Oscillation Criteria for Second-Order Quasilinear Neutral Delay Dynamic Equations on Time Scales

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

(1.1)

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

(1.2)

(1.3)

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

(1.4)

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

(1.5)

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

(2.1)

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:

(2.2)

Lemma 2.1 (see [3, Lemma 2.4]).

Assume that there exists sufficiently large, such that

(2.3)

Then

(2.4)

Lemma 2.2.

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

(2.5)

Proof.

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

(2.6)

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

(2.7)

which implies by (1.2) that

(2.8)

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

(2.9)

which implies by (1.2) that

(2.10)

Therefore, there exist and such that

(2.11)

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

(2.12)

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

(2.13)

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

(2.14)

From we obtain

(2.15)

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

(2.16)

By we get

(2.17)

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

(2.18)

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

(2.19)

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

(2.20)

By (1.1) and (2.5), we obtain

(2.21)

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

(2.22)

By Young's inequality

(2.23)

we have

(2.24)

By Lemma 2.1, we have

(2.25)

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

(2.26)

Thus

(2.27)

Set

(2.28)

Using the inequality

(2.29)

we obtain

(2.30)

Integrating the last inequality from to we obtain

(2.31)

which yields

(2.32)

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

(2.33)

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

(2.34)

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

(2.35)

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

(2.36)

Then for From (1.1) we have

(2.37)

Integrating the above inequality from to we obtain

(2.38)

that is,

(2.39)

Integrating the last inequality from to we get

(2.40)

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 ,

(2.41)

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

(2.42)

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

(2.43)

By Young's inequality (2.23), we have

(2.44)

By Lemma 2.1, we have

(2.45)

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

(2.46)

Thus

(2.47)

Set

(2.48)

Using the inequality (2.29), we obtain

(2.49)

Integrating the last inequality from to we obtain

(2.50)

which yields

(2.51)

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 ,

(2.52)

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

(2.53)

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

(3.1)

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

(3.2)

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.