Problem

1

0 0( )x t x R (IC)

A solution to the initial value problem (IVP) is a

differentiable function 1( ) : ( , )x t a b R

defined on a connected interval (a, b) such that

1. t0 (a, b),

2. x(t0 ) = x0 ,

3. (t, x(t)) D for all t (a, b),

4. x(t) satisfies (Σ) for all t (a, b).

( ) , ( )x t f t x t(Σ) {(IVP)

( , ) :f t x D R R R

Initial Value Problem

1R

t

x0

t0

)(tx

1

0 0( )x t x R (IC)

( ) , ( )x t f t x t(Σ) {(IVP)

( , ) :f t x D R R R

D

-3 -2 -1 0 1 2 3 4-20

-15

-10

-5

0

5

10

15

20

1 ,1

1)(

t

ttxl

2

( ) ( )x t x t (0) 1x

tt

txr 1 ,1

1)(

Initial Condition

1

0 0( )x t x R (IC)

( ) , ( )x t f t x t(Σ) {(IVP)

( , ) :f t x D R R R

)(tx

1R

t

x0

t0

D

2/3

( ) ( )x t x t 2/3( , )f t x x D R R

f(t, x) is continuous EVERYWHERE!!!

Theorem 1. If f: D ---> R is a continuous function on

a domain D and (t0, x0 )D, then there exists at least

one solution to the initial value problem (IVP).

( )y t

Existence Theorem

2/3

( ) ( )x t x t (0) 0x

1R

t

( ) 0x t

c

3

0,( )

([ ] / 3) ,

t cx t

t c t c

{

c

!! INFINITE NUMBER OF SOLUTIONS !!

Non-Uniqueness

Uniqueness Theorem

1R

t

x0

t0

)(txD

Theorem 2. If there is an open rectangle D

about (t0, x0) such that

are continuous at all points (t, x) , then there a

unique solution to the initial value problem (IVP).

( , )f t x ( , )x

f t x

and

First Order Linear

( ) ( ) ( ) ( )x t a t x t g t

( )

00

( ) ( )t

at a t sx t e x e g s ds

VARIATION OF PARAMETERS FORMULA

( )a t given ( )g tand

homogenous ( ) 0g t non-homogenous ( ) 0g t

constant coefficient ( ) constanta t a

( ) ( ) ( )x t ax t g t 0(0)x x

First Order Linear

( ) ( , ( )) ( ) ( ) ( )x t f t x t a t x t g t

( , ) ( ) ( )f t x a t x g t ( , ) ( )f t x a tx

( ) ( ) ( ) ( )x t a t x t g t

Note: If is continuous on an interval (a, b) and

t0 (a, b), then there a unique solution to the initial

value problem

( )a t

0 0( ) .x t x

( ) ( ) ( ) ( )x t a t x t g t

1( ) ( ) ( ) cos( )

( ) 1

tx t x t t

x

sin( )1( ) ( )t

t tx t

(0, )t

Gronwall – Reid – Bellman Inequality

0

0 ( ) ( ( )) ( )

t

btt a b s ds t ate Gronwall (1919)

Reid (1929) Dissertation - page 296

( ) 0, ( ) 0,

0 ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )exp ( )

k s s a s b

tt t k s s ds

a

t tt t s k s k d ds

a s

Re-discovered by Bellman in 1943

W. T. Reid, “Properties of solutions of an infinite system of ordinary differential equations of the first order4 with auxiliary boundary conditions”, Transactions of the AMS, vol. 32 (1930), pp. 284 – 318.

Bellman’s Version

Bellman in 1943

( )

( ) 0, 0,

0 ( ) ( ) ( )

( ) exp ( ) e

t

a

t k s ds

a

k s a s b

tt k s s ds

a

t k s ds

Peano - Reid Version

( ) 0, ( ) 0,

0 ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )exp ( )

k s s a s b

tt t k s s ds

a

t tt t s k s k d ds

a s

0 ( ) ( ) ( ) ( )

( ) ( )exp ( )

tt t k s s ds

a

tt t k s ds

a

and if is non-decreasing, then ( )s

Homework

PROVE THESE INEQUALITIES

EXISTENCE AND UNIQUENESS

SCALAR PROBLEMS

Proof of Main Theorems

Theorem 1. If f: D ---> R is a continuous function on

a domain D and (t0, x0 )D, then there exists at least

one solution to the initial value problem (IVP).

