TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.6, Jun
e
201
4, pp. 4882 ~ 4
8
9
2
DOI: 10.115
9
1
/telkomni
ka.
v
12i6.584
9
4882
Re
cei
v
ed
Jan
uary 28, 201
4
;
Revi
sed Ma
rch 2
2
, 2014;
Acce
pted April 6, 2014
Oscillation Criteria for Even-order Half-linear Functional
Differential Equations with Da
mping
Shouhua Liu
*
1
, Quanxin Zhang
2
, Li Gao
3
Dep
a
rtment of Mathematics a
nd Informa
tio
n
Scienc
e, Binzh
ou Un
iversit
y
,
Shand
on
g 25
6
603, Peo
p
l
e
'
s
Rep
ubl
ic of Chi
n
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: l316
92
07@
1
63.com
1
, 331
4
744
@16
3
.com
2
, gaolibz
x
y@
1
63.com
3
A
b
st
r
a
ct
In this pap
er, a class of eve
n
-ord
er half-l
i
n
ear
functio
n
a
l
differenti
a
l e
q
u
a
tions w
i
th da
mp
in
g
i
s
studie
d
. By us
ing t
he
ge
nera
l
i
z
e
d
Ricc
a
ti tr
ansfor
m
at
i
on
a
nd th
e i
n
tegr
al
aver
agi
ng t
e
c
hni
que, s
i
x n
e
w
oscill
atio
n crite
r
ias are
obtai
n
ed for al
l sol
u
tions of
th
e eq
uatio
ns. T
he r
e
sults o
b
tain
e
d
ge
nera
l
i
z
e
a
n
d
improve s
o
me
know
n results.
Ke
y
w
ords
:
oscill
atio
n criteria, da
mpin
g, half-lin
ear, fu
ncti
on
al differe
ntial e
quati
on,
integra
l
aver
agi
n
g
met
h
od
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
In this pa
per,
we
con
s
id
er the oscillato
ry
behavior
of solutio
n
s fo
r the n-th o
r
d
e
r half
-
linear fun
c
tio
nal differentia
l equation
with dampin
g
of the form:
0
)
1
(
)
1
(
,
0
)))
(
(
,
(
))
(
(
)
(
))
(
(
)
(
d
d
t
t
t
g
x
t
f
t
x
t
p
t
x
t
r
t
n
n
.
(1)
Whe
r
e
n
is ev
en,
u
u
u
1
|
|
)
(
,
is a re
al
numbe
r and
0
. For s
i
mplic
i
t
y
, we note :
0
I[
,
)
,
t
)
,
0
[
R
),
,
0
(
R
0
.
Thro
ugh
out this pa
per, we
assume that:
(H
1
)
)
R
I,
(
)
(
,
0
)
(
'
),
R
I,
(
)
(
0
1
C
t
p
t
r
C
t
r
.
(H
2
)
0
)
(
'
,
)
(
),
R
I,
(
)
(
0
t
g
t
t
g
C
t
g
,
)
(
lim
t
g
t
.
(H
3
)
)
R
R,
I
(
)
,
(
C
x
t
f
.
In order to di
scuss
c
onveniently in the following cont
ext, several definitions
will
firstly be
given.
Defini
tion 1
.
The func
tion
0
1
),
R
),
,
([
)
(
t
T
T
C
t
x
x
x
n
is called
a sol
u
tion of
(1), if
)
R
),
,
([
))
(
(
)
(
1
)
1
(
x
n
T
C
t
x
t
r
and
)
(
t
x
satisfy (1) on an inte
rval
)
,
[
x
T
.
Defini
tion 2.
A nontrivial
solution
of (1)
is
said
to
be
oscillatory
if it ha
s
arbit
r
aril
y larg
e
zeros; othe
rwise, it is called nono
scill
atory. (1) is
said to be o
s
cillatory if all its solution
s are
oscillatory.
Very few p
eople h
a
ve
studie
d
the
oscilla
tory b
ehavior
of e
v
en-o
r
de
r ha
lf-linea
r
function
al differential e
q
u
a
tions
with d
a
mping. So,
much
re
sea
r
ch, e
s
pe
ciall
y
some o
n
the
Philos o
s
cilla
tion crite
r
ia
of (1)
and th
e other re
l
a
ted re
sult
s, will be do
ne i
n
this p
ape
r by
referring to
[1-8]
. Moreover, functional i
nequ
alities in
this pape
r h
o
ld for all sufficient larg
e
t
if
there is n
o
sp
ecial expl
anat
ion.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Oscillation
cri
t
eria for Even
-order
Half-li
n
ear
Fu
nctio
n
a
l
Differential
Equation
s
… (Shouhu
a Liu
)
4883
2. Main Results
The followi
ng
lemma is a well-kno
w
n resu
lt; let us see
[1, Lemma 2.2.1] and [2].
