In this paper, we establish sufficient conditions for the oscillation of solutions of second order neutral delay dynamic equations [r (t )( x (t )  p (t ) x ( (t ))) ]  q (t ) f ( x ( (t ))) = 0   on an arbitrary time scale 핋.

1. International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 4 Issue 10 || December. 2016 || PP-01-08
www.ijmsi.org 1 | Page
Oscillation results for second order nonlinear neutral delay
dynamic equations on time scales
Hiba Jawaedeh, Ali Zein
Department of Applied Mathematics, Palestine Polytechnic University, Palestine.
Abstract: In this paper, we establish sufficient conditions for the oscillation of solutions of second order neutral
delay dynamic equations
0=)))((()(])))(()()()(([ txftqtxtptxtr  

on an arbitrary time scale 𝕋.
Keywords: Oscillation; Neutral; Dynamic equations; Time scale.
I. Introduction
The theory of time scales was introduced by Hilger in his Ph.D. thesis [1] in 1988 in order to unify continuous and
discrete analysis. Since then, many authors have considered the time scales theory and its usage in various aspects
of applied mathematics, see [2-9] and the references cited therein. In particular, we refer to the books of Bohner
and Peterson [3, 4] as detailed references for the time scales calculus.
Using the theory of time scales helps to avoid proving results twice, once for differential equations and once for
difference equations [3, 4]. A time scale𝕋 is a non empty closed subset of real numbers. In this theory, the so
called dynamic equations unify the classical theories for differential and difference equations if this time scale is
equal to the reals and to the integers, respevtively. Moreover, the new theory of dynamic equations is able to
extend these classical cases to cases ’in between’, for example to the so-called q-difference equations, see [3, 4].
During the last few years, there has been a growing interest in obtaining oscillation criteria for the neutral dynamic
equations on time scales, see [5-9] and the references cited therein.
This paper is concerned with the oscillation of the second order neutral delay dynamic equations of the form
),[0,=)))((()(])))(()()()(([ 0


tttxftqtxtptxtr  𝕋 (1)
Where𝕋is an arbitrary time scale unbounded from above, and ),[ 0
t 𝕋  ),[:= 0
t 𝕋.Throughout the paper it is
assumed that:
)( 1
A ),([ 0
 tCr rd 𝕋, ℝ) with 0>)(tr ;
)( 2
A ),([ 0
 tCp rd 𝕋, ℝ)and 1<)(0, 0
ptp  ;
)( 3
A ),([ 0
 tCq rd 𝕋, ℝ) and 0>)(tq ;
)( 4
A ),([ 0
1
 tC rd
 𝕋, 𝕋) with tt )( , 

=δ(t)lim
t
;
)( 5
A ),([ 0
1
 tC rd
 𝕋, 𝕋)with tt )( , 

=)(lim
t
t ;
)( 6
A f is continuous function with 0>
)(
K
x
xf
 for 0x , where K is a constant.
We construct sufficient conditions for the oscillation of equation (1) for both cases
,=)(
1



 ssr
t
(2)
and
.<)(
1



 ssr
t
(3)

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By a solution of (1) we mean a real-valued function x which satisfies (1) and 0>}|:)({|sup x
tttx  for any
0
tt x
 . Such a solution is said to be oscillatory if it has no constant sign eventually, and nonoscillatory otherwise.
There are numerous numbers of oscillation criteria obtained for oscillation for neutral differential equations, i.e.
when 𝕋=ℝ. These results have considered many different forms of equations with different conditions.
Relatively, few oscillation results for dyanmic equations on an arbitrary time scale𝕋 are known. Moreover,
considering more general forms of nonlinear dyanmic equations on an arbitrary time scale𝕋 is still a challenge.
Several authors have considered the oscillatory behaviour of (1) and related forms. In particular, Agarwal et al. [2]
obtained sufficient conditions for the oscillation of the second ordered nonlinear neutral delay dynamic equations
0,=))(,(]])()()([()([ 



txtftxtptxtr
where 0> is a quotient of odd positive integers,  and  are positive constants, )(tr and )(tp are
real-valued positive functions defined on 𝕋 and the condition (2) holds. Saker [7] also provided an oscillation
criteria for this equation considering the same assumption (2).
Oscillation results of the second order neutral dynamic equations
0=))()(]]))()()([()([ 



txtqtxtptxtr
were presented in Li et al. [6] considering the condition (2). Where 0> is a quotient of odd positive integers
with )(tr and )(tp are real-valued positive functions defined on𝕋.
Thandapan and Piramanantham [8] established an extended oscillation criteria for the dynamic equations
0,=))()(]]))()()([()([ 



txtqtxtptxtr
where 1 , and 0> are quotients of odd positive integers,  and  are positive constants. Both
conditions (2) and (3) were considered.
Zhang et al. [9] considered the oscillatory behaviour of delay dynamic equations
0,=))(()(])))(()()(()([ txtqtxtptxtr  