1

0 0( )x t x R (IC)

( ) , ( )x t f t x t(Σ) {(IVP)

1R

t

x0

t0

D

Peano Existence Theorem

Two step process (Coddington & Levinson: Pages 1-8)



ˆ :1, 2,...,S t p


( ) , ( )x t f t x t

An -approximate solution of () on a connected interval I is a PWS function

such that 1: I

( ) ( , ( )) , ,

( ) ( ) ( , ( )) , .

i t t D t I

ii t f t t t I S

1

0 0( )x t x R

1R

t

D x0

t0

Let R be a closed rectangle inside D as

shown below

R

0t a0t a

0x b

0x b

Peano Existence Theorem 1R

t

D x0

t0

R

0t a0t a

0x b

0x b

0 0 ( , ) : , t x t t a x x b R

max ( , ) : ( , )M f t x t x R

min ,b

aM


Lemma 1.1 Assume f: D ---> R1 is a continuous function on a domain D and

(t0, x0 )R D is the closed rectangle above, then there exists an

-approximate solution of (),

where . 0 I :t t t

1: I

Proof Since f: R ---> R1 is continuous, it is uniformly continuous so there

exists a = () >0 such

1 1 2 2 1 1 2 2( , ) ( , ) , if ( , ) , ( , )f t x f t x t x t x R R

1 2 1 2 ( ), ( ).t t x x and

t0 t0+ t2 t1 t3 tn t0+a …

1

( )max min ( ), k kt t

M


t0 t0+ t2 t1 t3 tn t0+a …

x0 + b

x0 - b

slope = M

slope = -M

x0

0 0slope ( , )f t x

1 1slope ( , )f t x

x1

x2

2 2slope ( , )f t x


0 0( )t x 1 1 1 1 1( ) ( ) ( , )( ), k k k k k kt t f t x t t t t t

1If , then ( ) andk k kt t t t t

0 0ˆ ˆ ˆ( ) ( ) , ,t t M t t t t t t

1 1 1 1( ) ( , ( )) ( , ) ( , ( )) ( , ( )) ( , ( ))k k k kt f t t f t x f t t f t t f t t

1 1 1 1 1

( )ˆ ˆ( ) ( ) ( , )( ) ( )k k k k kt t f t x t t M t t M

M

and hence

Thus, is an -approximate solution on . 1: I 0 0[ , ]t t

The same construction yields an -approximate solution on

and connecting the two pieces produces the piecewise smooth -

approximate solution 1

0 0:[ , ]t t

0 0[ , ]t t


A set of functions defined on a bounded interval I is equicontinuous on I if

for any > 0, there is a () > 0 such that if

then

1 2 1 2, , satisfy ( ) and ( )t t I t t

1 2 ( ) ( ) t t

Lemma 1.2 (Ascoli) Assume that for each n=1,2,3,… , n : I ---> R1 is a

sequence of uniformly bounded and equicontinuous functions defined on the

interval I. Then there is a subsequence

and a continuous function : I ---> R1 such that as

uniformly on I.

1:kn I

( ) ( )kn

k


Theorem 1. (Peano) If f: D ---> R is a continuous

function on a domain D and (t0, x0 )D, then there

exists at least one solution to the initial value

problem (IVP).

Proof Let and be the -approximate

solution of the IVP computed as above . Clearly, so the

sequence is uniformly bounded and

Thus, is a uniformly bounded equicontinuous

set.

1 2 1 2( ) ( ) 1/ .n n nt t M t t n

1/n n 1

0 0:[ , ]n t t

0( )n t x b

( ) : 1, 2,3,...n n

( ) : 1,2,3,...n n

Pick the subsequence uniformly convergent on

to the continuous function .

