Publicacions
Ma emá iques,
Vol
36
(1992),
3-18
.
Abs ac
OSCILLATIONS
OF
THE
SOLUTIONS
OF
NONLINEAR
HYPERBOLIC
EQUATIONS
OF
NEUTRAL
TYPE
D
.P
.
MISHEV
AND
D
.D
.
BAINOV
In
his
pape
nonlinea
hype bolic
equa ions
o
neu al
ypé
o
he
o m
2
(~
2
[
u
(
X
'
)
+
a( )u(x'
-
T)]
-
[Au(x,
)
+
11( )AU(x,
-
u)]
+
c(x,
,
u)
=
(X,
),

(x,
)
E
9
x
(0,
oo)
-
G,
a e
conside ed,
we e
,
a
=
cons
>
0,
wi h
bounda y
condi ions
an
+7(x,
)u
=
g(x,
),

(x,
)
E
as
x
[0,
oo)
o
u
=
0,

(x,
)
E
asl
x
[O,oo)
.
Unde
ce ain
cons ain s
on
he
coe icien s
o
he
equa ion
and
he
bounda y
condi ions,
su icien
condi ions
o
oscilla ion
o
he
solu ions
o
he
p oblems
conside ed a e
ob ained
.
1
.
In oduc ion
In
he
las
ew
yea s
esul s
ela ed
o
he
oscilla o y
p ope ies
and
asymp o ic
beha iou
o
he
solu ions
o
some
classes o
hype bolic
equa-
ions
we e
published
.
We
shall
men ion
especially
he
wo k
o
K
.
K ei h,
T
.
Kusano
and
N
.
Yoshida
[4]
in
which
su icien
condi ions
o
oscilla ion
o
he
solu ions
o
he
nonlinea
hype bolic
equa ion
u
-
Au
+
c(x,
,
u)
=
(x,
)
conside ed
in
a
cylind ical
domain
a e
ob ained
.
Oscilla o y
p ope ies
o
he
solu ions
o
hype bolic
di e en ial
equa ions
wi h
a
de ia ing
a -
gumen
we e
in es iga ed
in
he
wo ks
o
D
.
Geo giou,
K
.
K ei h
[2],
D
.
Geo giou
[3]
.
Hype bolic
di e en ial
equa ions
wi h
maxima
we e
in es-
iga ed
in
he
wo k
o
D
.
Mishe
[6]
and
some
condi ions
o oscilla ion
o
he
solu ions
o
hype bolic
equa ions
o
neu al
ype
we e
ob ained
by
D
.
Mishe and
D
.
Baino
in
[7],
[8]
.
The
p esen
in es iga ion
is
suppo ed
by
he
Minis y
o
Cul u e,
Science
and
Edu-
ca ion
o
People's
Republic
o
Bulga ia
unde
G an
61
.
4

D
.P
.
MISHE ,
D
.D
.
BAINOV
2
.
P elimina y
no es
In
he
p esen
pape
su icien
condi ions
o
oscilla ion o
he
solu ions
o
nonlinea
hype bolic
equa ions
o
neu al
ype
o
he
o m
a
(1)

e a
[u(x,
)
+
A( )u(x,
-
T)]
-
[DU(x,
)
+
p,( )DU(x,
-
Q)]+
+
c(x,
,
u)
=
(x,
),

(x,
)
E
SZ
x
(0,
oo)
-
G,
n
a e ob ained,
whe e
T,
Q
=
cons
>
0,
niu(x,
)
_

u
.,
.,
(x,
)
and
9
is
-i
a
bounded
domain
in
R'n
wi h
a
piecewise
smoo h
bounda y
.
Conside
bounda y
condi ions
o
he
o in
(2)

~n+
~(x,
)u
=
g(x,
),

(x,
)
E
&SZ
x
[0,
00)
(3)

u
=
0,

(x,
)
E
áQ
x
[0,
00)
We
shall
say
ha
condi ions (H) a e
sa is ied
i
he
ollowing
condi ions
hold
:
H1
.
~( )
E
C
a([0,
00)
;
[0,
00
)),
p,
( )
E
C([0,
00)
;
R),
H2
.

c(x,
,
u)
E
C(G
x
R
;
R),
113
.

c(x,
,
-u)
=
-c(x,
,
u),

(x,
,
u)
E
G
x
(0,
oo),
114
.

c(x,
,
u)
>
p( )
-
h(u),

(x,
,
u)
E
G
x
.
(0,
oo),
whe e
p( )
is
a
con inuous
and
posi i e
unc ion
in
he
in e al
(0,
00)
and
h(u)
is
a
con inuous,
posi i e
and
con ex
unc ion
in
he
same
in e al
(
0
,
00
)
.
H5
.

