Massera type theorems in vector-valued analytic

functions and hyperfunctions

Yasunori OKADA

Graduate School of Science,Chiba University

Aug.27,2013,

FASDE III,Bedlewo

table of contents

1

Introduction

Classical Massera type theorems

Brief overview

2

Bounded hyperfunctions at innity

Bounded hyperfunctions at innity

Operators for bounded hyperfunctions

Periodicity of bounded hyperfunctions and operators

3

Results

Massera type theorems

Idea of the proof

Notes for re exive valued cases

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 1/25

Introduction

Classical Massera type theorems

classical Massera theorem

J.L.Massera (1950,Duke Math.J.17) studied the existence of a

periodic solution to a periodic ordinary dierential equation of normal

form.In the linear case,he gave

Theorem (Massera,linear case)

Consider an equation

dx

dt

= A(t)x +f (t);

where A:R!R

mm

and f:R!R

m

are 1-periodic and continuous.

Then,the existence of a bounded solution in the future (i.e.,a solution

dened and bounded on a set ft > t

0

g with some t

0

) implies the existence

of a 1-periodic solution.

Note:Since periodic C

1

-functions are bounded,we have the equivalence:

9 a bounded solution in the future.() 9 a 1-periodic solution.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 2/25

Introduction

Classical Massera type theorems

some generalizations

Question

Do such phenomena appear commonly in periodic linear equations?

After Massera,many generalizations have appeared.

Refer,for ex.,to

functional dierential equations with delay:Chow-Hale (1974,FE.

17),Hino-Murakami (1989,Lect.Notes Pure Appl.Math.118),etc.

Banach valued,abstract settings:Shin-Naito (1999,JDE.153),

Naito-Nguyen-Miyazaki-Shin (2000,JDE.160),etc.

discrete dynamical systems in re exive Banach spaces and those in

sequentially complete locally convex spaces with the sequential

Montel property:Zubelevich (2006,Regul.Chaotic Dyn.11).

and also the references therein.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 3/25

Introduction

Brief overview

my interest

Interest

Is there any counterpart to the Massera type phenomenon in the

framework of hyperfunctions?

\Hyperfunctions":a notion of generalized functions,due to Sato (1959,

1960,J.Fac.Sci.Univ.Tokyo,8).

Note:There's NO notion of boundedness for hyperfunctions on ]t

0

;+1[.

Obstacles

1

What is\a bounded solution in the future"?

2

Does the periodicity imply the boundedness in the future?

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 4/25

Introduction

Brief overview

previous results

We constructed

B

L

1

:the sheaf of bounded hyperfunctions at innity on D

1

:= Rtf1g,

(an extension of the sheaf B of hyperfunctions on R).

1

We can interpret\a bounded solution in the future"as a solution u in

B

L

1 at t = +1.

2

Periodic hyperfunctions can be canonically identied with periodic

bounded hyperfunctions at t = +1.

By using B

L

1,we gave a Massera type theorem

Statement

for a class of

equations,(O.,2008.)

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 5/25

Introduction

Brief overview

previous results

The class contains,for ex.,

du

dt

= A(t)u +

Z

r

0

B(t;s)u(t s)ds +f (t);

A,B,f:square matrices and a column vector,continuous,and!-periodic

in t.Moreover,A,B are real-analytic in t,

r > 0:a constant,(representing the\nite delay").

Note:This result could be extended to some classes of equations,

containing integro-dierential equations with innite delay,under an

additional assumption,so-called the\fading memory condition",which we

mentioned in FASDE II,Aug.2011.

But,today,we do not focus on this direction.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 6/25

Introduction

Brief overview

vector valued cases

Let E be a sequentially complete locally convex space.

The result can be extended to E-valued case,when E admits the

sequential Montel property:

Denition (sequential Montel property)

(M) Any bounded sequence in E has a convergent subsequence.

or,when E is re exive.

