Pub
.
Ma
.
UAB
Nó
13,
Juliol
1979
CRITICAL
CASE
FOR
PERIODIC
SOLUTIONS
OF
A
CLASSOF
NEUTRAL
EQUATIONSWITH
A
SMALL
PARAMETER
Ped oMa ínez
Amo es
Secciónde
Ma emá icas
-
Uni e sidad
de
G anada
In oduc ion
In
his
palie
we s udy
a
class
o
neu al
unc io
nal
di e en ialequa ionswhich a ises
om
a
coupled
sys
em
o
di e en ial-di e ence
and
o dina y
di
e ence
equa
ions
ha
occu
in
a ious
applica ions,
as
elec ical
ci
cui swi hlossless
ansmision
lines,
[11
.
In
[101,
[151
is
s udied
he
s abili yo
such
sys em
and
in
[131
a e
gi en
-
condi ions
o
he
exis ence
o
pe iodic
solu ions
.
In
his
pape ,
we
conside
a
nonlinea
sys em
wi h
a
small
pa ame--
e
and
we
s udy
he
exis en e
o
pe iodic
solu ionswhen
-
he
co esponding
linea
sys em
can
ha e
pe iodic
solu ions
(c i ical
case)
.
This
is
done
by
applying
he
me hod
o
Ha-
le
[5,61
andi s
ex ension
o
delay
di e en ial
equa ions
-
(see
[141)
o
he
neu al
equa ions
ob ained
om
he
coupled
sys em
a e
aking
in o
accoun
he
esul so
[101
.
l
.
No a ion
and
summa y
o
known esul s
Le
E
n
be
a
complex
n-dimensional
linea ec o
space
wi h
no m
1
.1
andle
be
a
ixed
posi i e
numbe
.
C =
C

( [
-
,

o
1, E
n) is
he
spaceo
con inuous
unc ions
1
:
[- ,
0
1
--~ E
n
wi h
no m
I~¡=
sup~

~(9)1
:9 ,
o1
~~
Suppose
D,
L
a e
bounded
linea ope a o s om
C
o
E
n
,

0
=
H
(0)
-
[dl+
(A)~~
(9)
-
0
L
(
1
)
=
-
[d
1
(9)]
(8)
whe e

li

is
an
n
x
n
ma ix,
de
li
=~
0,
,
~
a e
n
xn
ma ix
Func ions
o
.
bounded a ia ion
on
[- ,
0] wi h
M
nona omic
a ze o
.
We
assume
N
has
no
singula pa
.
I
.

x

is
a
con inuous
unc ion
mapping
[a'-
,

)
in o
E
n
,
hen
o
any
E[0,w)we de ine
x
in
C
by
-

-
x
(9)
= x
(
+
A)
,9E[- ,

0]
.
An
au onomouslinea homoge-
-
neousneu al un ionaldi e en ial
equa ion
(N F D E
)
is
-
de
ined
o
be
d
d
D(x
)
=

L
(x
)
A
solu ion
x=
x
(¢)
o

(1 .]
.)
h ough
a
poin
JE
C

a

=
0
is
a
con inuous unc ion
aking
[- ,
A),

-
A>
0,
in o
E
n
such
ha

xo
=
,
D(x
)
is
con inuously
di--
e en iable
on
[0, A)
and
equa ion
(1
.1)
is sa i,s ied
on
-
his
in e al
.
I is
p o ed
in
[2,4]
ha
he e
is
a
unique
solu ion

x
(~)
h ough

and
x
(1)( )

is
con inuous
in

( ,
b
I
.
he
ans o ma ion
T( )
:
C
-+C
is
de ined
by
-
T
( )
~
= x
(1),
hen
i is
shown
in
[11]
ha
1
T
(
),
>,0
}
-
is
a
s onglycon inuoussemig oup
o
linea
ope a o s
wi h
he
in
ini esimal
gene a o A
:
-9(A)
-

C,

A

(
9)
_

(
9
),

-
and he
spec um

0'(A)
o
A
consis
o
hose
X
which
sa is y
de
62
0

0
O %)
=
0,

a(ñ)
-_
ñH
_~J

e
e
d~(9)
_
J

é6dj(9
_

-
The
undamen al
ma ixsolu ion
o
(1
.1)
is
de i-
ned
o be
he
n
x
n ma ix
solu ion
o
he
equa ion
D(X
)
=
I+J

