# Yasunori Okada, Massera type theorems in vector-valued ... - IM PAN

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8 Οκτ 2013 (πριν από 4 χρόνια και 7 μήνες)

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Massera type theorems in vector-valued analytic
functions and hyperfunctions
Aug.27,2013,
FASDE III,Bedlewo
1
Introduction
Classical Massera type theorems
Brief overview
2
Bounded hyperfunctions at innity
Bounded hyperfunctions at innity
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
Massera type theorems
Aug.27,2013,Bedlewo 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 dierential 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
mm
and f:R!R
m
are 1-periodic and continuous.
Then,the existence of a bounded solution in the future (i.e.,a solution
dened 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.
Massera type theorems
Aug.27,2013,Bedlewo 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 dierential 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.
Massera type theorems
Aug.27,2013,Bedlewo 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?
Massera type theorems
Aug.27,2013,Bedlewo 4/25
Introduction
Brief overview
previous results
We constructed
B
L
1
:the sheaf of bounded hyperfunctions at innity on D
1
:= Rtf1g,
(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 identied 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.)
Massera type theorems
Aug.27,2013,Bedlewo 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-dierential equations with innite delay,under an
mentioned in FASDE II,Aug.2011.
But,today,we do not focus on this direction.
Massera type theorems
Aug.27,2013,Bedlewo 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:
Denition (sequential Montel property)
(M) Any bounded sequence in E has a convergent subsequence.
or,when E is re exive.
Statement in re exive case
Massera type theorems
Aug.27,2013,Bedlewo 7/25
Bounded hyperfunctions at innity
boundedness for hyperfunctions
D
1
:= Rt f1g:a compactication 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 innity 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.
Massera type theorems
Aug.27,2013,Bedlewo 8/25
Bounded hyperfunctions at innity
boundedness for hyperfunctions
D
1
:= Rt f1g:a compactication 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 innity 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.
Massera type theorems
Aug.27,2013,Bedlewo 8/25
Bounded hyperfunctions at innity
Bounded hyperfunctions at innity
bounded hyperfunctions at innity
Denition (sheaves O
L
1 and B
L
1)
1
The sheaf O
L
1 on D
1
+i R is dened 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 dened 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 Frechet topology.
Massera type theorems
Aug.27,2013,Bedlewo 9/25
Bounded hyperfunctions at innity
Bounded hyperfunctions at innity
vector valued variants
E:a sequentially complete Hausdor locally convex space,
E
O:the sheaf of E-valued holomorphic functions on C.
Denition (sheaves
E
O
L
1
and
E
B
L
1
)
1
The sheaf
E
O
L
1
on D
1
+i R is dened 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 dened 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.
Massera type theorems
Aug.27,2013,Bedlewo 10/25
Bounded hyperfunctions at innity
Bounded hyperfunctions at innity
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
identied with the space B
L
1 of bounded hyperfunctions due to
Chung-Kim-Lee (2000,Proc.AMS.128).
Massera type theorems
Aug.27,2013,Bedlewo 11/25
Bounded hyperfunctions at innity
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
VU
:a family of linear continuous maps
P
V
:
E
O
L
1(V +K)!
E
O
L
1(V):
Denition (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
)
Massera type theorems
Aug.27,2013,Bedlewo 12/25
Bounded hyperfunctions at innity
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.
Massera type theorems
Aug.27,2013,Bedlewo 13/25
Bounded hyperfunctions at innity
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
dierential operator
P
m
j=0
a
j
(t)@
j
t
with coecients 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.
dierence 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.
Massera type theorems
Aug.27,2013,Bedlewo 14/25
Bounded hyperfunctions at innity
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
!-periodic boundary value representation.
Massera type theorems
Aug.27,2013,Bedlewo 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
Massera type theorems
Aug.27,2013,Bedlewo 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
Massera type theorems
Aug.27,2013,Bedlewo 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.
Massera type theorems
Aug.27,2013,Bedlewo 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
.
Massera type theorems
Aug.27,2013,Bedlewo 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
.
Massera type theorems
Aug.27,2013,Bedlewo 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
+f2!g.
  
+1
V
0
u
0
//
OO
We dene v
k
:=
1
k
P
k1
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.
Massera type theorems
Aug.27,2013,Bedlewo 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
+f2!g.
  
+1
V
0
u
0
:::::::
::
::
:
:
:
::
:
:
:
::
:
::::::::
T
!
u
0
!
oo
//
OO
We dene v
k
:=
1
k
P
k1
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.
Massera type theorems
Aug.27,2013,Bedlewo 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
+f2!g.
  
+1
V
0
u
0
:::::::
::
::
:
:
:
::
:
:
:
::
:
::::::::
T
!
u
0
:::::::
::
::
:
:
:
::
:
:
:
::
:
::::::::
T
2
!
u
0
!
oo
//
OO
We dene v
k
:=
1
k
P
k1
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.
Massera type theorems
Aug.27,2013,Bedlewo 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
+f2!g.
  
+1
V
0
u
0
:::::::
::
::
:
:
:
::
:
:
:
::
:
::::::::
T
!
u
0
:::::::
::
::
:
:
:
::
:
:
:
::
:
::::::::
T
2
!
u
0
:::::::
::
::
:
:
:
::
:
:
:
::
:
::::::::
T
j
!
u
0
!
oo
//
OO
We dene v
k
:=
1
k
P
k1
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.
Massera type theorems
Aug.27,2013,Bedlewo 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
+f2!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 dene v
k
:=
1
k
P
k1
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.
Massera type theorems
Aug.27,2013,Bedlewo 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.
Massera type theorems
Aug.27,2013,Bedlewo 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 Kothe 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...
Massera type theorems
Aug.27,2013,Bedlewo 22/25
Results
Notes for re exive valued cases
the Kothe duality
E:a re exive locally convex space,E
0
:its strong dual.
For a compact L  C and an open V  L,we dene 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 Kothe 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.
Massera type theorems
Aug.27,2013,Bedlewo 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 Kothe 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.
Massera type theorems
Aug.27,2013,Bedlewo 24/25
Results
Notes for re exive valued cases
idea of the proof (re exive case)
Re exive Banach case:
E
w
P
V
is sequentially closed in
E
w
O topologies.
Therefore,a similar proof for
E
w
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.
Massera type theorems
Aug.27,2013,Bedlewo 25/25
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