Massera type theorems in vectorvalued 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 1periodic 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 1periodic solution.
Note:Since periodic C
1
functions are bounded,we have the equivalence:
9 a bounded solution in the future.() 9 a 1periodic 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:ChowHale (1974,FE.
17),HinoMurakami (1989,Lect.Notes Pure Appl.Math.118),etc.
Banach valued,abstract settings:ShinNaito (1999,JDE.153),
NaitoNguyenMiyazakiShin (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 realanalytic in t,
r > 0:a constant,(representing the\nite delay").
Note:This result could be extended to some classes of equations,
containing integrodierential equations with innite delay,under an
additional assumption,socalled 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 Evalued 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 Evalued variant of O
L
1
,
E
B
L
1 the Evalued 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 Evalued variant of O
L
1
,
E
B
L
1 the Evalued 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 Evalued 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
ChungKimLee (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 Evalued 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 Evalued 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.
S.N.Chow and J.K.Hale.
Strongly limitcompact maps.
Funkcial.Ekvac.,17:31{38,1974.
S.Y.Chung,D.Kim,and E.G.Lee.
Periodic hyperfunctions and Fourier series.
Proc.Amer.Math.Soc.,128(8):2421{2430,2000.
Y.Hino and S.Murakami.
Periodic solutions of a linear Volterra system.
In Dierential equations (Xanthi,1987),volume 118 of Lecture Notes
in Pure and Appl.Math.,pages 319{326.Dekker,New York,1989.
J.L.Massera.
The existence of periodic solutions of systems of dierential equations.
Duke Math.J.,17:457{475,1950.
T.Naito,N.V.Minh,R.Miyazaki,and J.S.Shin.
A decomposition theorem for bounded solutions and the existence of
periodic solutions of periodic dierential equations.
J.Dierential Equations,160(1):263{282,2000.
Y.Okada.
Massera criterion for linear functional equations in a framework of
hyperfunctions.
J.Math.Sci.Univ.Tokyo,15(1):15{51,2008.
M.Sato.
Theory of hyperfunctions.I.
J.Fac.Sci.Univ.Tokyo.Sect.I,8:139{193,1959.
M.Sato.
Theory of hyperfunctions.II.
J.Fac.Sci.Univ.Tokyo Sect.I,8:387{437,1960.
J.S.Shin and T.Naito.
SemiFredholm operators and periodic solutions for linear
functionaldierential equations.
J.Dierential Equations,153(2):407{441,1999.
O.Zubelevich.
A note on theorem of Massera.
Regul.Chaotic Dyn.,11(4):475{481,2006.
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