under the assumption (2).
Yang et al. [10] and Zhang et al. [11] discussed oscillation of the second-order nonlinear neutral dynamic
equations with distributed deviating arguments.
The aim of this paper is to obtain new sufficient conditions for equation (1) to oscilate. Both cases (2) and (3) are
considered. One of our results includs a result of Al-Hamouri and Zein [12] when 𝕋=ℝ.
In this paper, we assume that the reader is familiar with the time scale calculus. For further reading we refer the
reader to [3,4]. To prove our results, we will use the following lemma which is one form of chain rules on time
scales.
Lemma 1.1 [[3], Theorem 1.93 ] Assume that g : 𝕋→ ℝ is strictly increasing, = g( 𝕋) is a
time scale and f : → ℝ.. If and g ∆
(t) exist for t ∈ then
).())((=)))(((
~
tgtgftgf

(4)
II. The Main Result
Theorem 2.1 Let (2) holds, and suppose that 0>)(t

 . If
,=)( 

ttq (5)

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then every solution of (1) is oscillatory .
Proof. Suppose to the contrary that )(tx is a nonoscillatory solution of equation (1). Without loss of generality,
we may assume that )(tx is eventually positive (the proof is similar when )(tx is eventually negative). That is,
let 0>)(tx , 0>))(( tx  , and 0>))(( tx  for 01
ttt  .
Set
))(()()(=)( txtptxtz  (6)
By )( 6
A we have
0.>)))((()))((( txKtxf   (7)
Using (7) together with (6) and (1) we get
0.<))(()()))((()(=))()(( txtKqtxftqtztr  

Thus )()( tztr

is an eventually decreasing function. Since )(tr is positive )(tz

is either negative or
positive eventuality. If 0<)(tz

for 12
ttt  , then by using the fact that )()( tztr

is decreasing, we have
),(
)(
)(
)( 2
2
tz
tr
tr
tz

 for 2
tt  .
Integrating this inequality and using (2), we get
.=)(
)(
)()()(lim 2
2
22


 



tz
tr
t
tztrtz
tt
This contradicts the fact that )(tz is eventually positive. Hence, 0>)(tz

eventually, say that
0>)(0,>))(( tztz

 and 0>))(( tz 

for 12
ttt  . In this case )(tz is increasing. Using this fact
with (6) and )( 2
A we get
))(()()(=)( txtptztx 
))(()()( tztptz 
))()(1( tptz 
).)(1( 0
ptz 
Then using the last inequality together with )( 6
A we obtain
.for)),(()(1)))((( 20
tttzpKtxf   (8)
Now consider the term

)))((( tz  , it can be differentiated by the chain rule (4) as
0>)())((=)))((( ttztz

 (9)
Define
,
))((
1
)()(=)(
tz
tztrtw


(10)
then 0>)(tw for 2
tt  .
By differentiating )(tw , taking into consideration (9) and the fact that 0>)(tw , we have the following
))(())((
))(()))(((
)))(((
))((
1
))()((=))(( tztr
tztz
tz
tz
tztrtw 







))((
))((
)))(((
))((
1
))()((= tw
tz
tz
tz
tztr 






.
))((
1
))((()(=
))((
1
))()((
tz
txftq
tz
tztr






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Using (8) in the last inequality we get that
).()(1)( 0
tqpKtw 

(11)
Integrating (11) and using the condition (5), we obtain
.=)()()(1)(lim 2
2
0
 


twttqpKtw
tt
This contradicts the fact that )(tw is positive and this completes the proof. □
In the next results besides conditions )( 1
A - )( 6
A we further assume that:
)( 7
A The deviating arguments are commute, i.e.   = .
Theorem 2.2 Let (3) holds, and there exists a positive constant 0
 with )(<0 0
t

  . If
,=)(
)(
1








tssQ
tr
(12)
where
))}((),({=)( tqtqmintQ  ,
then every solution of (1) is oscillatory or tends to zero eventually .
Proof. Assume that )(tx is a nonoscillatory solution of equation (1). Without loss of generality, assume that
)(tx is eventually positive. That is, let 0>)(tx , 0>))(( tx  , and 0>))(( tx  for 01
ttt  . Set
)(tz as in (6), also by )( 6
A we obtain (7).
Let
),()(=)( tztrtu

(13)
then
0.<))(()()))((()(=))(( txtKqtxftqtu  

(14)
Thus )(tu is strictly decreasing and so with constant sign eventually. Since )(tr is positive )(tz

is
eventuality of one sign. Hence, either
,for0,>)( 12
ttttz 

(15)
or
.for0,<)( 12
ttttz 

(16)
If (15) holds, then equation (1) implies that
0.=))))(((())(())(( txftqtu  

Using the chain rule (4), also taking into consideration that   = together with )( 5
A and )( 6
A we
obtain
0.)))((())(())](([ 0
0
0


txtqKptu
p


Adding this inequality to (14) we get
0,)))((())(())](([))(()())(( 0
0
0


txtqKptu
p
txtKqtu 


this leads to
0.))](([)))]((())(()[())((
0
0
0


tu
p
txptxtKQtu 

 (17)