L
(
X)
ds,
>
0
0
s
0
,
- c9<0
Xo (0)

0
(1
.6)

x
-X
o
G( )
=
T( - )[
-X
oG(C)]+

+JT
T( -s)X
o
F(s)ds-
J
e
.
(d
sT( -s)X
o
IG(s)
o
> ,
E
C,
whe e
i
is always
unde s ood
ha
he
in e
g alsin
(1
.6)
a e
ac uallyan
in eg als
in E
n
.
Fo mula
(1
.6)
sugge
he
changeo
a iables
x
-
Xo
G( )
=
z
,

-X
o
G
(T)
=
om
C
-
P
C
.

I
his
is
done,

equa ion
(1
.6
)
becomes

(1
.7)
z
=T
( -T) +

T
( -s)X
F(s)ds-
[d
s
T( -s),XJG(s)
.

T

o
De ini ion
1
.1
.
The
ope a o D
is
said
o be
s able
i
he-
e is
a
-4
>
0

such
ha
all
oo so
he
equa ions
de
D
(e
x .
I)
=
0
s is y
ReÁ<-a
.
I
D(+)
= H
j(0)
-
M«- ),
hen
D
is
s able
i
he
oo so
he
polynomialequa ion
de
(H
-PM)
=
0
sá is y

-
An
impo an
p ope yo
equa ion
(1
.1)
when
D
is
s ableis
he
ollowing
(see
[3])
:
I
D
is
s able,
hen
-
he e
is
a
cons an

aD<
0
such
ha
o
any
a
>
a
D
,
he e
-
a e
only
a
ini enimbe
o
oo so
de
(j(ñ)
=
0
wi h
-
-
Reñ)
a
.
Le
D
be
s able
.

I
A
=
~
:
de
á
(X)
=
0,

Re
X>
.0
},
hen
A
is
a
ini e
se
and
i
is
ollows
om
[11]
ha
he
space
C
can
be
descomposed
as
C=P
49
Q,
whe e
P,
Q
a e
-
subspaces
o
C
in a ian
unde
T( ),
he
space
P
is
ini e
dimensional
and
co esponda
o
he
ini ial alues
o
all
-
hose
solu ions
o
(1
.1)
which
a e
o
he
o m
p( )e
x
,

-
whe e

p( )
is
a
polynomial
in
and
ÁEA
.
I
í

is
a
basis
o
P
hen
o
e e
y
+E
P
he eexis a
a
ec o
aE
E

such
ha
+=í
a
.

En
his
case,
we
can
de ine
T( )+=
l
e
B
a,

-
6
3
I

F,
G
:
[0,
~)

--~
En
a e
con inuous,
a
nonhoimo-
:
geneous
linea
N
FD
E

is
de ined
as
(1
.3)

d

ID
(x )-G( »=

L(x
)+
F( )
.
A
solu ion
h ough
a

=
T

o
(1
.3)
is
de i-
ned
as
be o e
and
is known o exis on
[o'- ,
c
o)
.
The
a ia ion
o
cons an s o mula
o
(1
.3)
(see
[9])
s a es ha
he
solu ion
o
(1
.3)
h ough
( ,
)
is
gi-
en
by

+
(1
.4)
x( )
=
T( -(
-
)
1(0)
+
ue
X( -s)F(s)ds-
..
[d
s
X( -s)]G(s)-G( ),
o
>
l
T,
whe e
X
is
he
undamen al
ma ix
solu ion
gi en
by
(1
.2)
.
Equa ion
(1
.4)
can
be
w i en
as
(1
.5)

x( )
-X(o)G( )=T( -Q»)~(0)-X( -T)G(T)

o
>,('
.
Now,
le
PC
be
he
space
o
unc ions aking

-
[-
,0]
in o
E
n
which
a e
uni o mly
con inuouson
E
-
.0)
and
may
be
discon inuous
a ze o
.
Wi h
he
ma ixX
as
de ined
0
be o e,

