Received 9 May 2016; accepted 13 June 2016; published 16 June 2016
The theory of time scales was first proposed by Hilger  in order to unify continuous and discrete analysis. Several researchers have made greater contributions to various aspects of this new theory; see  -  . The new theory of dynamic equations on time scales not only unifies the theories of differential equations and difference equations, but also extends these classical cases to cases “in between”, e.g., to so-called q-difference equations where.
In recent years, there has been much research involving the oscillation and nonoscillation of solutions of various equations on time scales such as  -  . In this paper we study and give the sufficient conditions for oscillation of the second-order neutral delay dynamic equation
where and is unbounded time scale. Besides that, we will have hypotheses as follows throughout the paper:
(H1) is the ratio of two positive odd integers and.
(H2) are positive rd-continuous functions with satisfying.
(H3) is a strictly increasing and differentiable function such that,
(H4) is a continuous function which satisfies for all where L is a positive constant.
In addition, for the sake of clearness and convenience,we will use the notation
in the following narrative.
It is well known by reserchers in this field that an dynamic equation is called oscillatory in case all its solutions are oscillatory, and a solution of the equation is said to be oscillatory if it is neither eventually positive nor eventually negative. We only discuss those solutions x of Equation (1.1) that are not eventually zero in this paper. Moreover we refer to   for general basic background, ideas and more details on dynamic equations.
Because of, we shall consider Equation (1.1) respectively based on the case
and the other case
2. Several Lemmas
In this section, we present and prove three lemmas which play important roles in the proofs of the main results.
Lemma 1. (  ) Assume that is strictly increasing, is a time scale and
. Let. If and exist for, then exists, and
Lemma 2. (  ) Assume that x is delta-differentiable and eventually positive or eventually negative, then
We give the below lemma and prove it similar to that of Q. Zhang and X. Song (  , Lemma 3.5).
Lemma 3. Based on (1.2), assume that (H1)-(H4) hold. If x is an eventually positive solution of (1.1), there exists such that
Proof. Assume is an eventually positive solution of (1.1). That is, there exists such that
and for Because of and, we get
esaily for. At the same time for, from equation (1.1) we obtain that
so is decreasing. From (2.4), we know that is either eventually positive or eventually negative. Now we assert that.
Suppose to the contrary that there exits such that for all. Because is decreasing,
for, where. Based on the above inequality (2.5), we get
After integrating the two sides of inequality (2.6) from t2 to, we have
When, we get from (1.2) and the above (2.7), which is contradictory to. So the above hypothesis of is false. In other words, we get for. This completes the proof. □
3. Main Results
Now we state and prove our main results in this section.
Theorem 1. Based on (1.2), assume that the conditions (H1)-(H4) hold. If there exists a positive nondecreasing D-differentiable function such that for every
then (1.1) is oscillatory on.
Proof. Assume that (1.1) has a nonoscillatory solution x on. We may assume that and
for all. By the definition of, it follows. From (H3) we
know, by Lemma 3 we have, so, and, we obtain
The proof that x is eventually negative is similar. By Lemma 3 we have for all,
, and by Lemma 1 and (H3), there exists such that for all.
Using (2.2) and (2.3), we have
is unbounded above, which implies. Furthermore, from Lemma 1 we get
Thus, by (H3),
Next we define the function by
Then on, we have. From the basic knowledge of the time scale calculus that you can see in  , we obtain
From (1.1) and (H4), we get
On the other hand, because
Using (3.7) in (3.6), we have
At last, integrating (3.8) from T to t, we obtain
which creates a contradiction to (3.1). This completes the proof. □
Remark 1. From Theorem 1, we can obtain different conditions for oscillation of all solutions of (1.1) with different choices of.
Next, we give the conditions that guarantee every solution of (1.1) oscillates when (1.3) holds.
Theorem 2. Based on (1.3), assume that the conditions (H1)-(H4), (3.1) and (3.2) hold. If for every
then (1.1) is oscillatory on.
Proof. Assume that (1.1) has a nonoscillatory solution x on, then it is neither eventually positive nor eventually negative. Without loss of generality, we may assume that, then for all, , it follows and
The proof is similar when x is eventually negative. Since is decreasing for all and, it is eventually of one sign and hence is eventually of one sign. So we shall distinguish the following two cases to discuss:
(I) for; and
Case (I). The proof that is eventually positive is similar to that in Theorem 1, so it is omitted here.
Case (II). For, we have
Integrating (3.11) from t (t ≥ T) to u (u ≥ t) and letting, we have
where. Applying (3.12) to Equation (1.1), we find
Integrating (3.13) from T to t, we have
Next integrating (3.14) from T to t, we obtain
By (3.9), we have, which contradicts. This completes the proof. □
Remark 2. By Theorem 2, we get the sufficient condition of oscillation for Equation (1.1) when the condition (1.3) is satisfied, while the usual result existing is that the conditions (1.3) was established, then every solution of the Equation (1.1) is either oscillatory or converges to zero on.
Theorem 3. Based on (1.2), assume (H1)-(H4) hold and. If there exists a positive D-
differentiable function such that for every
where is as the same as that in (3.2), then (1.1) is oscillatory on.
Proof. Assume that (1.1) has a nonoscillatory solution x on. Without loss of generality, we can assume that and for all,. By the definition of, it follows. The proof when x is eventually negative is similar. Proceeding as the proof of Theorem 1, we obtained (3.3) and (3.5). Using (3.3) in (3.5), we have that on
Also, since, we have
Substituting (3.17) into (3.16), we obtain on
Now using inequality (3.7), we get
Hence, we have
This implies that on
Using (3.19) in (3.18), we have on that
Integrating both sides of this inequality from T to t, taking the limsup of the resulting inequality as and applying condition (3.15), we obtain a contradiction to the fact that for. This completes the proof. □
Using the same ideas as in the proof of Theorem 2, we can now obtain the following result based on (1.3).
Theorem 4. Under the condition (1.3), assume that the conditions (H1)-(H4), (3.9) and (3.15) hold, then (1.1) is oscillatory on. □
Now we shall reformulate the above conditions which are sufficient for the oscillation of (1.1) when (1.2) holds on different time scales:
If, Equation (1.1) becomes
and then conditions (3.1) and (3.15), respectively, become
The conditions (4.2) and (4.3) are new.
If, Equation (1.1) becomes
and conditions (3.1) and (3.15), respectively, become
At same time, the Theorems 1 and 3 are new for the case.
Example 1. Consider the second-order nonlinear delay 2-difference equations
where. This gives
The conditions (H1)-(H3) are clearly satisfied, and (H4) holds with L = 1. Because
(1.2) is satisfied. Now let for all, and then
Thus when, we have
It is easy to see that (3.1) is satisfied as well. Altogether, the Equation (4.7) is oscillatory by Theorem 1.
We thank the Editor and the referee for their comments. Research of Q. Zhang is funded by the Natural Science Foundation of Shandong Province of China grant ZR2013AM003. This support is greatly appreciated.
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