Lemma 1
. Let
u
be
a
po
sit
i
ve and
n
-
times
differe
ntiable fun
c
tion
o
n
an
inte
rval
)
,
[
T
with its n-th derivative
)
(
n
u
non-positive on
)
,
[
T
a
nd n
o
t ide
n
tically ze
ro
on
any interval
o
f
the
form
),
,
'
[
T
T
T
'
. Then there
exist
s
a
n
intege
r
l
,
1
0
n
l
, with
l
n
odd a
nd
su
ch that f
o
r
some
'
*
T
T
:
)
1
,
,
1
,
(
),
,
[
,
0
)
1
(
*
)
(
n
l
l
j
T
t
u
j
j
l
;
)
1
,
,
2
,
1
(
),
,
[
,
0
*
)
(
l
i
T
t
u
i
, when
1
l
.
Lemma 2
[
8
]
. Ass
u
me that
)
(
t
x
satisfies all
the
con
d
ition
s
in
Lemm
a
1
and
x
n
n
t
t
t
x
t
x
,
0
)
(
)
(
)
(
)
1
(
; then there e
x
ists co
nsta
nts
)
1
,
0
(
, and
0
M
s
u
c
h
t
hat:
)
(
)
(
'
)
1
(
2
t
x
Mt
t
x
n
n
,
For all suffici
ent large
t
.
Theorem 1
. If the following
conditio
n
s a
r
e true:
(H
4
)
t
t
r
t
E
t
d
)]
(
)
(
[
/
1
0
, where
u
u
r
u
p
t
E
t
t
d
)
(
)
(
exp
)
(
0
1
;
(H
5
) Su
ppo
se
that there exists
)
R
I,
(
)
(
0
C
t
q
,
)
R
R,
(
)
(
C
x
F
s
u
c
h
that
:
)
sgn(
)
(
)
(
)
sgn(
)
,
(
x
x
F
t
q
x
x
t
f
,
x
x
k
x
F
x
F
1
|
|
)
(
)
(
,
Then
0
,
0
,
0
k
x
;
(H
6
)
s
s
q
s
E
t
t
t
d
)
(
)
(
sup
lim
0
.
Then (1) is o
s
cillatory.
Proof
. A
s
su
me that
is
an eventu
a
lly po
sitive sol
u
tion of (1), t
hen the
r
e
exists
0
1
t
t
,
such t
hat
:
0
))
(
(
,
0
)
(
t
g
x
t
x
,
for all
1
t
t
.
From (1
) an
d
(H
5
), we obtain:
1
)
1
(
,
0
))
(
(
)
(
)
(
))]'
(
(
)
(
)
(
[
t
t
t
g
x
t
q
t
kE
t
x
t
r
t
E
n
.
(2)
Hen
c
e,
1
)
1
(
,
0
))]'
(
(
)
(
)
(
[
t
t
t
x
t
r
t
E
n
.
(3)
From (3) a
nd
(H
4
), then the
r
e exist
s
1
2
t
t
su
ch
t
hat
:
2
)
1
(
,
0
)
(
t
t
t
x
n
.
(4)
From (4) a
nd
(1), we obtain
:
2
)
1
(
,
0
]'
))
(
)(
(
[
t
t
t
x
t
r
n
.
(5)
)
(
t
x
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4882 – 4
892
4884
It follows
from
that:
2
)
(
,
0
)
(
t
t
t
x
n
.
(6)
From L
e
mma
1, we obtain:
2
,
0
)
(
'
t
t
t
x
.
(7)
From (3) a
nd
(4), we obtain
:
2
)
1
(
,
0
))
(
(
)
(
)
(
]'
))
(
)(
(
)
(
[
t
t
t
g
x
t
q
t
kE
t
x
t
r
t
E
n
.
(8)
In view of
,
0
)
(
'
,
0
)
(
t
x
t
x
th
en there exi
s
ts
2
t
T
and
0
, for all
T
t
, we have
))
(
(
t
g
x
.
Hen
c
e,
T
t
t
q
t
kE
t
x
t
r
t
E
n
,
0
)
(
)
(
]'
))
(
)(
(
)
(
[
)
1
(
.