i is
clea ha
PC=
C+
(
X0
),
whe e
(X
0
)
is
he
-
-
span
o
X
0
;
ha
is
any1,EP
C

is
gi en
as
^+
=
1+
X
0
b,

1
E
C,
n
bEE
.

We make
P
Cano med
ec o space
by
de ining
he
no m
1+1= max{/41,
b)
.
Le
us
de ine
x
(~
)
=
T
( )
~
,
whe e
iEP C

and

-
x(^ )

is
he
solu ion
o
(1
.1)
h ough
+
.
The
ope a o
-

-
T( )
:
PC
-
a
( unc ions
on
[- ,
0]
)
is
linea ,
bu
T( )
does
64
no
ake
P
C
--j
PC
.
I
is an ex ensiono
he
o iginal
semi
g oup
T
.( )
on
C
.
I
we
use
his
no a ion,
hen
he
a ia ion
o
cons an s
o mula
(1
.5)
can
be
w i en
as
whe e

B

is
an
n
x
n
g a ix
de ined
by

Aí
=
B
.

The
spec um
o B
is
A
.
I

C

is
decomposed
by
n
as

C =
P
G
Q
hen
equa--
ion
(1
.6)
is
equi alen
o
x -
Xp
G( )
=
T( -T)C,~P-
XpG(Q')]+
. .
T( -s)XpF(s)ds
-
-
[ds
T( -s)XP]G(s)
0
whe e
he
supe sc i s
P
and
Q
designa e
he
p ojec ions
o
-
he
co espondine unc ions
on o
he
subspaces
P
and
Q,
es-
pec i ely,
and
hey
can
be
de e minedby meanso adjoin
di-
e en ial
equa ion
o
(1
.1),
see
(111
.
2
.
The
li
nea
p oblem
x -
XQ
G( )
=
T( -~)[¢Q-
XQ
G(c)J
+
T
T( -s)
XQ
F(s)ds
-
(ds
T( -s)XQG(S)
0
1
In
his
sec ion,
we conside
he
sys em
a)
z( )
=
A1x( )+A2y( - )
b)
y( )-A
3
x( )+A
4
y( - )=0
whe e
x,
y
a e
n- ec o
andall
ma ices
a e
cons an s
.
Fo
any aEE
n,-- EC,
onecan
de ine
a
solu ion
o
(2
.1)
wi h

-
ini ial
alue
x
(o)
=
a,

y

=
^~
.

I we
de
ine,

C
=
C
([- ,
0],
E
n
)
0
(2
.2)
D
1
L
1
D, L
:

Enx

C
---->
E
nx
n'

D=
[D

L
-
0
2
7
D
1
(a,
- )=
a
D
2 (a,^ )
=' (0)
-
A
3
a -
A4
L
1
(a,°%)
=
A
l
a+A
2
+(- )

65

henequa ion
(2,1)
is
a
specialcase
o
he
NF
D E
(2
.3)
d
d
andone
ob ain
he
sys em
(2,1)
by
equi ing
ha
is
a
s cngly
con inuous
semig oupo
The
in ini esimal
gene-
_es o
e
o
T
(+)
í_-
0
0
D
(x (
),

y
)

=

L(x (
),
y
)
D
2
(a,^%)=
0
Equá ion
(2-3)
de inesa
semig oup
T( )
on
Enx
C
,
I
we
de-
ine
(E
n
xC)
0
=
«
a, )EE
n
xC
:
D2(a,^ )=0}
hen
(E
n
xC)
0
can
be
conside ed
as
a
Banach
spacee
Fu he mose,
o any
-

-
(a,
-+)E(E
n
xC)
0
,
he
solu ion
o

(2,3)
h ough
(a,- )

will
-
be
in
(E
nx
C)
0

since
i
co esponds
o
he
solu ion
o

(2
.1)
h ough(a,- )o
Consequen ly,
T

( )

de

T ( )
(

,

(E
n.
C)

-~

(En
.
C
)
0

0

0
n

1
(E
x
C)
0
ni esimal
gene a o
o
T( )¿One
shows
ha
T(A
0)=Ii1E4
::
Obse e
ha
=
A
l

whe e
A
is
he
in i
¡(E
nx
C) 0
~
I-A
1
de ~(ñ)=0},

o
(ñ)
=

-A
3
a

I
0 a
0 0
a
D(
a
,^~
.
~(O)
-A
3
a
-
A4
Y(- )