Statement in re exive case

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 7/25

Bounded hyperfunctions at innity

boundedness for hyperfunctions

D

1

:= Rt f1g:a compactication of R.Consider

O C = R+i R D

1

+i R

E

O

L

1

[ [

B R = ]1;+1[ D

1

= [1;+1]

E

B

L

1

Here,

O the sheaf of holomorphic functions on C,

B the sheaf of hyperfunctions on R,

O

L

1

the sheaf of bounded holomorphic functions on D

1

+i R,

B

L

1

the sheaf of bounded hyperfunctions at innity on D

1

,

and for a sequentially complete Hausdor locally convex space E,

E

O

L

1

the E-valued variant of O

L

1

,

E

B

L

1 the E-valued variant of B

L

1.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 8/25

Bounded hyperfunctions at innity

boundedness for hyperfunctions

D

1

:= Rt f1g:a compactication of R.Consider

O C = R+i R D

1

+i R

E

O

L

1

[ [

B R = ]1;+1[ D

1

= [1;+1]

E

B

L

1

Here,

O the sheaf of holomorphic functions on C,

B the sheaf of hyperfunctions on R,

O

L

1

the sheaf of bounded holomorphic functions on D

1

+i R,

B

L

1

the sheaf of bounded hyperfunctions at innity on D

1

,

and for a sequentially complete Hausdor locally convex space E,

E

O

L

1

the E-valued variant of O

L

1

,

E

B

L

1 the E-valued variant of B

L

1.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 8/25

Bounded hyperfunctions at innity

Bounded hyperfunctions at innity

bounded hyperfunctions at innity

Denition (sheaves O

L

1 and B

L

1)

1

The sheaf O

L

1 on D

1

+i R is dened by

O

L

1

(U) = ff 2 O(U\C) j 8L b U;f is bounded on L\Cg

for any open set U D

1

+i R.

2

The sheaf B

L

1

on D

1

is dened as the sheaf associated with the

presheaf

D

1

open

7!lim

!

U

O

L

1(U n

)

O

L

1(U)

:

Here U runs through complex neighborhoods of

.

The space O

L

1

(U) is endowed with a natural Frechet topology.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 9/25

Bounded hyperfunctions at innity

Bounded hyperfunctions at innity

vector valued variants

E:a sequentially complete Hausdor locally convex space,

E

O:the sheaf of E-valued holomorphic functions on C.

Denition (sheaves

E

O

L

1

and

E

B

L

1

)

1

The sheaf

E

O

L

1

on D

1

+i R is dened by

E

O

L

1(U) = ff 2

E

O(U\C) j 8L b U;f is bounded on L\Cg

for open sets U D

1

+i R.

2

The sheaf

E

B

L

1 on D

1

is dened as the sheaf associated with the

presheaf

D

1

open

7!lim

!

U

E

O

L

1(U n

)

E

O

L

1(U)

:

Here U runs through complex neighborhoods of

.

The space

E

O

L

1(U) is endowed with a natural locally convex topology.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 10/25

Bounded hyperfunctions at innity

Bounded hyperfunctions at innity

properties of bounded hyperfunctions

B

L

1j

R

= B.

B

L

1 is abby.

u 2 B

L

1

(]a;+1]) admits a boundary value representation.

There exists a natural embedding L

1

(]a;+1[),!B

L

1(]a;+1]).

The space B

L

1

(D

1

) of the global sections of our sheaf B

L

1

can be

identied with the space B

L

1 of bounded hyperfunctions due to

Chung-Kim-Lee (2000,Proc.AMS.128).

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 11/25

Bounded hyperfunctions at innity

Operators for bounded hyperfunctions

operators for bounded hyperfunctions

K = [a;b]:a closed interval in R (including the case K = fag),

U:an open set in D

1

+i R,

P = fP

V

g

VU

:a family of linear continuous maps

P

V

:

E

O

L

1(V +K)!

E

O

L

1(V):

Denition (Operators of type K)

P is said to be an operator of type K for

E

O

L

1 on U,if the diagram below

commutes for any pair of open sets V

1

V

2

in U.

