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Oscillation Criteria for Second-Order Quasilinear Neutral Delay Dynamic Equations on Time Scales
Advances in Difference Equations volume 2010, Article number: 512437 (2010)
Abstract
We establish some new oscillation criteria for the second-order quasilinear neutral delay dynamic equations on a time scale
, where
,
. Our results generalize and improve some known results for oscillation of second-order nonlinear delay dynamic equations on time scales. Some examples are considered to illustrate our main results.
1. Introduction
In this paper, we are concerned with oscillation behavior of the second order quasilinear neutral delay dynamic equations

on an arbitrary time scale where
and
are quotient of odd positive integers such that
,
, and
are rd-continuous functions on
and
, and
are positive,
the so-called delay functions
satisfy that
for
and
as
for
and there exists a function
which satisfies that
and
as
Since we are interested in the oscillatory and asymptotic behavior of solutions near infinity, we assume that and define the time scale interval
by
We will also consider the two cases


Recently, there has been a large number of papers devoted to the delay dynamic equations on time scales, and we refer the reader to the papers in [1–17].
Agarwal et al. [1], Sahiner [10], Saker [11], Saker et al. [12], and Wu et al. [15] studied the second-order nonlinear neutral delay dynamic equations on time scales

where and (1.2) holds. By means of Riccati transformation technique, the authors established some sufficient conditions for oscillation of (1.4).
Sun et al. [14] considered (1.1), where and (1.2) holds. The authors established some oscillation results of (1.1). To the best of our knowledge, there are no results regarding the oscillation of the solutions of (1.1) when (1.3) holds.
We note that if (1.1) becomes the second-order Emden-Fowler neutral delay differential equation

Chen and Xu [18] as well as Xu and Liu [19] considered (1.5) and obtained some oscillation criteria for (1.5) when Qin et al. [20] found that some results under the case when
in [18, 19] are incorrect.
The paper is organized as follows. In the next section, by developing a Riccati transformation technique some sufficient conditions for oscillation of all solutions of (1.1) on time scales are established. In Section 3, we give some examples to illustrate our main results.
2. Main Results
In this section, by employing the Riccati transformation technique, we establish some new oscillation criteria for (1.1). In order to prove our main results, we will use the formula

which is a simple consequence of Keller's chain rule [21, Theorem 1.90]. Also, we need the following lemmas.
It will be convenient to make the following notations:

Lemma 2.1 (see [3, Lemma 2.4]).
Assume that there exists sufficiently large, such that

Then

Lemma 2.2.
Assume that (1.2) holds; Furthermore,
is an eventually positive solution of (1.1). Then there exists
such that

Proof.
Let be an eventually positive solution of (1.1). Then there exists
such that
and
for
,
From (1.1), we have

for all and so
is an eventually decreasing function.
We first show that is eventually positive. Otherwise, there exists
such that
; then from (2.6) we have
for
and so

which implies by (1.2) that

and this contradicts the fact that for all
Hence, we have that (2.5) holds and completes the proof.
Lemma 2.3.
Assume that (1.2) holds, and
Furthermore, assume that there exists
such that
and
Then an eventually positive solution
of (1.1) satisfies eventually (2.5) or
Proof.
Suppose that is an eventually positive solution of (1.1). Then there exists
such that
and
for
,
From (1.1), we have that (2.6) holds for all
and so
is an eventually decreasing function.
We first show that is eventually positive. Otherwise, there exists
such that
; then from (2.6) we have
for
and so

which implies by (1.2) that

Therefore, there exist and
such that

Thus, we can choose some positive integer such that
for
and

The above inequality implies that for sufficiently large
which contradicts the fact that
is eventually positive. Hence
is eventually positive. Consequently, there are two possible cases:
(i) is eventually positive, or
(ii) is eventually negative.
If there exists a such that case (ii) holds, then
exists, and
; we claim that
Otherwise,
We can choose some positive integer
such that
for
and we obtain

which implies that and so
which contradicts
Now, we assert that
is bounded. If it is not true, then there exists
with
as
such that

From we obtain

which implies that ; it contradicts
Therefore, we can assume that

By we get

which implies that so
Hence,
The proof is complete.
Theorem 2.4.
Assume that (1.2) holds, , and
Furthermore, assume that there exist positive rd-continuous
-differentiable functions
and
such that, for all sufficiently large
for

Then every solution of (1.1) is oscillatory.
Proof.
Suppose that (1.1) has a nonoscillatory solution We may assume without loss of generality that
,
for all
By Lemma 2.2, there exists
such that (2.5) holds. Define the function
by

Then By the product rule and the quotient rule, noteing (2.19), we have

By (1.1) and (2.5), we obtain

In view of from (2.1), we have
By (2.20), we obtain

By Young's inequality

we have

By Lemma 2.1, we have

Hence, by (2.19) and (2.22), we obtain

Thus

Set

Using the inequality

we obtain

Integrating the last inequality from to
we obtain

which yields

which leads to a contradiction to (2.18). The proof is complete.
Theorem 2.5.
Assume that (1.2) holds, , and
Furthermore, assume that there exist positive rd-continuous
-differentiable functions
and
such that, for all sufficiently large
for