-A
3
I

^Y(0
)

0

A4
^¡'(-
)
de
=

H
~(O)
-M
4(-
)
a

A1 0
a
ío
A
L(a,~)
--
-
Y(O
)
-A3a-A4

~,(- )

0

0

.~((0)
+ 0

0
66
d-

Ni
(0)
+
P
~(- )o
-A
2
e

Á
I-A,e
X
duli
less han
1,
hen
D
is
s able
.
ix
solu ion
X( )
o
(2
.3)
as
I

X
=
(
X
11

X
121
~
_whe e

X
i
.,

i, j
=
1
9
2
1
a e
nxn
ma i
21
22

j

_
ces,
hen
X
mus be
a
solu ion
o
(2
.3)
wi h
he
ini ial
-
da a
speci ied
abo e
.
The e o e,
he
ma ices
X,
.
mus
sa--
1sJ
is y
(2
.4)
No ice
ha

(2
.4a)

impliesX
11
,

X
21
a e
solu ions
o

(2
.1)
.
The
unc ions
X12,
X22
do
no
sa is y
(2 .1 b)
.
This
implies
ha
he
a ia ion
o
X( )
sa is ies
he
sys em
(2
.1)
.
UsingLaplace
ans o m
o
he
same
ype
o
a gu--
men s
as in
Hale
()
pag
.
303,
onecan
p o e
he
ollowing
Lemma
2
.1
.

I
he
eigen alues
o
A
4
ha e
moduliless
han
1
and
all
oo s
o
de
á
(ñ)
=
0
sa is y
Reñ<-d<0,
hen
-
he e
a e
posi i econs an s
K,
o(
such ha
Ix
11
( )1,

1x21( )ls

I
X
ij
( )j_4
Ke
C(
s
a.e

>,0,

i,j=1,2
.
Now
we conside
henonhomogeneous
sys em
(2
.5)
Thus,
i
he
eigen alues
o
he
ma ix
A
4
ha e
mo-
Al so,
om
(1
.2)
.
we
can
de ine
he
undamen al
ma
I
0
X(9)
= H
1
o

(A
3Ie
=
0,
x0
(e)
=
o,

-
i6
9
<0
.
a)
D2
(x
11
( ),
x
21 .
)
=
o
,
>
"
o
b)
D2
(x
12 ( ),
x
22 .
)
=

,
>,
o
.
X( )=A
1
x( )+A
2
y( - )+ ( )
y( )
-A
3
x( )
-A
4
y( - )
-
g( )
=0
n
whe e
,
g
a e
con inuous
unc ions
om
[0,
-0)
o
E
.
Wi h
D,
L
de ined
as in
(2
.2),
sys em
(2
.5)
is
a
specialcase
o
he
N
F
D
E
(2
.6)

c {D
(w )-G( )}-
L(w )+F( )
whe e

w

=
col
(x
(
),

y

),

w

_
¢

col
(a,
^~
)E
E
nxC
=

,
-

o
G
=
col
(0,g)EEnxn$
F
=
col( ,
O)EE
nxn
and
one
ob ains
he
sys em
(2
.5)
om
(2
.6)
by
equi ing
ha
D
2
(a,y)=
g(0)
.
As in
Sec ion
1,
i
we ex end
he
de ini ion
o
-
T( ) o
E
n
x
(C
+
(
X_
)
)
de
Y,

hen
he
gene alsolu ion
o
.

