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Based on Riccati transformation and the inequality technique, we establish some new sufficient conditions for oscillation of the second-order neutral delay dynamic equations on time scales. Our results not only extend and improve some known theorems, but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation on time scales. At the end of this paper, we give an example to illustrate the main results.

The theory of time scales was first proposed by Hilger [

In recent years, there has been much research involving the oscillation and nonoscillation of solutions of various equations on time scales such as [

where

(H_{1})

(H_{2})

(H_{3})

as

(H_{4})

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 [

Because of

and the other case

In this section, we present and prove three lemmas which play important roles in the proofs of the main results.

Lemma 1. ( [

Lemma 2. ( [

We give the below lemma and prove it similar to that of Q. Zhang and X. Song ( [

Lemma 3. Based on (1.2), assume that (H_{1})-(H_{4}) hold. If x is an eventually positive solution of (1.1), there exists

Proof. Assume

so

Suppose to the contrary that there exits

for

After integrating the two sides of inequality (2.6) from t_{2} to

When

Now we state and prove our main results in this section.

Theorem 1. Based on (1.2), assume that the conditions (H_{1})-(H_{4}) hold. If there exists a positive nondecreasing D-differentiable function

where

then (1.1) is oscillatory on

Proof. Assume that (1.1) has a nonoscillatory solution x on

_{3}) we

know

The proof that x is eventually negative is similar. By Lemma 3 we have

_{3}), there exists

Using (2.2) and (2.3), we have

Thus, by (H_{3}),

Next we define the function

Then on

From (1.1) and (H_{4}), we get

i.e.,

On the other hand, because

we get

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 (H_{1})-(H_{4}), (3.1) and (3.2) hold. If for every

where

then (1.1) is oscillatory on

Proof. Assume that (1.1) has a nonoscillatory solution x on

The proof is similar when x is eventually negative. Since

(I)

(II)

Case (I). The proof that

Case (II). For

then

Integrating (3.11) from t (t ≥ T) to u (u ≥ t) and letting

and thus

where

Integrating (3.13) from T to t, we have

Therefore,

Next integrating (3.14) from T to t, we obtain

By (3.9), we have

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 (H_{1})-(H_{4}) hold and

differentiable function

where

Proof. Assume that (1.1) has a nonoscillatory solution x on

Also, since

i.e.,

Substituting (3.17) into (3.16), we obtain on

i.e.,

Now using inequality (3.7), we get

Hence, we have

This implies that on

Using (3.19) in (3.18), we have on

Integrating both sides of this inequality from T to t, taking the limsup of the resulting inequality as

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 (H_{1})-(H_{4}), (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

and then conditions (3.1) and (3.15), respectively, become

and

The conditions (4.2) and (4.3) are new.

If

and conditions (3.1) and (3.15), respectively, become

and

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

The conditions (H_{1})-(H_{3}) are clearly satisfied, and (H_{4}) holds with L = 1. Because

(1.2) is satisfied. Now let

Thus when

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.

Quanxin Zhang,Xia Song,Li Gao, (2016) On the Oscillation of Second-Order Nonlinear Neutral Delay Dynamic Equations on Time Scales. Journal of Applied Mathematics and Physics,04,1080-1089. doi: 10.4236/jamp.2016.46112