(x,
)
E
C(G
;
IEB)
116
.

g
(x,
)
E
C
(8Q
x
[0,
oc)
;
IR)
H7
.
-
y(x,
,)
E
C(09
x
[0,
oc)
;
[0,
o0))
.
De ini ion
1
.

The
solu ion
u(x, )
E
C
a
(G)
n
Cl(G)
o
p oblem
(1),
(2)
((1,
(3))
is
said o
oscilla e
in
he
domain
G
i
o
any
posi i e
numbe
p,
he e
exis s a
poin
(xo,
o)
E
9
x
[p,,
0o),
such
ha
he
equali y
u(xo,
o)
=
0
holds
.
In
he
subsequen
heo ems
su icien
condi ions
o
oscilla ion
o
he
solu ions
o
p obem
(1),
(2)
and
(1),,
(3)
in
he
domain
G
a e
ob ained
.
We
shall
no e
ha
in
he
wo k
o
K
.
K ei h,
T
.
Kusano,
N
.
Yoshida
[4]
condi ions
a e
ob ained
o
he
oscilla ion
o
he
solu ions
only
o
p oblem
(1),
(2)
in
he
case
when
A
( ) -- 0,
p( )
--
0
and
-y(x,
)
-
0
.
NONLINEAR
HYPERBOLIC
EQUATIONS
OF
NEUTRAL
TYPE

5
In oduce
he
ollowing
no a ion
:
whe e
191
=
sz
dx
.
Wi h
any
solu ion
u(x, )
E
C
2
(G)
1
C
l
(G)
o
p oblem
(1),
(2)
we
associa e
he
unc ion
Lemma
1
.
Le ,
condi ions
(H)
hold
and
le
u(x,
)
be
a
posi i e
solu-
ion
o
p oblem
(1),
(2)
in
he
domain
G
.
Then
he
unc ion
( )
de ined
by
(5)
sa is ies
he
di e en ial
inequali y o
neu al
ype
2
(6)

d 2
[ ( )
+
A( ) (
-
,
)]
+
p( )h( ( ))
<
G( )
+
p( )G(
-
a)
+
F( ),
_>
o,
whe e
o
is
a
su icien ly
la ge
posi i e
numbe
.
P oo£- Le
u(x,
)
be a
posi i e
solu ion
in
he
domain
G
o
p oblem
(1),
(2)
and
o
=
max{T, a}
.
Then
u(x,
-
- )
>
0and
u(x,
-
a)
>
0
o
(x, )
E
52
x
[ o,
oc)
.
We
in eg a e
bo h
sides
o
equa ion
(1)
wi h
espec
o
x
o e
he
domain
9
and
ob ain
o
?
o
:
( )
=
191
1

-
1~u
(x,
)
dx,

>
0
.
j
2

u(x,
)
dx
+
A
( )

u(x,
-
- )
dx
J
-
~~
áu(x,
)
dx+
+
p,
( )

Du(x,
-
a)
dx]
+

c(x,
,
u)
dx
=

,
(
.x,
)
dx
.
in

9
Rom
G een's o mula
and
condi ion
H7
i
ollows
ha
Du(x,
)
dx
=
ao
án
ds
=
~~
[g(x,
)
-
-y(x, )u]
ds
<
~~
g(x,
)
ds
(9)

áu(x,
-
a)
dx
=

&u
(x,
-
a)
ds
=
9

lasa
8n
=
~~[g(x,
-
a)
-
7(x,
-
a)
-
u(x,
-
a)]
ds
<_<
~
sa
g(x
'
-
a)
ds
(4)
F
( )
1
=
(x,
)
dx,
>
0,
(5)
G
( )
=
IS2I
~~
g
(x,
)
ds,
>
0,
6