(9)
We get that:
))
(
)(
(
)
(
d
)
(
)
(
))
(
)(
(
)
(
)
1
(
)
1
(
T
x
T
r
T
E
s
s
q
s
E
k
t
x
t
r
t
E
n
t
T
n
.
Hen
c
e, we ha
ve a contra
di
ction to the condition (H
6
).
The proof is complete.
Theorem 2.
Assu
me cond
itions (H
4
) an
d (H
5
) h
o
ld, a
nd the followi
ng co
ndition i
s
true
(H
7
) Sup
p
o
s
e that there e
x
ists
0
)
(
'
),
R
I,
(
)
(
1
t
C
t
, and
0
s
u
c
h
that:
s
s
s
q
s
E
s
k
t
t
t
d
)]
(
'
)
(
)
(
)
(
[
sup
lim
0
.
Then (1) is o
s
cillatory.
Proof.
A
s
su
me that
)
(
t
x
is a
n
eventually
positive
solut
i
on
of (1), proce
edin
g
a
s
the
proof of The
o
r
em 1, we o
b
tain (8
) hold
s
. Con
s
id
er the
function:
0
)
1
(
,
))
(
(
)
(
)
(
)
(
)
(
)
(
t
t
t
g
x
t
x
t
r
t
E
t
t
W
n
.
(10)
Then
0
)
(
t
W
, and:
))
(
(
)
(
'
))
(
(
'
))
(
)(
(
)
(
)
(
))
(
(
]'
))
(
)(
(
)
(
)[
(
))
(
(
)
(
)
(
)
(
)
(
'
)
(
'
1
)
1
(
)
1
(
)
1
(
t
g
x
t
g
t
g
x
t
x
t
r
t
E
t
t
g
x
t
x
t
r
t
E
t
t
g
x
t
x
t
r
t
E
t
t
W
n
n
n
))
(
(
]'
))
(
)(
(
)
(
)[
(
))
(
(
)
(
)
(
)
(
)
(
'
)
1
(
)
1
(
t
g
x
t
x
t
r
t
E
t
t
g
x
t
x
t
r
t
E
t
n
n
.
(11)
From (3), (7
),
(8) an
d (11
)
, we obtai
n:
0
)
1
(
,
))
(
(
))
(
)(
(
)
(
)
(
'
)
(
)
(
)
(
)
(
'
t
T
T
g
x
T
x
T
r
T
E
t
t
q
t
E
t
k
t
W
n
.
(12)
Let
))
(
(
)
(
)
(
)
(
)
1
(
T
g
x
T
x
T
r
T
E
n
, we get:
)
(
'
)
(
)
(
)
(
)
(
'
t
t
q
t
E
t
k
t
W
.
(13)
0
)
(
'
t
r
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Oscillation
cri
t
eria for Even
-order
Half-li
n
ear
Fu
nctio
n
a
l
Differential
Equation
s
… (Shouhu
a Liu
)
4885
Integrating th
e above from
T
to
t
, we obtain:
s
s
s
q
s
E
s
k
T
W
t
W
t
T
d
]
)
(
'
)
(
)
(
)
(
[
)
(
)
(
.
(14)
In (14), let
t
.Bec
ause
0
)
(
t
W
, we have a contradictio
n to condition
(H
7
). The proof i
s
c
o
mplete.
Theorem 3
.
Assu
me cond
ition (H
4
) and
(H
5
) h
o
ld, an
d the followin
g
con
d
ition is
true
(H
8
) Suppo
se that there e
x
ists
)
R
I,
(
)
(
1
C
t
s
u
c
h
that:
0
1
1
,
]d
))
(
)
(
(
)
1
(
)
(
)
(
))
(
'
(
)
(
)
(
)
(
[
sup
lim
t
T
s
s
G
s
s
r
s
E
s
s
q
s
E
s
k
t
T
t
,
(15)
Whe
r
ein
)
s
(
)
(
'
)
(
2
n
g
s
Mg
s
G
,
M
,
in Lemma 2,
)
(
s
E
in (H
4
). Then (1) is oscillatory.
Proof
. Ass
u
me that
)
(
t
x
is
an eventu
a
lly positive
sol
u
tion of (1),
pro
c
ee
ding
a
s
the
proof of The
o
r
em 1, we o
b
tain (8
) hold
s
. Con
s
id
er the
function:
0
)
1
(
,
))
(
(
)
(
)
(
)
(
)
(
)
(
t
t
t
g
x
t
x
t
r
t
E
t
t
W
n
.