E

O

L

1(V

1

+K)

P

V

1

//

restriction

E

O

L

1(V

1

)

restriction

E

O

L

1(V

2

+K)

P

V

2

//

E

O

L

1(V

2

)

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 12/25

Bounded hyperfunctions at innity

Operators for bounded hyperfunctions

properties of operators of type K

An operator P of type K for

E

O

L

1 on U induces a family of linear maps

P

:

E

B

L

1

(

+K)!

E

B

L

1

(

);for open sets

D

1

\U;

commuting with restrictions.

An operator of type K = f0g induces a local operator.

An operator of type K = [r;0] induces an operator of nite delay r.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 13/25

Bounded hyperfunctions at innity

Operators for bounded hyperfunctions

typical examples of our operators

U:= D

1

+i ]d;d[,U

:= U\C = R

1

+i ]d;d[:strip domains,

!> 0:a constant.

Example

dierential operator

P

m

j=0

a

j

(t)@

j

t

with coecients a

j

2 O

L

1(U)

is an operator of type K = f0g.

translation operator T

!

:u(t) 7!u(t +!)

is an operator of type K = f!g.

dierence operator T

!

1:u(t) 7!u(t +!) u(t)

is an operator of type K = [0;!].

integral operator with nite delay u(t) 7!

R

r

0

k(t;s)u(t s)ds

is an operator of type K = [r;0],

if the kernel k(w;s) belongs to (C\L

1

)(U

[0;r]),

and is holomorphic in w.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 14/25

Bounded hyperfunctions at innity

Periodicity of bounded hyperfunctions and operators

periodicity for bounded hyperfunctions

T

!

:the!-translation operator u(t) 7!u(t +!),(!> 0).

We introduce the notion of!-periodicity

for bounded hyperfunction u by the equation (T

!

1)u = 0,and

for operators P of type K by the commutativity P T

!

= T

!

P.

Then,we have,

Fact

Every!-periodic hyperfunction f 2

E

B(R) has the unique!-periodic

extension

^

f 2

E

B

L

1(D

1

).

Every!-periodic bounded hyperfunction f 2

E

B

L

1

(D

1

) admits an

!-periodic boundary value representation.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 15/25

Results

Massera type theorems

scalar valued case

!> 0,K = [a;b] R,and U = D

1

+i ]d;d[.

P:an!-periodic operator of type K for O

L

1

on U,

f 2 B(R):an!-periodic hyperfunction,

its unique!-periodic extension in B

L

1

(D

1

) is also denoted by f,

(B

L

1)

+1

= lim

!

R

B

L

1(]R;+1]):the stalk of B

L

1 at +1.

Theorem

Pu = f has an!-periodic B(R)-solution if and only if it has an

(B

L

1)

+1

-solution.

back

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 16/25

Results

Massera type theorems

re exive valued cases

!> 0,K = [a;b] R,and U = D

1

+i ]d;d[.

E:a re exive locally convex space.

P:an!-periodic operator of type K for

E

O

L

1 on U,

f 2

E

B(R):an!-periodic E-valued hyperfunction,

its unique!-periodic extension in

E

B

L

1(D

1

) is also denoted by f,

(

E

B

L

1

)

+1

= lim

!

R

E

B

L

1

(]R;+1]):the stalk of

E

B

L

1

at +1.

Theorem (re exive case)

Pu = f has an!-periodic

E

B(R)-solution if and only if it has an

(

E

B

L

1)

+1

-solution.

back

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 17/25

Results

Massera type theorems

analytic solutions

!> 0,K = [a;b] R,and U = D

1

+i ]d;d[.

E:a sequentially complete Hausdor locally convex space.

P:an!-periodic operator of type K for

E

O

L

1

on U,

f 2

E

O(R):an!-periodic E-valued analytic function,

its unique!-periodic extension in

E

O

L

1(D

1

) is also denoted by f,

(

E

O

L

1

)

+1

= lim

!R

E

O

L

1

(]R;+1]):the stalk of

E

O

L

1

at +1.

