Advances in Pure Mathematics, 2011, 1, 49-53
doi: 10.4236/apm.2011.13011 Published Online May 2011 (http://www.SciRP.org/journal/apm)
New Results on Oscillation of even Order Neutral
Differential Equations with Deviating Arguments
Lianzhong Li1, Fanwei Meng2
1School of Mathematical and System Sciences, Taishan College, Tai’an, China
2School of Mathematical Sciences, Qufu Normal University, Qufu, China
Received January 19, 2011; revised March 15, 2011; accepted Ma rc h 20, 2011
Abstract
In this paper, we point out some small mistakes in [6] and revise them, we obtain some new oscillation re-
sults for certain even order neutral differential equations with deviating arguments. Our results extend and
improve many known oscillation criteria because the article just generalizes Meng and Xu’s results.
Keywords: Oscillation, Neutral Differential Equation, Deviating Argument
1. Introduction
Oscillation of some even order differential equations
have been studied by many authors. For instance, see
[1-7] and the references therein. We deal with the oscil-
latory behavior of the even order neutral differential eq-
uations with deviating arguments of the form
  

  

1
1
0
0,
n
m
ii
i
l
jj j
j
xtp txt
qtfx t
tt











 (1)
where 2n is even, throughout this paper, it is as-
sumed that:
(A1)








0
,,,,,,0
ijj j
pqC tRfCRR ufu

 
for 0uand

j
f
uis non-decreasing on R,
1, 2,,,1, 2,,;imjl
(A2)






0,, , ii
CtRttt 

 and


lim ,
i
tt




1, 2,,im;
(A3)






1,, ,,lim
jjj
t
CtRt tt 


 
and

0
jt

, 1, 2,,;jl
(A4) There exists a constant M0 such that
 
sgn M
j
f
xx x for 0,1,2, ,
x
jl;
(A5)
 
1
,0,1
m
i
i
pt pp


 and there exists a func-
tion
 


0,,qtC tR

such that,




min :
j
qtq t

1, 2,,jl.
By a solution of Equation (1) we mean a function


x
twhich has the pro perty that
  
1
m
ii
i
x
tptxt 






,,
nx
Ct R for some 0x tt and satisfies Equa-
tion (1) on


,
x
t

. We restrict our attention to those
solutions


x
t of Equation (1) which exist on some
half-line


,
x
t

with


sup :0xttT for any
x
Tt. A nontrivial solution of Equation (1) is called
oscillatory if it has arbitrarily large zeros, otherwise it is
said to be nonoscillatory. Equation (1) is said to be os-
cillatory if all of it’s nontrivial solutions are oscillatory.
Recently, Meng and Xu [6] studied Equation (1) and
obtained some sufficient conditions for oscillation of the
Equation (1), we list the main results of [6] as follows.
Following Philos [5], we say that a function


,
H
Hts belongs to a function class W, denotes by
H
W

, if


,
H
CDR

, where




0
,:Dtstst,
which satisfies: (H1)


,0Htt and

,0Hts for
0
tst

; (H2)
H
has a continuous non-positive
partial derivative
H
S


satisfying the condition:

  

,,,
H
tsk s
hts Hts
Sks



for some


,
loc
hL DR
   
10
,,,0,kCt is a
non-decreasing function.
Theorem A ([6, Theo re m 2.1]).
Assume that (A1) - (A5) hold, let the functions
,,
H
hksatisfy (H1) and (H2), suppose
 
12
1
limsup ,,
4
tCFtrGtr
C






(2)

L. Z. LI ET AL. 50
holds fo r every 01 2 ,0,0,rtC C where
  
 
 
 
1
2
2
1
,,d,
,
,
1
,d
,,
l
t
j
rj
t
n
rjj
F
trH tsk sqss
Htr
kshts
Gtr s
Htr Htss s 







and 1p

 , then every solution of Equation (1) is
oscillatory.
Theorem B ([6, The orem 2.2]).
Assume that (A1)-(A5) hold, and ,,
H
hk are the same
as in Th eorem A, suppose that


00
,
inf 0
,
liminf
st t
Hts
Htt







(3)
and


0
limsup ,
tGtt   (4)
If there exists a function




0,,mCt R such that
for all 0
tTt.
 