(16)
Then
0
)
(
t
W
. From (8) and L
e
mm
a 2, we obtai
n:
T
t
t
W
t
r
t
E
t
t
G
t
W
t
t
t
q
t
E
t
k
t
W
),
(
)]
(
)
(
)
(
[
)
(
)
(
)
(
)
(
'
)
(
)
(
)
(
)
(
'
/
)
1
(
/
1
.
(17)
By using the inequ
ality:
B
A
Bu
Au
1
1
/
)
1
(
)
1
(
.
(18)
Then
0
,
0
,
0
u
B
A
, we have:
T
t
t
G
t
t
r
t
E
t
t
q
t
E
t
k
t
W
,
))
(
)
(
(
)
1
(
)
(
)
(
))
(
'
(
)
(
)
(
)
(
)
(
'
1
1
.
(19)
Integrating th
e above from
T
to
t
, we get:
]
))
(
)
(
(
)
1
(
)
(
)
(
))
(
'
(
)
(
)
(
)
(
[
)
(
)
(
1
1
t
T
s
G
s
s
r
s
E
s
s
q
s
E
s
k
T
W
t
W
s
d
.
Bec
a
us
e
0
)
(
t
W
, we have:
)
(
]d
))
(
)
(
(
)
1
(
)
(
)
(
))
(
'
(
)
(
)
(
)
(
[
sup
lim
1
1
T
W
s
s
G
s
s
r
s
E
s
s
q
s
E
s
k
t
T
t
.
(20)
Hen
c
e, we ha
ve a contra
di
ction to the condition (H
8
).
The proof is complete.
Theorem 4
.
Assu
me the condition (H
4
) and (H
5
) hol
d
,
and the following
con
d
ition is true
(H
9
) Supp
ose
that there exists
)
R
I,
(
)
(
1
C
t
s
u
ch that:
s
s
G
s
s
r
s
E
s
s
q
s
E
s
k
s
t
t
t
T
n
n
t
]d
))
(
)
(
(
)
1
(
)
(
)
(
))
(
'
(
)
(
)
(
)
(
[
)
(
1
sup
lim
1
1
,
(21)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4882 – 4
892
4886
Whe
r
e
n
> 1 and fun
c
tion
)
(
),
(
s
G
s
E
is given by (H
4
) and (H
8
). Then (1
) is o
scillatory.
Proof.
Assu
me that
)
(
t
x
is
an eventu
a
lly po
sitive sol
u
tion of
(1),
pro
c
ee
ding
a
s
th
e
proof of The
o
r
em 3, and fu
nction
)
(
t
W
is given by (16), we get (19
)
hold
s
.
From (19
)
, we obtain:
.
Multiplying two side
s by
n
s
t
)
(
an
d integratin
g the above fro
m
T
to
t
(
t
>
T
) , we get:
s
s
W
s
t
s
s
G
s
s
r
s
E
s
s
q
s
E
s
k
s
t
t
T
n
t
T
n
d
)
(
'
)
(
d
]
))
(
)
(
(
)
1
(
)
(
)
(
))
(
'
(
)
(
)
(
)
(
[
)
(
1
1
.
Since:
n
t
T
n
t
T
n
T
t
T
W
s
s
W
s
t
n
s
s
W
s
t
)
)(
(
d
)
(
)
(
d
)
(
'
)
(
1
,
We get:
s
s
W
s
t
t
n
t
T
t
T
W
s
s
G
s
s
r
s
E
s
s
q
s
E
s
k
s
t
t
t
T
n
n
n
t
T
n
n
d
)
(
)
(
)
)(
(
d
]
))
(
)
(
(
)
1
(
)
(
)
(
))
(
'
(
)
(
)
(
)
(
[
)
(
1
1
1
1
.
Therefore:
n
t
T
n
n
t
T
t
T
W
s
s
G
s
s
r
s
E
s
s
q
s
E
s
k
s
t
t
)
)(
(
d
]
))
(
)
(
(
)
1
(
)
(
)
(
))
(
'
(
)
(
)
(
)
(
[
)
(
1
1
1
.
Then:
s
s
G
s
s
r
s
E
s
s
q
s
E
s
k
s
t
t
t
T
n
n
t
]d
))
(
)
(
(
)
1
(
)
(
)
(
))
(
'
(
)
(
)
(
)
(
[
)
(
1
sup
lim
1
1
.
Hen
c
e, we ha
ve a contra
di
ction to the condition (H
9
). The proof is complete.
By Philos i
n
tegral
average
conditions, th
e new
oscillation theorems are
given for
Equation (1). Con
s
id
er the
sets:
}
:
)
,
{(
D
0
0
t
s
t
s
t
,
}
:
)
,
{(
D
0
t
s
t
s
t
.