Then every solution of (1.1) is oscillatory.
Proof.
Suppose that (1.1) has a nonoscillatory solution We may assume without loss of generality that
,
for all
By Lemma 2.2, there exists such that (2.5) holds. Defining the function
as (2.19), we proceed as in the proof of Theorem 2.4, and we get (2.20). In view of
using (2.1), we have
From (2.20) we obtain

The remainder of the proof is similar to that of Theorem 2.4, and hence it is omitted.
Theorem 2.6.
Assume that (1.3) holds, and
Furthermore, assume that there exist positive rd-continuous
-differentiable functions
, and
such that
then for all sufficiently large
for
one has that (2.18) holds, and

Then every solution of (1.1) is either oscillatory or converges to zero.
Proof.
We proceed as in Theorem 2.4, and we assume that ,
for all
From the proof of Lemma 2.2, we see that there exist two possible cases for the sign of
If is eventually positive, we are then back to the proof of Theorem 2.4 and we obtain a contradiction with (2.18).
If ,
then there exist constants
such that
,
,
and
Since
is bounded, we let
,
From definition of
noting
we have
; hence, we have
On the other hand, ; hence,
Assume that
Then there exist a constant
and
such that
for
Define the function

Then for
From (1.1) we have

Integrating the above inequality from to
we obtain

that is,

Integrating the last inequality from to
we get

We can easily obtain a contradiction with (2.35). Hence, This completes the proof.
From Theorem 2.6, we have the following result.
Theorem 2.7.
Assume that (1.3) holds, ,
, and
Furthermore, assume that there exist positive rd-continuous
-differentiable functions
, and
such that, for all sufficiently large
for
one has that (2.33) and (2.35) hold. Then every solution of (1.1) is either oscillatory or converges to zero.
The proof is similar to that of the proof of Theorem 2.6; hence, we omit the details.
In the following, we give some new oscillation results of (1.1) when
Theorem 2.8.
Assume that (1.2) holds, ,
, and
Furthermore, there exists
such that
and
If there exist positive rd-continuous
-differentiable functions
and
such that, for all sufficiently large
for
,

then every solution of (1.1) is oscillatory or tends to zero.
Proof.
Suppose that (1.1) has a nonoscillatory solution We may assume without loss of generality that
,
for all
By Lemma 2.3, there exists
such that (2.5) holds, or
Assume that (2.5) holds. Define the function
as (2.19), and then we get (2.20). By (1.1), we obtain

In view of from (2.1), we have
By (2.20), we obtain

By Young's inequality (2.23), we have

By Lemma 2.1, we have

Hence, by (2.19) and (2.43), we obtain

Thus

Set

Using the inequality (2.29), we obtain

Integrating the last inequality from to
we obtain

which yields

which leads to a contradiction with (2.41). The proof is complete.
Theorem 2.9.
Assume that (1.2) holds, ,
, and
Furthermore, there exists
such that
and
If there exist positive rd-continuous
-differentiable functions
and
such that, for all sufficiently large
for
,

then every solution of (1.1) is oscillatory or tends to zero.
Proof.
Suppose that (1.1) has a nonoscillatory solution We may assume without loss of generality that
,
for all
By Lemma 2.3, there exists
such that (2.5) holds, or
Assume that (2.5) holds.
Define the function as (2.19), and then we get (2.20). In view of
using (2.1), we have
From (2.20) we obtain

The remainder of the proof is similar to that of Theorem 2.8, and hence it is omitted.
Remark 2.10.
One can easily see that the results obtained in [1, 10–12, 15] cannot be applied in (1.1), so our results are new.
3. Examples
In this section, we will give some examples to illustrate our main results.
Example 3.1.
Consider the second-order quasilinear neutral delay dynamic equations on time scales

where and we assume that
Let ,
,
,
and
Take
It is easy to show that (2.18) and (2.35) hold. Hence, by Theorem 2.6, every solution of (3.1) oscillates or tends to zero.
Example 3.2.
Consider the second-order quasilinear neutral delay dynamic equations on time scales

where and we assume there exists
such that
and
Let ,
,
,
,
,
Take
It is easy to show that (2.41) holds. Hence, by Theorem 2.8, every solution of (3.2) oscillates or tends to zero.
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Acknowledgment
This research is supported by the Natural Science Foundation of China (60774004, 60904024), China Postdoctoral Science Foundation funded project (20080441126, 200902564), Shandong Postdoctoral funded project (200802018), the Natural Science Foundation of Shandong (Y2008A28, ZR2009AL003), and also the University of Jinan Research Funds for Doctors (B0621, XBS0843).
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Sun, Y., Han, Z., Li, T. et al. Oscillation Criteria for Second-Order Quasilinear Neutral Delay Dynamic Equations on Time Scales. Adv Differ Equ 2010, 512437 (2010). https://doi.org/10.1155/2010/512437
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Keywords
- Positive Integer
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Equation
- Differentiable Function