Theorem

Assume the sequential Montel property or the re exivity for E.Then

Pu = f has an!-periodic

E

O(R)-solution if and only if it has an

(

E

O

L

1

)

+1

-solution.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 18/25

Results

Idea of the proof

idea of the proof (simplest case),1

We give the idea of the proof of the theorem for analytic solutions,of the

part

\9 a solution u

0

in (

E

O

L

1

)

+1

) 9 an!-periodic solution u in

E

O(R)",

in the case that P is of type K = f0g,(that is,P is a local operator).

We can nd a neighborhood V

0

D

1

+i R of +1,such that

u

0

2

E

O

L

1(V

0

) and that Pu

0

= f on V

0

.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 19/25

Results

Idea of the proof

idea of the proof (simplest case),1

We give the idea of the proof of the theorem for analytic solutions,of the

part

\9 a solution u

0

in (

E

O

L

1

)

+1

) 9 an!-periodic solution u in

E

O(R)",

in the case that P is of type K = f0g,(that is,P is a local operator).

We can nd a neighborhood V

0

D

1

+i R of +1,such that

u

0

2

E

O

L

1(V

0

) and that Pu

0

= f on V

0

.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 19/25

Results

Idea of the proof

idea of the proof (simplest case),2

u

0

is a solution on V

0

,and bounded there.

T

!

u

0

is a solution on V

0

+f!g.

T

2

!

u

0

= T

2!

u is a solution on V

0

+f2!g.

+1

V

0

u

0

//

OO

We dene v

k

:=

1

k

P

k1

j=0

T

j

!

u

0

,and consider a bounded sequence fv

k

g

k

in

E

O

L

1(V) on a domain V = V

1

+[0;!] V

0

\C with some convex

domain V

1

V

0

\C.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 20/25

Results

Idea of the proof

idea of the proof (simplest case),2

u

0

is a solution on V

0

,and bounded there.

T

!

u

0

is a solution on V

0

+f!g.

T

2

!

u

0

= T

2!

u is a solution on V

0

+f2!g.

+1

V

0

u

0

:::::::

::

::

:

:

:

::

:

:

:

::

:

::::::::

T

!

u

0

!

oo

//

OO

We dene v

k

:=

1

k

P

k1

j=0

T

j

!

u

0

,and consider a bounded sequence fv

k

g

k

in

E

O

L

1(V) on a domain V = V

1

+[0;!] V

0

\C with some convex

domain V

1

V

0

\C.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 20/25

Results

Idea of the proof

idea of the proof (simplest case),2

u

0

is a solution on V

0

,and bounded there.

T

!

u

0

is a solution on V

0

+f!g.

T

2

!

u

0

= T

2!

u is a solution on V

0

+f2!g.

+1

V

0

u

0

:::::::

::

::

:

:

:

::

:

:

:

::

:

::::::::

T

!

u

0

:::::::

::

::

:

:

:

::

:

:

:

::

:

::::::::

T

2

!

u

0

!

oo

//

OO

We dene v

k

:=

1

k

P

k1

j=0

T

j

!

u

0

,and consider a bounded sequence fv

k

g

k

in

E

O

L

1(V) on a domain V = V

1

+[0;!] V

0

\C with some convex

domain V

1

V

0

\C.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 20/25

Results

Idea of the proof

idea of the proof (simplest case),2

u

0

is a solution on V

0

,and bounded there.

T

!

u

0

is a solution on V

0

+f!g.

T

2

!

u

0

= T

2!

u is a solution on V

0

+f2!g.

+1

V

0

u

0

:::::::

::

::

:

:

:

::

:

:

:

::

:

::::::::

T

!

u

0

:::::::

::

::

:

:

:

::

:

:

:

::

:

::::::::

T

2

!

u

0

:::::::

::

::

:

:

:

::

:

:

:

::

:

::::::::

T

j

!

u

0

!

oo

//

OO

We dene v

k

:=

1

k

P

k1

j=0

T

j

!

u

0

,and consider a bounded sequence fv

k

g

k

in

E

O

L

1(V) on a domain V = V

1

+[0;!] V

0

\C with some convex

domain V

1

V

0

\C.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 20/25

Results

Idea of the proof

idea of the proof (simplest case),2

u

0

is a solution on V

0

,and bounded there.