A
ssu
me t
hat
)
R
D,
(
C
H
sat
i
sf
ie
s t
he f
o
llowin
g
co
nd
it
ions:
(i)
0
,
0
)
(
t
t
t,t
H
;
0
D
)
,
(
,
0
)
(
s
t
t,s
H
;
(ii)
H
has
a n
on-p
o
sitive
continuo
us
pa
rtial de
rivative with
re
spe
c
t to the
se
con
d
variable in
0
D
.
Then the fun
c
tion
H
has the
prop
erty
P
(Den
oted as
P
s
t
H
)
,
(
).
Theorem 5.
Assu
me the condition (H
4
) and
(H
5
) hol
d
,
and the following
con
d
ition is true
(H
10
)
P
t,s
H
)
(
, and that there exist
s
functio
n
s
and
)
R
I,
(
)
(
1
C
t
su
ch t
hat
:
0
)
1
/(
D
)
,
(
),
(
)
,
(
)
(
)
(
)
(
'
)
(
s
t
t,s
H
s
t
h
t,s
H
s
s
s
t,s
H
,
(22)
T
s
s
W
s
G
s
s
r
s
E
s
s
q
s
E
s
k
),
(
'
))
(
)
(
(
)
1
(
)
(
)
(
))
(
'
(
)
(
)
(
)
(
)
1
(
)
1
(
)
R
,
D
(
)
,
(
0
C
s
t
h
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Oscillation
cri
t
eria for Even
-order
Half-li
n
ear
Fu
nctio
n
a
l
Differential
Equation
s
… (Shouhu
a Liu
)
4887
And,
s
s
G
s
t
h
s
r
s
E
s
s
q
s
E
s
k
s
t
H
t
t
H
t
t
t
]d
)
(
)
1
(
|
)
,
(
|
)
(
)
(
)
(
)
(
)
(
)
(
)
,
(
[
)
,
(
1
sup
lim
1
1
0
0
,
(23)
Whe
r
e fun
c
tions
)
(
),
(
s
G
s
E
are given
by (H
4
) an
d (H
8
). Then (1) is oscillatory.
Proof.
Assu
me that
)
(
t
x
is
an eventu
a
lly po
sitive sol
u
tion of
(1),
pro
c
ee
ding
a
s
th
e
proof of The
o
r
em 3, and fu
nction
)
(
t
W
is given by (16), we get (17
)
hold
s
.
From (17
)
, we obtain:
.
(24)
Repl
aci
ng
t
by
s
, multiplying two sides by
)
(
t,s
H
and integ
r
a
t
ing the abov
e from
T
to
t
(
t
>
T
),
we get:
s
s
W
s
r
s
E
s
s
G
s
W
s
s
s
W
s
t
H
s
s
q
s
E
s
k
s
t
H
t
T
t
T
d
)]
(
)]
(
)
(
)
(
[
)
(
)
(
)
(
)
(
'
)
(
'
)[
,
(
d
)
(
)
(
)
(
)
,
(
/
)
1
(
/
1
s
s
W
s
r
s
E
s
s
t
H
s
G
s
W
s
s
s
t
H
s
s
t
H
T
W
T
t
H
t
T
d
)}
(
)]
(
)
(
)
(
[
)
,
(
)
(
)
(
]
)
(
)
(
'
)
,
(
)
,
(
{[
)
(
)
,
(
/
)
1
(
/
1
s
s
W
s
r
s
E
s
s
t
H
s
G
s
W
s
t
H
s
t
h
T
W
T
t
H
t
T
d
)}
(
)]
(
)
(
)
(
[
)
,
(
)
(
)
(
)]
,
(
|
)
,
(
{[|
)
(
)
,
(
/
)
1
(
/
1
)
1
/(
(25
)
The rig
h
t end
of (25) i
s
int
egra
b
le fun
c
ti
ons fo
r usi
n
g
the inequ
ality (18), then f
o
r
T
s
t
, we
have:
)
(
)
1
(
|
)
,
(
|
)
(
)
(
)
(
)
(
)]
(
)
(
)
(
[
)
,
(
)
(
)
(
)
,
(
|
)
,
(
|
1
1
/
)
1
(
/
1
)
1
/(
s
G
s
t
h
s
r
s
E
s
s
W
s
r
s
E
s
s
t
H
s
G
s
W
s
t
H
s
t
h
.