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By (6) we get that ))(()()( 0
txptxtz  and this with (17) lead to
0.))](([))(()())((
0
0


tu
p
tztKQtu 

 (18)
Using the fact that )(tz is increasing, one can find 23
tt  such that
ctztz =)())(( 3
 , for all 3
tt  ,
where c is a positive constant, then from (18) we get
.
1
))](([))(()(
0
0
Kc
tu
p
tutQ












(19)
Integrating this inequality from ttot3
, we have
.
1
)))(())((()()()( 3
0
0
3
3 Kc
tutu
p
tututtQ
t
t 







 

Since )(tu is positive, we get
.=
1
))(()()( 03
0
0
3
3
C
Kc
tu
p
tussQ
t
t 







 

(20)
Note that 0
C is a positive constant. Dividing by )(tr and integrating, we obtain
.<
)(
)(
)(
1
3
0
33


 

tr
t
CtssQ
tr t
t
tt
Which contradicts the condition (12).
Suppose that (16) holds, i.e. 2
for0,<)( tttz 

, then )(tz is decreasing. Since 0>)(tz we have
0.=)(lim 

Ltz
t
Now, recall the definition of )(tz then
))(()()(=)( txtptztx 
))(()()( tztptz 
.
))((
)(
))(( 0 







 p
tz
tz
tz

 (21)
Let 0>L . Following [13], since )(tz is decreasing, for every 0> there exists 23
tt  such that
  LtztzL ))(()(
for all 3
tt  . From this we can conclude that
.for,
))((
)(
3
tt
L
L
tz
tz




Using this in ( 21) we get
.))(()( 0 







 p
L
L
tztx

 (22)

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Choose 0> such that 0>= 00
Lp
L
L

 
then
.for,))(()( 30
ttLtztx   (23)
Substituting (23) in (14) we have
0.)))((()())(()))((()())(( 00


tztqKLtutztQKLtu  (24)
For 34
ttt  , Ltz )))(((  , then (24) becomes
).())()(( 0
tLQKLtztr 

(25)
Integrating this inequality from 4
t to t we get
).()()()()( 44
4
0
tztrssQLKLtztr
t
t

 
Hence
.)(
)(
1
)(
4
0
ssQ
tr
LKLtz
t
t
 

Integrating again we get
).()(
)(
1
)( 4
44
0
tzvssQ
vr
LLKtz
v
t
t
t
 
Taking limit as t and using (12), we get that 

=)(lim
t
tz . This contradicts the fact that 0>)(tz .
If 0=L , i.e 0=)(lim
t
tz

. Since )()(<0 tztx  taking the limit as t implies that 0)( tx eventually and
this completes the proof. □
Theorem 2.3 Let (3) holds and there exists a positive constant 0
 with )(<0 0
t

  . If
,=)( 

ttQ (26)
and there exists a positive function )(t with 0>)(t

 such that
,=)()(
)()(
1











tssQs
ttr
t


(27)
where )(tQ is defined as in Theorem 2.2, then every solution of (1) is oscillatory or tends to zero eventually .
Proof. Assume that equation (1) has a non-oscillatory solution )(tx . Without loss of generality, we assume that
there exists a 01
tt  such that 0>)(tx , 0>))(( tx  , and 0>))(( tx  for all 1
tt  .
Proceeding as in the proof of Theorem 2.2, we conclude that )()( tztr

is decreasing and )(tz

is eventually
of one sign. Hence, either (15) or (16) holds.

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If (15) holds, by proceeding as in the proof of Theorem 2.2 we obtain (20) which reads
.=
1
))(()()( 03
0
0
3
3
C
Kc
tu
p
tussQ
t
t 







 

Taking the limit as t , we have 

<)(
3
ssQ
t
, and this contradicts with (26).
If (16) holds, proceed as in the proof of Theorem 2.2 to get (25), which reads
40
,)())()(( ttfortLQKLtztr 

(28)
Define
),()(=)( tuttw  (29)
then by using the fact that )(tu

is negative we obtain

 ))()(()()(=)( tuttuttw 

.))()((<

tut
Using (28) with this inequality, we get
).()(<)( 0
tQtLKLtw 

(30)
Integrating inequality (30) from 4
t to t we get
.)()()(<)(
4
04
ssQsLKLtwtw
t
t
  
By using the definitions of )(tw and )(tu , with the fact that 0<)(tw , we obtain
,)()(<)()()(
4
0
ssQsLKLtztrt
t
t
 


or
.)()(
)()(
)(
4
0
ssQs
trt
LKL
tz
t
t


 



(31)
Integrating (31) we get
)()()(
)()(
1
)( 4
44
0
tzussQs
uru
LKLtz
u
t
t
t





  

Taking the limit as t and by using (27), a contradiction with 0>)(tz is obtained.
If 0=L , i.e 0=)(lim tz
t 
. Since )()(<0 tztx  taking the limit as t implies that 0)( tx eventually and
this completes the proof. □
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