T

!

u

0

is a solution on V

0

+f!g.

T

2

!

u

0

= T

2!

u is a solution on V

0

+f2!g.

+1

V

0

u

0

:::::::

::

::

:

:

:

::

:

:

:

::

:

::::::::

T

!

u

0

:::::::

::

::

:

:

:

::

:

:

:

::

:

::::::::

T

2

!

u

0

:::::::

::

::

:

:

:

::

:

:

:

::

:

::::::::

T

j

!

u

0

fv

k

g

k

:9subseq.

!periodic sol.

V

!

oo

//

OO

We dene v

k

:=

1

k

P

k1

j=0

T

j

!

u

0

,and consider a bounded sequence fv

k

g

k

in

E

O

L

1(V) on a domain V = V

1

+[0;!] V

0

\C with some convex

domain V

1

V

0

\C.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 20/25

Results

Idea of the proof

idea of the proof (simplest case),3

Scalar (or sequential Montel) case:

We can show a Montel type lemma for

E

O(V).

By applying it,we can choose a convergent subsequence from fv

k

g

k

,

and its limit is an!-periodic solution.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 21/25

Results

Notes for re exive valued cases

notes on re exive locally convex spaces

Consider the case that E is a re exive locally convex space.We denote

by E

w

,the space E endowed with the weak topology,and

by E

0

,the dual space of E endowed with the strong topology.

We can use the following facts,(V C):

Fact

E

O =

E

w

O,algebraically.

E

O admits (a weak form of) the Kothe duality.

P

V

is sequentially closed as a map

E

w

O(V +K)!

E

w

O(V).

E

O(V) admits (a weak form of) Montel type lemma

Here...

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 22/25

Results

Notes for re exive valued cases

the Kothe duality

E:a re exive locally convex space,E

0

:its strong dual.

For a compact L C and an open V L,we dene a bilinear form

h;i

L

:

E

0

O(V n L)

E

O(L)!C;h';vi

L

:=

Z

'(w)(v(w))dw;

by taking a suitable contour .

Theorem (a weak form of the Kothe duality)

The bilinear form h;i

L

induces the isomorphisms between vector spaces:

E

0

O(V n L)

E

0

O(V)

!(

E

O(L))

0

;

E

0

O

(P

1

n L)

!(

E

O(L))

0

:

Here

E

0

O

(P

1

n L) denotes f'2

E

0

O(Cn L) j lim

jwj!1

'(w) = 0g.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 23/25

Results

Notes for re exive valued cases

Montel type lemma

E:re exive,V C:open,h;i

L

:

E

0

O

(P

1

n L)

E

O(L)!C:a bilinear

form in the Kothe duality.

Lemma (a weak form of the Montel type lemma)

Let ff

n

g

n

be a bounded sequence in

E

O(V).Then,there exists f 2

E

O(V)

satisfying the following property:For any compact L b V and any

F 2

E

0

O

(P

1

n L),we can take a subsequence fn(k)g

k

such that

lim

k!1

hF;f

n(k)

i

L

= hF;f i

L

:

Note:When E is a re exive Banach space,the subsequence fn(k)g

k

can

be taken independently of L and F.But in the general case,it may

depend on the choice of L and F.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 24/25

Results

Notes for re exive valued cases

idea of the proof (re exive case)

Re exive Banach case:

E

w

admits the sequential Montel property.

P

V

is sequentially closed in

E

w

O topologies.

Therefore,a similar proof for

E

w

O(V) instead of

E

O(V) can be applied.

Re exive case:

We can not expect the sequential Montel property for E

w

.But nevertheless

we can bypass that part by using the weak form of Montel type lemma.

Y.Okada (Chiba Univ.)

Massera type theorems

Aug.27,2013,Bedlewo 25/25

Thank you for your attention.

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