(26)
Form (25) and (26), we have:
)
(
)
,
(
)
(
)
,
(
]d
)
(
)
1
(
|
)
,
(
|
)
(
)
(
)
(
)
(
)
(
)
(
)
,
(
[
0
1
1
T
W
t
t
H
T
W
T
t
H
s
s
G
s
t
h
s
r
s
E
s
s
q
s
E
s
k
s
t
H
t
T
.
(27)
Therefore:
s
s
G
s
t
h
s
r
s
E
s
s
q
s
E
s
k
s
t
H
t
t
]d
)
(
)
1
(
|
)
,
(
|
)
(
)
(
)
(
)
(
)
(
)
(
)
,
(
[
1
1
0
s
s
G
s
t
h
s
r
s
E
s
s
q
s
E
s
k
s
t
H
T
t
t
T
]d
)
(
)
1
(
|
)
,
(
|
)
(
)
(
)
(
)
(
)
(
)
(
)
,
(
[
}
{
1
1
0
0
0
0
,
)
(
)
,
(
d
)
(
)
(
)
(
)
,
(
0
t
t
T
W
t
t
H
s
s
q
s
E
s
k
t
t
H
T
t
,
Whi
c
h implies:
s
s
G
s
t
h
s
r
s
E
s
s
q
s
E
s
k
s
t
H
t
t
H
t
t
t
]d
)
(
)
1
(
|
)
,
(
|
)
(
)
(
)
(
)
(
)
(
)
(
)
,
(
[
)
,
(
1
sup
lim
1
1
0
0
)
(
d
)
(
)
(
)
(
0
T
W
s
s
q
s
E
s
k
T
t
.
Hen
c
e, we ha
ve a contra
di
ction to the condition (H
10
). The proof is
compl
e
te.
If condition
(H
10
)
doe
s n
o
t
hold, then
we
can
use the follo
wing
oscillatory th
eore
m
to
Equation (1).
T
t
t
W
t
r
t
E
t
t
G
t
W
t
t
t
W
t
q
t
E
t
k
),
(
)]
(
)
(
)
(
[
)
(
)
(
)
(
)
(
'
)
(
'
)
(
)
(
)
(
/
)
1
(
/
1
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4882 – 4
892
4888
Theorem 6.
Assu
me
the con
d
ition
(H
4
) and
(H
5
) h
o
ld,
P
t,s
H
)
(
, and th
e followin
g
conditions is true.
(H
11
)
)
,
(
)
,
(
inf
lim
inf
0
0
0
t
t
H
s
t
H
t
t
s
;
(H
12
)
s
s
G
s
t
h
s
r
s
E
s
t
t
H
t
t
t
d
)
(
)
1
(
|
)
,
(
|
)
(
)
(
)
(
)
,
(
1
sup
lim
0
1
1
0
, where
)
(
),
(
s
G
s
E
is given by (H
4
) and
(H
8
);
(H
13
) That the
r
e exist
s
)
R
I,
(
)
(
C
t
s
u
ch that:
)
,
[
,
d
)]
(
)
(
)
(
[
)
(
)
(
0
/
1
/
)
1
(
t
T
s
s
r
s
E
s
s
s
G
T
,
Then
}
0
),
(
max{
)
(
s
s
; and:
(H
14
)
)
(
]d
)
(
)
1
(
|
)
,
(
|
)
(
)
(
)
(
)
(
)
(
)
(
)
,
(
[
)
,
(
1
sup
lim
1
1
T
s
s
G
s
t
h
s
r
s
E
s
s
q
s
E
s
k
s
t
H
T
t
H
t
T
t
.
Then (1) is o
s
cillatory.
Proof.
Assu
me that
)
(
t
x
is
an eventu
a
lly po
sitive sol
u
tion of
(1),
pro
c
ee
ding
a
s
th
e
proof of The
o
r
em 3, and fu
nction
)
(
t
W
is given by (16), we get (25
)
hold
s
.
From (25
)
, we obtain:
s
s
q
s
E
s
k
s
t
H
T
t
H
t
T
d
)
(
)
(
)
(
)
,
(
)
,
(
1
s
s
W
s
r
s
E
s
s
t
H
s
G
T
t
H
s
s
W
s
t
H
s
t
h
T
t
H
T
W
t
T
t
T
d
)
(
)]
(
)
(
)
(
[
)
,
(
)
(
)
,
(
d
)
(
)
,
(
|
)
,
(
|
)
,
(
1
)
(
/
)
1
(
/
1
)
1
/(
.