12
1
,,
4
liminf
tCFtTGtTm T
C






(5)
and
 



0
22
limsupd,1,2,,
n
tjj
t
t
m
s
jl
k
sss
s





(6)
where





max ,0mtmt
, then every solution of
Equation (1) is oscillatory.
In Theorem A and B, function


,Gtr should be

,
j
Gtr, so each of the condition (2), (4), (5) and (6) has
as many as l conditions. Meanwhile, the Riccati func-
tion


t 
is not well-defined and there exist some
small errors in the proof of the theorems. The purpose of
this paper is further to strengthen oscillation results ob-
tained for Equation (1) by Meng and Xu [6]. In our paper,
we redefine the functions



,,,,tr tFrGt  and
provide some new oscillation criteria for oscillation of
Equation (1).
2. Main Results
In the sequel, we need the following lemmas:
Lemma 2.1 ([1]).
Let

x
t be a n times differentiable function on


0,t of one sign,


0
n
xt on


0,t which
satisfies
 
() 0 n
xtxt. Then:
(I1) There exists a 10
tt such that


,1,2,,1
i
x
ti n are of one sign on
 
1,t;
(I2) There exists a number

1, 3, 5,,1hn when
n is even, or

2,4,6, ,1hn when n is odd,
such that



0
i
xtxt for 0,1, ,ih,1;tt
 


1
10
ni i
xtx t


for 1 1,2,,,ihhn tt.
Lemma 2.2 ([1]).
If


x
t is as in Lemma 2.1 and




10 nn
xxtt


for 0 tt, then for every

01  , there exists a
constant 0N, such that
 


1
1n n
Nt txt X 



for all large t.
Lemma 2.3([7]).
Suppose that


x
t is an eventually positive solution
of Equation (1), let
  
1
m
ii
i
zx pxttt t 


,
then there exists a number 10 tt such that


0,zt



1
0, 0
n
zztt


and


1
0,
nttzt.
Theorem 2.1
Assume that (A1) - (A5) hold, let the functions ,,
H
hk
satisfy (H1) and (H2), suppose
 
limsup ,,
4
tMFtrGt r
N







(7)
holds fo r every 0
rt and for some 1

, where
    
 
 
 
2
2
1
1
,,d
,
,
1
,d
,,
t
r
t
l
rn
jj
j
FtrHtsksqs s
Htr
ksh ts
Gtr s
Htr Htsss








and 1p


, then every solution of Equation (1) is
oscillatory.
Proof. Suppose to the contrary that


x
t is a
nonoscillatory solution of Equation (1) and that


x
t is
even- tually positive (when

x
t is eventually negative,
the proof is similar).
Let


zt be defined as in Lemma 2.3, then following
the proof of Theorem 2.1 in [6], without loss of
generality, assume there exists a 10 tt such that




  


(1)
0, 0,0,
0, 0
njj
xt tt
tt
z
zzt
z
z 


 







1
2n n
jj
tNtzt z  


 (by lemma 2.2) and

  
1
l
njj
j
ttzMqzt 

  for all 1 tt.
Let
 




1
1
n
l
j
j
z
k
z
t
tt
t





 (not as [6]),
then we have

L. Z. LI ET AL. 51
 
   
 
2
11
2,,
ln
jj
j
N
k
Mk qt
tt
t
tt ttt
kt kt t
 
 




 
 (not as [6]).
Multiplying the above equation, with t replaced by
s
, by

,Hts and integrating it from T to t, for all 1
tTt, for some 1

, we obtain
  
 
  
  
  
 
  
  
  
2
12
2
212
2
1
2
1
2
1
,, ,
,
,
,
,
d, dd
,d d
41
,4
ln
jj
ttt
j
TTT
ln
jj
tt
j
l
TT
n
jj
j
ln
jj
j
ln
jj
j
M
HkqsHtTTh sNHs
k
NH
kh
H
tT Tss
k
NH
ss
ts s stsstss
s
tss s
sts s
s
tss s
ts sNH kh ss
st
stss s kNH







 













 

 













 
 
2
2
2
1
d
,
,
,
,d
4
t
T
t
l
Tn
jj
j
s
k
s
sh
Ht ts
ts s
TT s
NH s
















Hence, we have
 
,,
4
M
FtTGtT T
N




for all 1 tTt, this gives
 
limsup ,,
4
tMFt rGtr
N







which contradicts (7). This completes the proof of the
Theorem.
The assumption (7) in Theorem 2.1 can fail, conse-
quently, Theorem 2.1 does not apply. The following re-
sults provide some essentially new oscillation criteria for
Equation (1).
Theorem 2.2
Assume that (A1)-(A5) hold, the functions ,,,
H
hkF
and G be the same as in Theorem 2.1, suppose that


00
,
inf 0
,
liminf
st t
Hts
Htt







(8)
If there exists a function




0,,mCt R such that
for all 0
tTt and for some 1

,
 
l,m,isup4
tMtT tTFmTG
N







(9)
and
 

0
22
1
limsup d
ln
jj
tj
t
t
msss
ss k








 (10)
where






max ,0mt mt
. Then every solution of
Equation (1) is oscillato ry.
Proof. Assume to the contrary that (1) is non-oscil-
latory. Following the proof of Theo rem 2.1, without loss
of generality, assume for all 0
tTt and for some
1