D
.P
.
Mis¡¡ ,
D
.D
.
BAINOV
Mo eo e ,
om
condi ion
H4
and
Jensen's
inequali y
i
ollows
ha
(10)

c(x,
,
u)
dx
->
p( )
h(u(x,
))
dx
>_
s

-
>_
p( )h
( o
u(x,
)
dx

dx)
-1/

dx
=
p( )
-
h( ( ))
-
1521
Using
(8)-(10)
and
condi ion
H1,
om
(7)
we
ob ain
z
d 2
[ ( )
+
( ) (
-
,
)]
<_
G( )
+
c( )G(
-
)
+
F( )
-
p( )

h( ( )),
which
p o es
Lemma
1
.
3
.
Main
esul s
Theo em
1
.
Le condi ions
(H)
hold
and
le
he
di e en ial
inequal-
i ies
o
neu al
ype
(11)
2
d 2
[ ( )
+
A( ) (
-
- )]
+
p( )
-
h( ( ))
<_-
G( )
+
u( )G(
-
o,)
+
F( )
(12)
z
d 2
[ ( )
+
A( ) (
-
T)]
+
p( )
-
h( ( ))
<_
G( )
-
h( )
-
G(
-
Q)
-
F( )
ha e
no
e en ually
posi i e
solu ions
.
Then
each
solu ion u(x,
)
o
p ob-
lem
(1),
(2)
oscilla es
in he
domain
G
.
P oó%
Le
p >
0
be
a
posi i e
numbe
.
Suppose
ha
he
asse ion
o
he
heo em
is
no
ue
and
le
u(x,
)
be
a
solu ion
o
p oblem
(1),
(2)
wi hou
ze oes
in
he
domain
G,
=
9
x
[p,,
oo)
.
I
u(x,
)
>
0
o
(x,
)
E
G
p
,
hen om
Lemma
1
i
ollows
ha
he
unc ion
( )
de ined
by
(5)
is
a
posi i e solu ion
o
inequali y
(11)
o
>--
o
+
ju,
Le
i is
an
e en ually
posi i e solu ion
o
(11)
which
con adic s
he
assump ion
o
he
heo em
.
I
u(x,
)
<
0 o
(x,
)
E
G,,,
hen
he
unc ion
-u(x,
) is
a
posi i e solu ion
o
he
p oblem
2
á 2
[u
+
A( )u(x,
-
T)]
-
[Du
+,
( )DU(x,
-
Q)]+
+
c(x,
,
u)
=-
(x,
),
(
.x,
)
E
G
ón
+
-y(x,
)u
=-
g(x,
),

(x,
)
E
óg
x
[0,
oo)
.
NONLINEAR
HYPERBOLIC
EQUATIONS
OF
NEUTRAL
TYPE

7
F~om
Lemma
1 i
ollows
ha
he
unc ion
Tg-'T
ez(-u(x,
))
dx
is
a posi-
i e
solu ion
o
inequali y
(12)
o
_>-
o
+
c
which
also
con adic s
he
assump ion
o
he
heo em
.
Thus
Theo em
1is
p o ed
.
Now
we
shall
in es iga ge
he
oscilla o y
p ope ies
o
he
solu ions
o
p oblem
(1),
(3)
.
Conside
in
he
domain
9
l e
ollowing
Di ichle
p oblem
:
DU
+
caU
=
0
in
S2
UlaQ
=
0
whe e
a =
cons
.
I is
well
known
[1]
ha
he
smalles
eigen alue
ao
is
posi i e
and
he
co esponding
eigen unc ion
W(x)
can
be
chosen
o
sa is y
he
inequali y
W(x)
>
0
o
x
E
9
.
Wi h
any
solu ion
u(x, )
E
C
2
(G)
1
Cl
(G)
o
p oble i
(1),
(3)
we
associa e
he
unc ion
-I
(13)

w
( )

u(x,
)W(x)
dx

w(x)
dx)

,

>
0
sz

sz
We
shall
no e ha
a
simila
a e aging
was
i s
used
by
N
.
Yoshida
in
he
wo k
[10]
.
Lemma
2
.
Le
condi ions
Hl-H6
hold
and
le
u(x,
)
be
a
posi i e
solu ion
in
he
domain
G
o
p oblem
(1),
(3)
.
Then
he
unc ion
w( )
de zned
by
(13)
sa is aes
he
di e en ial
inequali y o
neu al
ype
2
(14)

T
2
[w( )
+, ( )w(
-
T)]
+
aol ( )
+
aoh( )
-
w(
-
o-)+
+
p( )
-
h(w( ))
:5
J
,
(x,
)w(x)
dx

~~
W
(x)
dx)
-
'

-->
o,
sz

sz

-
whe e
o is
a
su
cien ly
la ge
posi i e
numbe
.
P oo
..
Le
u(x,
)
be a
posi i e solu ion
in
he
,
domain
G
o
p oblem
(1),
(3)
and
o
=
max{T,
u}
.