Let,
s
s
W
s
t
H
s
t
h
T
t
H
t
A
t
T
d
)
(
)
,
(
|
)
,
(
|
)
,
(
1
)
(
)
1
/(
,
(28)
s
s
W
s
t
H
s
R
T
t
H
t
B
t
T
d
)
(
)
,
(
)
(
)
,
(
)
(
/
)
1
(
,
(29)
Whi
c
h,
/
1
)]
(
)
(
)
(
[
)
(
)
(
s
r
s
E
s
s
G
s
R
.
(30)
Then,
)
(
)
(
)
(
d
)
(
)
(
)
(
)
,
(
)
,
(
1
t
B
t
A
T
W
s
s
q
s
E
s
k
s
t
H
T
t
H
t
T
.
(31)
From (27
)
, we have:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Oscillation
cri
t
eria for Even
-order
Half-li
n
ear
Fu
nctio
n
a
l
Differential
Equation
s
… (Shouhu
a Liu
)
4889
)
(
]d
)
(
)
1
(
|
)
,
(
|
)
(
)
(
)
(
)
,
(
[
)
,
(
1
sup
lim
1
1
T
W
s
s
R
s
t
h
s
q
s
E
s
k
s
t
H
T
t
H
t
T
t
.
(32)
From (32
)
an
d (H
14
), we have:
)
,
[
,
)
(
)
(
0
t
T
T
T
W
,
(33)
And,
)
(
d
)
(
)
(
)
(
)
,
(
)
,
(
1
sup
lim
T
s
s
q
s
E
s
k
s
t
H
T
t
H
t
T
t
.
(34)
Joint (3
1) a
n
d
(34) to p
r
odu
ce:
)
(
)
(
d
)
(
)
(
)
(
)
,
(
)
,
(
1
sup
lim
)
(
)]
(
)
(
[
inf
lim
T
T
W
s
s
q
s
E
s
k
s
t
H
T
t
H
T
W
t
A
t
B
t
T
t
t
.
(35)
We cl
aim that
:
s
s
W
s
R
T
d
)
(
)
(
/
)
1
(
.
(36)
Otherwise, if:
s
s
W
s
R
T
d
)
(
)
(
/
)
1
(
.
(
3
7
)
From (H
11
), then there exists
0
can be u
s
ed in:
0
)
,
(
)
,
(
inf
lim
inf
0
0
t
t
H
s
t
H
t
t
s
.
(38)
Let
be an arbitrary con
s
ta
nt from (37
)
, then there exists
can be u
s
ed in:
1
/
)
1
(
,
d
)
(
)
(
T
t
s
s
W
s
R
t
T
.
(39)
Thus,
]
d
)
(
)
(
d[
)
,
(
)
,
(
d
)
(
)
,
(
)
(
)
,
(
)
(
/
)
1
(
/
)
1
(
u
u
W
u
R
s
t
H
T
t
H
s
s
W
s
t
H
s
R
T
t
H
t
B
t
T
s
T
t
T
s
s
s
t
H
u
u
W
u
R
T
t
H
s
s
s
t
H
u
u
W
u
R
T
t
H
t
T
s
T
t
T
s
T
1
d
)
)
,
(
](
d
)
(
)
(
[
)
,
(
d
)
)
,
(
](
d
)
(
)
(
[
)
,
(
/
)
1
(
/
)
1
(
1
1
,
)
,
(
)
,
(
d
)
)
,
(
(
)
,
(
1
T
t
T
t
H
T
t
H
s
s
s
t
H
T
t
H
t
T
.
(40)
From (38
)
, then there exi
s
t
1
2
T
T
, can be u
s
ed
:
2
0
1
,
)
,
(
)
,
(
T
t
t
t
H
T
t
H
.
(
4
1
)
Joint (4
0) a
n
d
(41) to p
r
odu
ce:
0
T
T
1
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4882 – 4
892
4890
2
,
)
(
T
t
t
B
.
For
is arbitra
r
y then,
)
(
lim
t
B
t
.
(42)
Con
s
id
er nex
t seque
nce
n
n
n
n
t
t
t
lim
),
,
[
}
{
0
1
, can b
e
used i
n
:
)]
(
)
(
[
inf
lim
)]
(
)
(
[
lim
t
A
t
B
t
A
t
B
t
n
n
n
.
From (35
)
, then there exi
s
ts
M
ca
n be u
s
ed in:
.
(43)
From (42),
.
(44)
From (43
)
, we have:
.
(45)
From (43
)
an
d (44
)
, whe
n
n
is
s
u
ffic
i
ently large, we have:
.
Therefore wh
en
n
is
suffic
i
ently large,
2
1
)
(
)
(
n
n
t
B
t
A
.