-
(2
.6)
is
gi enby
he
a ia ion
o
cons an s
o mula
(2
.7)
w
-Xo
G( )
=
T( «-
X
0
G(0),+

T( -s)X
0
F(s)ds
-
0
-
[d
s
T( -s)
X0
]G(s)
.
o
This
las
o mula
sugges s
he
change
o
a iables
w
-
X0
G( )
=
z
s

J-
X0
G(O)
=5,

1
om'E
nx C

o

Y
.
I
his
is
done,
o mula
(24)
becomes

L
(2
.8)

z
T( )
+
T( -s)
X0
F(s)ds-
[ds
T( -s)X
0
]G(s~
>,0
.
Onecan
gi e
an
explici
decomposi ion
o
(2
.8)
by
using
he
adjoin equa ion
o
(2
.3)
.
Fo
his,
we w i e
- -
(2
.3)
in
he
o m
d
á ~H
w
( )
-
Mw( - )
)=
Nw( )
+P
w( - ),
w0
=JEE
nx
C
which
is
equi alen
o
(2
.9)
d
{w( )-Mw
( - )}=
Ñw( )+Pw( - )
lince
H- 1
M=M
and
whe e
H
1
N=N,
H
1
P=P
.
We de ine
he
adjoin
equa ion
o
(2
.9)
as
(2
.10)
d
j ( )-
.( + )
M}=
- ( )Ñ- ( + )
P,
O
= <GE
n
XC~
In
he
lame way,
we
may
w i e
(2
.6)
in
he
o m
68
(2
.11)

d
{w( )-Mw( - )-H
1
G( )}=
Nw( )+Pw( - )+H
1
F( )
Using
he
samea gumen s
as in
[7,13),
i
is
-
--
easily
shown
ha
i
he
eigen alues
o
A4
ha e
moduli
less
han
1
and
EnxC

is
-decomposed
by
/ =
I%
:
de ,&(a)
=0
,

Rein>
.0
as
P
eQ
henequa ion
(2
.8)
is
equi alen
o
(2
.12)

-
a)

z =
T( )

P+
O

T( -s)
Xó
H
1
F(s)
ds

-
-
[d

T( -s)
X
P
]
H
1
G(s)
0
s

0
b)
zQ=T( )3
Q
+
T( -s)
XóH
1
F(s)
ds
-
[d
s
T( -s)
Xo]H
_1 G(s)
0
whe e

JS

=
4
-XoG
(0),

4

=
o

i

z
=
íu
(
),

whe e

is
a
basis
o
P
and
T( )
~_

eB ,

-
he
spec um
o
B
is
11 .,
hen

u

sa is ¡es
he
equa ion
(2
.13)

ü( )=Bu
( )+^1(0)H
1
F( )+B
i(0)
H
1
G( ),
-oo4 coo
whe e
IP
is
a
basis
o he
ini ial alues
o
hose
solu-
-
ions
o
(2
.10) o
he
o m
p( )e
-
~
,
p a
polynomial,IEA
.
I 9-
is
he
Banachspaceo con inuous
and

-
T-pe iodic
unc ions
wi h
no m
1

=

sup
1
( (
)j
,
E[0,
T]},
-
hen
onecan
s a e
he
heo em
on
he
F edholm
al e na i e
o
pe iodic
solu ions
as
:
Theo em
2
.1,03]
I
he
eigen alues
o
A
4
ha e
moduliless
han
1
and
,
gE
-9,
hensys em
(2
.5)
has
a
solu ion
in
i
and
only
i
(2
.14)

0
( )H
.
1
F(s)ds-
jd
( )]H
1
G(s)=0
.
I
he
unc ion
z
*
(a,
£
)
in
Lemma
3.1

is
di e en iable

-
wi h espec
o
a,
we
can
apply
he
implici
unc ion
heo
em
o
(3
.9)
o
ha e
a =
a(6
)
.
g
=
£
g,
whe e

( ,

g( ,

)
ble
in
~
and
we de ine
In
pa icula ,
i£
=
F
-F
.,
a e
con inuoslydi e en ia
(3
.
.10)

F
1
(a,
E )
=
T
X11
(- )
H
1
( 9
w
(a,
F
.)
)

d
+
o
TX
12
Fi
1
g
( ,

w*
(a,
E
)
)
d
=

0
0
F
2(a,
E
.
)

=
,
jT X
21
(-
)
H

1
( ,
w
(a,
E)

)

d

+
o
T
1
+
0

X
22
H

g
( ,
w
(
a,
b)
d
=
0
1
hen,
we ha e
he
ollowing heo em
o
he
i s app oxi-
ma ion
Theo em
3
.2
.
Le
,
g
sa is y
he
aboyé
condi ions
.
I
-
he e
is
an

ao
, I
í
eB ao
1
4
T
,

such
ha
Ó(F,F)
l
(3
.11)

F1(a0,
0)=0,
F
2
(a
o
,
0)=0,

de

1

2
(a
0
,
0)
J
9E
0
áa
hen
he e
is an
E
.7
0
such ha sys em'(3
.1)
has
a
T-pe io
o

-
dic
solu ionw (a
0
,
E
)
,
01IE)
<
E
0
,
con inuous
in
6
:
and
B
w
(a
0
,
0
)
_

e

a
0
.
P oo
.