Then
u(x,
-
T)
> 0 and
u(x,
-
a)
> 0
o (x,
)
E
9
x
( o,
oo)
.

Mul iply
bo h
sides o
equa ion
(1)
by
he
eigen unc ion
W(x) o
he
Di ichle
p oblem
and
in eg a e
wi h
espec
o
.x
o e
he
domain
9
.
Fo
->
o
we
ob ain
(15)
d
2
U
(
x,
)
w(x)
dx
+
A
( )

u(x,
-
- )cp(x)
dx]
-
d
-
~~
Du(x,
)W(x)
dx
+
p( )
n

2
nu(x,
-
u)p(x)
dx]
+
+
.~s
c(x,
,
u)
W(x)
dx
=
.~
(x,
)
~o(x)
dx
.

8

D
.P
.
Misü ,
D
.D
.
BAINOV
F o n
G ee 's
o iula
i
ollows
ha
(16)

L
Du(x,
)W(x)
dx
=
i
u(x,
)
AW(x)
dx
=
s

in
=-
ao
-

u(x,
)W(x)
dx
=
-aow( )
-
1
W(x)
dx
sz

sz
(17)

L
Du(x,
-
u)cp(x)
dx
=
1
u(x,
-
o,)Ocp(x)
dx
=
sz

sz
=-
ao
u(x,
-
u)W(x)
dx
=
-cxow(
-
a)
-
~p(x)
dx,
sz

sz
whe e
ao
is
he
smalles
eigen alue
.
Mo eo e ,
om
condi ion
H4
and
Jenscn's
inequali y
i
ollows
ha
(18)

,ISZ
c(x,
,
u)
W(x)
dx
~
p( ) 19
h(u)~o(x)
dx
_1
p( )
-
1¿

u(x,
)
W(x)
dx
-
C
W(x)
dx
/
l

~,
co(x)
dx
=
S2

S2

S2
Using
(16)-(18)
a ad
condi ion
H1,
om
(15)
we
ob ain
2
2
~1A1( )
+
ñ( )1l1(
-
T)
1
<
-00
[ID
( )
+
/b( )1U(
-
u)~-
_
p( )
.
h(w( ))
+
ig
(x'
)~o(x)
dx
-
(I 2
W(x)
dx)
-1
which
comple es
he
p oo
o
Lemma
2
.
In oduce he
no a ion
=
p( )
-
h(w( ))
-
~,
cp(x)
dx
sz
(19)

Fi
( )
_

(x,
)~(x)
dx

~
W(x)
dx)
-1
,

>
0
.
sz

Sa
Analogously
o
Theo em
1
he
ollowing
heo em
is
p o ed
.
Theo em
2
.
Le ,
condi ions
Hl-H5
hold
and
le
he
di e en ial
in-
equali ies
o
neu al,
ype
z
(20)

22
~1U( )
+
~( )1U(
-
T))
+
a
o
[w( )
+
~( )w(
-
~)
1+
+
p( )
-
h(w( ))
<_
Fl( ),

?
o,
NONLINBAR
HYPERBOLIC
EQUATIONS
O
NEUTRAL
TYP13

9
2
(21)

d
2
[w( )
+
( )W(
-
T)]
+
c1!o
[w( )
+
P( )w(
-
o,)]+
+
p( )
-
h(w( ))
<-
-
FI
( ),

?
o
ha e
no
e en ually
posi i e
solu ions
.
Then
each
solu ion
u(x,
)
o p ob-
lem
(1),
(3)
oscillla es
in
he
domain
G
.
F om
he
heo ems
p o ed
abo e
i
ollows
ha
he
inding
o
su iicien
condi ions
o
oscilla ion
o
he
solu ions
o
equa ion
(1)
in
he
domain
G
is
educed
o
he
in es iga ion
o
he
oscilla o y
p ope ies
o
di e en ial
inequali ies o
neu al
ype
o
he
o m
2
(22)

d 2
[x( )
+
( )x(
-
T)]
+
go
( )x( )
+
q( )x(
-
o

)+
We
shall
say
ha
condi ion (A) a e
sa is ice
i
lic
'ollowing
condi ions
hold
:
A1
.
~ ( )
E
C
2
([ o,
00)
;
[0,
00
)),
A2
.
qo( ),
q( )
E
C([ o,
co)
;[
0
,
oo)),
A3
.
p( )
E
C([ o,
oo)
;
[0,
oo)),
A4
.
h(u) E
C(R
;
R),
h(u)
>
0
o
u
>
0,
A5
.
H
( )
E
C([
o
,
oo)
;
H)
.
+
p( )
-
h(x( ))
<_
H( ),