From (45
)
, we have:
)
(
)
(
lim
1
n
n
n
t
B
t
A
.
(46)
On the othe
r hand u
s
in
g the Hold
er ine
q
uality, we have:
s
s
W
s
t
H
s
R
s
R
s
t
h
T
t
H
t
A
n
t
T
n
n
n
n
d
)
(
)
,
(
)
(
)
(
|
)
,
(
|
)
,
(
1
)
(
)
1
/(
)
1
/(
)
1
/(
)
1
/(
/
)
1
(
)
1
/(
1
1
]
d
)
(
)
,
(
)
(
)
,
(
1
[
]
d
)
(
|
)
,
(
|
)
,
(
1
[
s
s
W
s
t
H
s
R
T
t
H
s
s
R
s
t
h
T
t
H
n
n
t
T
n
n
t
T
n
n
.
Therefore:
]
d
)
(
)
,
(
)
(
)
,
(
[
]
d
)
(
|
)
,
(
|
)
,
(
1
[
1
)
(
/
)
1
(
/
1
1
/
)
1
(
s
s
W
s
t
H
s
R
T
t
H
s
s
R
s
t
h
T
t
H
t
A
n
n
t
T
n
n
t
T
n
n
n
.
Noted that
)
(
t
B
is definedby th
e above eq
ua
tion wa
s wh
e
n
n
is
s
u
ffic
i
ently large.
0
,
2
,
1
,
)
(
)
(
n
M
t
A
t
B
n
n
)
(
lim
n
n
t
B
)
(
lim
n
n
t
A
2
1
)
(
1
)
(
)
(
n
n
n
t
B
M
t
B
t
A
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Oscillation
cri
t
eria for Even
-order
Half-li
n
ear
Fu
nctio
n
a
l
Differential
Equation
s
… (Shouhu
a Liu
)
4891
/
1
1
/
)
1
(
]
d
)
(
|
)
,
(
|
)
,
(
1
[
1
)
(
)
(
s
s
R
s
t
h
T
t
H
t
B
t
A
n
t
T
n
n
n
n
.
That is,
s
s
R
s
t
h
T
t
H
t
B
t
A
n
t
T
n
n
n
n
d
)
(
|
)
,
(
|
)
,
(
1
)
(
)
(
1
1
.
(47)
From (38),
)
,
(
)
,
(
inf
lim
0
t
t
H
s
t
H
t
.
Then the
r
e e
x
ists
T
T
3
, can be
use
d
in:
3
0
,
)
,
(
)
,
(
T
t
t
t
H
T
t
H
.
Therefore wh
en
n
is
suffic
i
ently large,
)
,
(
)
,
(
0
t
t
H
T
t
H
n
n
.
(48)
From (47
)
an
d (48
)
, we get
:
s
s
R
s
t
h
t
t
H
t
B
t
A
n
t
t
n
n
n
n
d
)
(
|
)
,
(
|
)
,
(
1
)
(
)
(
0
1
0
1
.
(49)
From (46
)
an
d (49
)
, we get
:
s
s
R
s
t
h
t
t
H
n
t
t
n
n
n
d
)
(
|
)
,
(
|
)
,
(
1
lim
0
1
0
,
Whi
c
h implies:
s
s
R
s
t
h
t
t
H
t
t
t
d
)
(
|
)
,
(
|
)
,
(
1
sup
lim
0
1
0
.
Notice (30
)
, this is
c
ontrary to condition
(H
12
). Therefore, ou
r asse
rti
on (3
6) is e
s
tabli
s
he
d.
Ho
wever, by (36
)
and (33
)
:
s
s
W
s
R
s
s
s
R
T
T
d
)
(
)
(
d
)
(
)
(
/
)
1
(
/
)
1
(
.
Notice (30
)
, this is
c
ontrary to condition
(H
13
).The p
r
oof is co
mple
te.
Ackn
o
w
l
e
dg
ements
This
wo
rk is
sup
porte
d by
a grant fro
m
the Natural
Scien
c
e F
o
u
ndation
of Shand
ong
Province of
Chin
a (Z
R2
0
11AL00
1, Z
R
20
13AM0
03
) an
d Sci
ent
ific Re
se
arch
Foun
dation
of
Binzho
u Univ
ersity (BZXY
L100
6).
Referen
ces
[1]
RP Agar
w
a
l,
SR Grace, D
ORega
n.
Oscil
l
atio
n T
heory f
o
r Differenc
e
and D
i
fferenti
a
l Equ
a
tions
.
Evaluation Warning : The document was created with Spire.PDF for Python.