, we obtain
   
 
 
 
 
2
2
1
2
12
,
,
,
,
d,
d
4
d
1
t
T
t
l
Tn
jj
j
ln
jj
tj
T
tsss
sts
tss s
tss s
s
s
MHk qsHtTT
kh s
NH
NH
s
k



















So, we get
 
1
,, ,
4
M
FtTGtT TNBtT
N





where
 
 
 
2
12
0
,
,,
1d,
ln
jj
tj
r
H
Bs
Hk
tss s
tr s
tr s
rt










L. Z. LI ET AL. 52
then
 
 
,,
4
1limin
ms
,
lup
f
i
t
t
M
FtT GtT
N
TNBtT













For all 0 Tt and for any 1

, by (9) we have
 
1liminf ,
t
TmTN BtT






So
 
TmT   (11)
and especially
 
000
liminf ,(1)
tBttt mt
N


 


 (12)
Now, we claim that
 
 
0
2
12
limsup d
ln
jj
tj
t
t
ss
ss
sk








 (13)
Suppose to the contrary that
 
 
0
2
12
limsup d
ln
jj
tj
t
t
ss
ss
sk








 (14)
By (8), there is a positive constant

satisfying


00
,
inf liminf0
,
st t
Hts
Htt








(15)
Let

be any arbitrary positive number, from (14)
there exists a 10 tt such that,
 
 
0
2
12d
ln
jj
tj
t
ss
s
ss k









 for all 1
tt
Then, for 1 tt, we have


 
 

 
 


10
10
2
12
2
0
0
0
1
12
0
,
,
,
,
,
1d
1
,
,
dd
ln
jj
ts
j
tt
ln
jj
ts
j
tt
tt
vv
tsd v
tt v
vv
ts v
tt v
t
B
H
v
Hk
H
vs
t
Hk
H
tHt































By (15), there exists a 21
tt such that, for all
2
tt,



1
0
,
,
Htt
Htt

, which implies

0
,Btt

 for all
2
tt. Since

is arbitrary, we have


00
liminf,lim ,
tt
Btt Btt
 


which contradicts (12), thus (13) holds. Then by (11) and
(13) we get
 
 
 
 
0
0
2
12
2
12
limsu p
limsu p
d
d,
ln
jj
tj
t
ln
jj
t
t
t
j
t
ms
k
s
k
ss
s
s
ss
s
s



















which contradicts (10). This completes the proof.
Remark 1 Let 1


in Theorem 2.1, Theorem 2.1
reduces to Theorem A [6]; we obtain the same result in
Theorem 2.2 in which we omit the assumption (4) in
Theorem B [6]. Therefore, Theorem 2.1 and 2.2 are gen-
eralizations and improvements of the results obtained in
[6].
Remark 2 With an appropriate choices of the func-
tions ,
H
h and k, one can derive a number of oscilla-
tion criteria for Equation (1) from our theorems.
Let ()1,0kt


 is a constant,

,tssHt

 ,
 
1
,,ts sht



 0 tst , and we have




00
,,
limlim 1
,,
tt
Hts ts
Htt tt


 

 for any 0
s
t.
Consequently, let 2


, using Theorem 2.2, we
have:
Corollary 2.1 Assume that (A1)-(A5) and (8) hold,
suppose that there exists a function




0,,mCt R
such that, fo r some 1

,
 

2
2
0
2
1
1
limsup d
,
l
tn
jj
j
t
T
M
tsqs s
tNss
mTt Tt



 


















(16)
and (10) (with


1ks

) hold. Then every solution of
Equation (1) is oscillatory.
Example 1 Let


4,t

, consider the following
second order neutral differential equation
 

 

0xt ptx tqtxt 




 (17)

L. Z. LI ET AL. 53
where
  


1,max21sin,0,
2
ptqtttf xx  ,
  
4
4d
12sin
t
ts
ss
 
, in this case 1
M

, Let
1, 2N

, by direct calculation, we get
 
 

2
2
2
2
1
limsup d
1
limsup1sin12 sind
cossincos 1
t
T
t
t
T
t
Mt s qss
Ns
t
tsssss s
t
mTTT TT















 


It is easy to verify that (10) holds, therefore, Equation
(17) is oscillatory by Corollary 2.1. However, we can
easily find that
  
0
2
01
limsup limsupd,12sin
t
t
tt
Gstts s
t
 


so condition (4) in Theorem B is not satisfied, these
show that Theorem B cannot be applied to Equatio n (17 ).
Obviously our results are superior to the resu lts obtained
before.
3. Acknowledegments
The authors are very grateful to the referee for his/her
valuable suggestions.
4. References
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