>-
11
Theo em
3
.
Le
condi ions
(A)
hold
as
well
as
he
condi ion
(23)

lim
in

1
-00
-
1
o
i >-
o
.
Then
he
di 'e en ial
inequali y
(22)
has
no
e en ualllly
posi-
i e
solu ions
.
P oo
::
Suppose
ha
his
is
no
ue
and
le
x( )
be
a posi i e
solu ion
o
inequali y
(22)
de ined
in
he
in e al
[ 1,oo),
whe e
1
>__
o
.
Then
in
i ue
o
condi ions
A2-A4
wc
ob ain
o
,
>- 2
( 2
>- 1
+
nax{a,T})
2
d 2
[x( )
+
( )x(
-
T)]
<_ H( )
-
go( )x( )
-
q( )x(
-
o-)-
-
p( )
-
h(x( ))
<-
H( )
.
We
in eg a e
wice
he
abo e
inequali y
o e
he
in e al
[ 2,
.],
.
>
2
and
ob ain
(
-
s)
-
H(s)
ds
=
-
n

1
x
( )
+
A( )x(
-
T)
<
C1
+
C2
(
-
2)
+'12'
[£

H(s)
ds
]
dp,
Z
10

D
.P
.
MISHEV,
D
.D
.
BAINOV
whe é
Cl,
C2
=
cons
.
Since
(24)

x( )
+
~(
)x(
-
T)

C

+
C2
+

1

J
~(
-
s)H(s)
ds
- 2

-
- 2

- 2
,
ha

1

e
H
(s)
ds
J
dp
=

(
-
s)H(s)
ds,
2
2

2
di iding
bo h
sides o
las
inequali y
by
-
2
>
0,
we
ob ain
Then
o
->
oo
om
(24),
making
use
o
condi ion
(23),
we
ob ain
x( )
+, ( )x(
-
T)
_
(25)

l m
i0

-

-
2
On
he
o he
hand,
using
condi ion
A1
and
he
ac
ha
x( )
>
0,
x( -
T)
>
0 o
_>
2,
we
ob ain
ha
li a
in

1

[x( )
+
A( )X(
-
T)J
>- 0,
-00
-
op

-
which
con adic s equali y
(25)
.
This
comple es
he
p oo
o
Theo em
3
.
Thé
ollowing
su icien
condi ion
o
oscilla ion
o
he
solu ions
o
p oblem
(1),
(2)
is
a
co olla y
o
Theo em
1
and
Theo em
3
.
Theo em
4
.
Le
condi ions
(H)
hold
as well as
he
condi ions
o
any_su cien ly
la ge
numbe
e,
whe e
he
unc ions
G( )
and
F( )
a e
de ined
by
(4)
.

Then
each
solu ion
u(x,
)
Q
p oblem
(I),
(2)
oscil-
la es
in
he
do nain
G
.
The
.
ollowing
su ñcien
condi ion
o
oscilla ion
o
he
solu ions
o
p oblem
(1),
(3)
is
a
co olla y
o
Theo em
2
and
Theo em
3
.
(26)
lim
in
£
(1
-
')
(G(s)
+
(s)G(s
-
a)
+
F(s))
ds
=
-oo,
(27)
limsu )
~
0
(1
-
)
(G(s)
+
p(s)G(s
-
a)
+
F(s))
ds
=
+oo
NONLINEAR
HYPERBOLIC
EQUATIONS
O
NEUTRAL
TYPE

11
Theo em
5
.
Le condi ions
Hl-H5
hold
as
well
as he
condi ions
(28)

M( )
?
0
o
>
0
(29)

lim
in
£
(1
-
)
Fi
(s)
ds
=
-oo
-oo
(30)

lim
sup
J
(1
-
S)
F
(s)
ds
=
+oo
---
oo


o
any
su icien ly la ge
numbe
o,
whe e
he
unc ion
FI
( )
is
de ined
by (19)
.
Then
each
solu ion
u(x,
)
o
P oblem
(1),
(i)
oscilla es
in he
domain
G
.
Example
1
.
Conside
he
equa ion
(31)

u
+
u
(x,
-
7 )
-
u
.
x
+u
=
2e
cos
x(sin
.
+
cos
í
-
e'
-
cos
),
(x,
)
E
(0,
2
)
x
(0,
oo)
-
G,
and
he
bounda y
condi ions
(32)