The
hypo hesis
(3
.11)
and he
implici unc ion
heo
em
imply he e
is an

i
o>
0,
such
ha
equa ions
(3
.10) ha
-
e
a
solu ion

a
'
(E),1
a(6)1-<a,

0-IEI<¿
0
.
Theo em
3.1
.

-
implies
he
esse ions
o
he
heo em
.

REFE
RENCES
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R
.
Small
-
signal
s abili y
c i e ion
o
elec--
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lossles
ansmision
lines
.
I
.
B
.M
.
J
.
Res
.
De elop
.
12(1968),
431-440-
[2]
C uz,
M
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and
J
.
Hale
.
Exis ence,
uniqueness
and
con i-
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nuousdependence
o
he edi a ysys ems
.
Annaly
Ma h
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Pu
a
Appl
.
85
(1970),
63-82
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C uz,
11
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and
J
.
Hale
.
Exponen ial
es ima es
and he
-
saddle
poin
o
neu al
unc ional
di e en ialequa-
ions
.
J
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Ma h
.
Anal
.
Appl
.
34
(1971),
267-288
.
Hale,
J
.
Fo wa d
and
backwa d
con inua ion
o
neu al
di
e en ial
equa ions
.
J
.
Di
e en ial
Equs
.
9(1971)
168-181
.
Hale,
J
.
O dina y
di
e en ial
equa ions
.
In e science,
1969
.
(6j
Hale,
J
.
Oscilla ionsin nonlinea
sys ems
.
McG aw-Hill,
1963
.
Hale,
J
.
Theo yo
unc ional
di
e en ial
equa ions
.
Appl
.
Ma h
.
Se¡
.
Vol
.
Sp inge -Ve lag,
1977
.
(8)
Hale,
J
.
C i ical
cases
o
neu al
unc ional
di e en-
ial
equa ions
.
J
.
Di e en ialEqus
.
10(1971),
59-82
.
Hale,
J
.
and
M
.
C uz
.
Asymp o ic
beha io o neu al

-
unc ional
di e en ial
equa ions
.
A ch
.
Ra
.
Mech
.
Anal
.
34(1969),
331-353-
[10)
Hale,
J
.
and
P
.
Ma inez-Amo es
.
S abili y
in
neu al
equa ions
.
J
.
Nonl
.
Anal
.
-
Theo y
Ma h
.
Appl
.
1(1977),
-
161-173
"
[111
Hale,
J
.
and
K
.
Meye
.

A
class
o
unc ional
di
e en-
ial
equa ions
.
Memoi s
Am
.
Ma h
.
Soc
.
No,76,
1967-
[121
Hen y,
D
.

Linea
au onomous
neu al unc ional
di
e-
en ial
equa ions
.
J
.
Di
e en ial
Equs
.
15(1974),
106
-128
.
(13)
Ma inez-Amo es,
P
.
Pe iodicsolu ions
o
coupled
sys ems
o
di
e en ial-di
e ence
and
di
e ence
equa ions
.
To
appea
in
Annly
Ma h
.
Pu a
Appl
.
(14)
Pe elló,
C
.

Pe iodicsolu ionso
di
e en ial
equa--
ionswi h
ime
lag
con aining
a
small
pa ame e
.
J
.
-
Di
e en ial
Equs
.
4(1968),
160-175
.
(153
Ras an,
V
.
Absolu e
s abili y
o
a
class
o
con ol
-
sys em
desc ibed
by
unc ionaldi
e en ial
equa ions
o
neu al
ype
.
Equa ionsDi e en ielles
e
Fan
. ione
lles
Nonlinai es
.
Iie man,
1973
"