( ) ( )kn 0 0[ , ]t t

( )


Let ( ) ( ) and ( ) ( )zk nx x

( ) ( , ( )) , kk k nx t f t x t t I S

0

0( ) [ ( , ( )) ( )] where

t

k k k

t

x t x f s x s s ds ( ) ( , ( )) ( ) on k k kx t f t x t t I S

( )kk nt

( ) ( ) uniformly ( , ( )) ( , ( )) uniformlyk kx x f t x t f t x t

0 0 0

0 0 0( ) ( , ( )) [ ( , ( )) ( )] - ( , ( ))

t t t

k k k

t t t

x t x f s x s ds x f s x s s ds x f s x s ds

0

[ ( , ( )) ( , ( )) ( )]

t

k k

t

f s x s f s x s s ds

0

( , ( )) ( , ( ))k

t

k n

t

f s x s f s x s ds


0

0 0

0( ) ( , ( )) ( , ( )) ( , ( )) 0k

tt

k k n

t t

x t x f s x s ds f s x s f s x s ds

( ) ( ) uniformly ( , ( )) ( , ( )) uniformlyk kx x f t x t f t x t

0

0( ) ( ) and ( ) ( , ( ))

t

k k

t

x x x x f s x s ds

0

0( ) ( , ( ))

t

t

x x f s x s ds

Since is continuous is differentiable and ( )x 0

0 ( , ( ))

t

t

x f s x s ds

( ) ( , ( ))x t f t x t


The function f: D ---> R is said to satisfy a Lipschitz

condition on D if there is a constant k > 0 such that for

every (t, x1 ) and (t, x2 ) in D,

1 2 1 2( , ) ( , ) .f t x f t x k x x

Theorem 2. Assume that the function f: D ---> R

satisfies a Lipschitz condition on D. If

are two solutions to the (IVP) on the interval (a, b),

then

In particular, solutions are unique.

1 2( ) and ( )x x

1 2( )= ( ), for all ( , ).x t x t t a b


Proof first note that f: D ---> R1 is continuous, so there exists at least one

solution. Assume that are two solutions defined on (a,b) so

that a < t0 < b and

then

which implies

00( ) ( ) ( , ( )) , 1,2, .

t

i i it

x t x t f s x s ds i a t b

1 2( ) and ( )x x

0 01 2 1 2( ) ( ) ( , ( )) ( , ( ))

t t

t tx t x t f s x s ds f s x s ds

01 2[ ( , ( )) ( , ( ))]

t

tf s x s f s x s ds

01 2( , ( )) ( , ( ))

t

tf s x s f s x s ds

01 2( ) ( )

t

tk x s x s ds


01 2 1 2( ) ( ) ( ) ( )

t

tx t x t k x s x s ds

Define so that 1 2( ) ( ) ( )t x t x t

0 0

0 ( ) ( ) 0 ( ) , ( , ).t t

t tt k s ds k s ds t a b

0( ) 0, 0 0, k t k t t b

Apply the Gronwall Inequality with

0 0

0

( ) ( ) ( ) exp ( ) e 0e 0

t

k t t k t t

t

t k s ds

First consider the case 0t t b


0 0

0

( ) ( )

0 ( ) exp ( ) e 0e 0,

t

k t t k t t

t

t k s ds t t b

Hence,

1 2 0( ) ( ) ( ) 0, x t x t t t t b

and the solution is unique on . 0t t b

Homework: Complete the proof and show that

1 2 0( ) ( ) ( ) 0, x t x t t a t t

so that 1 2( ) ( ), x t x t a t b

Continuation of Solutions

1R

t

x0

t0

)(tx

1

0 0( )x t x R (IC)

( ) , ( )x t f t x t(Σ) {(IVP)

( , ) : Df t x R R R

DD

a b a1 b1

Can’t extend beyond a certain point

am bM


Theorem 2. Assume that the function f: D ---> R is

continuous on D and there is a constant such that

If

If then the solution

may be extended to the left of a [or to the right of b].

M

( , ) , for all ( , ) .f t x M t x D R R

( ) is a solution to the (IVP) on the interval ( , ), thenx a b

lim ( )= ( ) and lim ( )= ( ) exist.t a t b

x t x a x t x b

( , ( ) ) [or ( , ( )) ], a x a D b x b D ( )x t

00( ) ( ) ( , ( )) , .

t

tx t x t f s x s ds a t b

00( ) ( ) ( , ( ))

b

tx b x t f s x s ds

00( ) ( ) ( , ( ))

a

tx a x t f s x s ds


1R

t

x0

t0

)(tx

DD

a b a1 b1

R

R

Theory of Ordinary Differential Equations of Ordinary Differential Equations Basic Existence and Uniqueness John A. Burns C enter for O ptimal D esign A nd C ontrol I nterdisciplinary

Documents