-u
x
(0,
)
=
0,

u
y (
7 ,
) ,_
-e
-
sin
,

?
0
A
s aigh o wa d
e i ica ion
shows
ha
he unc ions
c(x,
,
u)
=
u,
(x,
)
=
2e
-
cos
x
-
(sin
+
cos
,
-
e'
cos
),
g(0,
)
=
0,
g (
2
,
)
_-e

sin
,
A( )
-
1,
p( )
-
0,
-Y
(x,
)
-
0
sa is y
condi ions
(H)
.
Mo eo e ,
om
(4)
we
ob ain ha
By
s aigh o wa d
calcula ions
we
ind
ha
I( )
.=
~
(1
-
)
(G(s)
+
p(s)G(s
-
a)
+
F(s))
ds
o
=
e
-
( i )
-I
-
(2
sin
-
2e
-
' sin
-
cos
)
+
C,
1
8

D
.P
.
MISHEV,
D
.D
.
BAINOV
Re e en es
1
.

V
.S
.
VLADIMIROV,
"Equa ions
o
Ma hema ical
Physics,"
Moscow,
Nauka,
1981
(in
Russian)
.
2
.

D
.
GEoRGiou,
K
.
KREITH,
Func ional
cha ac e is ic
ini ial
alue
p oblems,
J
.
Ma h
.
Anal
.
Appl
.
10
7
(1985),
414-424
.
3
.

D
.
GEORGIOU,
"Ex ema
solu ions
o
une ional
hype bolic
ini ial
alue p oblems,
Di e en ial
equa ions
:
quali a i e
heo y,"
ol
.
I,
11
(Szeged,
1984),
Colloq
.
Ma h
.
Soc
.
János
Bolyai
47,
No h-Holland,
1987
.
4
.

K
.
KREITII,
T
.
KUSANO,
N
.
YOSHIDA,
Oscilla ion
p ope ies
o
no llinea
hype bolic
equa ions,
Siam
J
.
Ma h
.
Anal
.
15,
3
(1984),
570-578
.
5
.

T
.
KUSANO,
M
.
NAITO,
Oscilla ion
c i e ia
o
a
class
o
pe u bed
Scl ódinge
equa ions,
Canad
.
Ma h
.
Bull
.
25,
1
(1982),
71-77
.
6
.

D
.P
.
MISHEV,
Oscilla o y
p ope ies
o
he
solu ions
o
hype -
bolic
di e en ial
equa ions
wi h
"maximun ",
Hi oshi na
Ma h
.
J
.
16
(1986),
77-83
.
7
.

D
.P
.
MISHEV,
D
.D
.
BAINOV,
Oscilla ion
p ope ies
o
he
solu-
ions
o
a
class
o
hype bolic
equa ions
o
neu a'
ype,
Funkcialaj
Ek acioj
29
(1986),
213-218
.
8
.

D
.P
.
MISHEV,
D
.D
.
BAINOV,
"Oscilla ion
p ope ies
o
he
solu-
ions
o
hype bolic
equa ions
o
neu al
ype,
Di e en ial
equa ions
:
quali a i e
heo y,"
ol
.
1,
11
(Szeged,
1984),
771-780,
Colloc
.
Ma h
.
Soc
.
János
Bolyai 47,
No h-Holland,
1987
.
9
.

D
.P
.
MISHEV,
Oscilla ion
o
he
solu ions
o
non-linea
pa abolic
equa ions
o
neu al
ype
( o
appea )
.
10
.

N
.
YOSHIDA,
Oscilla ion
o
nonlinca pa abolic equa ions
wi h
une-
ional
a gunien s,
Hi oshima
Ma h
.
J
.
16,
2
(1986),
305-314
.
11
.

A
.I
.
ZAHARmV,
D
.D
.
BAINOV,
Oscilla ing
p ope ies
o
he
solu-
ions
o
a
class
o
neu al
ype
une ional
di e en ial
equa ions,
Bull
.
Aus al
.
Ma h
.
Soe
.
22,
3
(1980),
365-372
.
P
.O
.
Box
45
1504
So ia
BULGARIA
Rebu
el
23
d'Agos
de 1990