`Abstract and Applied AnalysisVolume 2014 (2014), Article ID 321764, 11 pageshttp://dx.doi.org/10.1155/2014/321764`
Research Article

On the Oscillation for Second-Order Half-Linear Neutral Delay Dynamic Equations on Time Scales

Department of Mathematics, Binzhou University, Shandong 256603, China

Received 14 January 2014; Accepted 26 February 2014; Published 28 April 2014

Copyright © 2014 Quanxin Zhang and Xia Song. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We discuss oscillation criteria for second-order half-linear neutral delay dynamic equations on time scales by using the generalized Riccati transformation and the inequality technique. Under certain conditions, we establish four new oscillation criteria. Our results in this paper are new even for the cases of and .

1. Introduction

In recent years, the research results relevant to oscillation of second-order dynamic equations on time scales are emerging, such as [17]. The research results of oscillation for the second-order linear, nonlinear, or half-linear dynamic equations can be found in [823]. On the basis of the above work, we will study the oscillatory behavior of all solutions of second-order half-linear neutral delay dynamic equation in this paper, which is given as follows: where . In this paper, we give the following hypotheses.(H1) is a time scale (i.e., a nonempty closed subset of the real numbers ) which is unbounded above and for with ; we define the time scale interval of the form by .(H2) are positive rd-continuous functions such that .(H3) is a strictly increasing and differentiable function such that (H4) is a continuous function such that, for some positive constant ,

By a solution of (1), we mean a nontrivial real-valued function satisfying (1) for . A solution of (1) is called oscillatory if it is neither eventually positive nor negative; otherwise, it is called nonoscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory. Our attention is restricted on those solutions of (1) which are not eventually identically zero.

The purpose of this paper is to establish the oscillation criteria of Philos [24] for (1). When the two famous results of Philos [24] about oscillation of second-order linear differential equations are extended to (1) in this paper. At the same time, when we obtain two criteria of (1) about that each solution is either oscillatory or converges to zero.

The paper is organized as follows. In Section 2, we present some basic definitions and useful results about the theory of calculus on time scales. In Section 3, we give six lemmas. Section 4 introduces the main results of this paper. We establish four new oscillatory criteria when the condition (4) or (5) holds, respectively, for the solutions of (1).

2. Some Preliminaries

On the time scale we define the forward and backward jump operators by

A point is said to be left-dense if it satisfies , right-dense if it satisfies , left-scattered if it satisfies , and right-scattered if it satisfies . The graininess of the time scale is defined by . For a function , the (delta) derivative is defined by if is continuous at and is right-scattered. If is right-dense, then the derivative is defined by provided this limit exists. A function is said to be rd-continuous if it is continuous at each right-dense point and if there exists a finite left limit at all left-dense points. Denoted by the set of rd-continuous functions on and the set of differentiable function on , whose derivative is rd-continuous. The derivative of , the shift of , and the graininess are related by the formula

We will make use of the following product and quotient rules for the derivative of the product and the quotient of two differentiable functions and :

For , the Cauchy integral of is defined by

The integration by parts formula reads and the infinite integral is defined by

For more details, see [8, 9].

3. Several Lemmas

In this section, we present five lemmas that will be needed in the proofs of our results in Section 4. Lemma 1 is the theorem 1.93 of [8]; Lemma 2 is the simple corollary of theorem 1.90 in [8]; Lemma 3 is the theorem 41 in [25]; and Lemma 4 is the theorem 3 in [26].

Lemma 1. Assume that is strictly increasing and is a time scale. Let . If and exist on , where
then

Lemma 2. If is differentiable, then

Lemma 3. Assume that and are nonnegative real numbers, then where the equality holds if and only if .

Lemma 4. Let with . Then for positive rd-continuous functions we have where and .

Lemma 5. Assume that (H1)–(H4) and (4) hold. Let be an eventually positive solution of (1). Then there exists such that

Proof. Suppose that is an eventually positive solution of (1). There exists such that and for . From the definition of , we get for , and at the same time for , from (1) we get
Hence,   is decreasing. So,   is either eventually positive or eventually negative. Therefore, for arbitrary , we have
Otherwise, we assume that (22) is not satisfied, then there exits such that for all . By (21) we have for , where . By (23), we get that is
After integrating the two sides of inequality (25) from to , we have
Nextly, we find the limits of the two sides of (26) when . From (4), we get . Therefore, is eventually negative, which is contradictory to . So the inequality (22) holds. This completes the proof.

4. Main Results

Firstly, the two famous results of Philos [24] about oscillation of second-order linear differential equations are extended to (1) when condition (4) is satisfied.

Theorem 6. Assume that (H1)–(H4) and (4) hold. Let be rd-continuous function, such that and has a nonpositive continuous -partial derivative with respect to the second variable and satisfies where is a rd-continuous function. If there exist a positive and differentiable function such that for , and a real rd-continuous function such that where , then (1) is oscillatory on .

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality we may assume that there exists a , such that and for all . By the definition of , it follows that
Since , there exists , such that , for all . Then for , we have
By Lemma 5 and (H3), we obtain that on (where is short hand for ), and holds. Moreover, using Lemmas 2 and 5, it follows that
In Lemma 1, let , and is unbounded above by (H1), so , and by (H3); using Lemma 1, we get
Thus
By the above inequality and the first inequality in (35), we obtain that holds on . Now we define the function by
Then we have on , and and then we obtain on , where . Thus, for every with , by (13), we get where . So using Lemma 3, let
Using the inequality (18), we have where . Thus
From (44) and (47), we obtain that is,
From condition (32), we have
By (44), we have In the above inequality, take , and write and meanwhile noting that (50), we obtain
Now we assert that holds. Suppose to the contrary that by (28), there exists a constant such that
From (55), there exists a for arbitrary real number such that for . By (13), we have
From (56), there exists a such that for . So . Since is arbitrary, we have
Selecting a sequence with satisfying and then there exists a constant such that for sufficiently large positive integer . From (59), we can easily see and (61) implies that
From (61) and (62), we have that is, for sufficiently large positive integer , which together with (63) implies
On the other hand, by Lemma 4, we obtain
The above inequality shows that
Hence, (66) implies which contradicts (30). Therefore (54) holds. Noting for , by using (54), we obtain which is contradicting with (31). This completes the proof.

Remark 7. From Theorem 6, we can obtain different conditions for oscillation of all solutions of (1) with different choices of and . For example, or .

Theorem 8. Assume that (H1)–(H4), (4), (28)-(29), and (31) hold. Suppose that , , , and are defined in Theorem 6. Assume that where , . Then (1) is oscillatory on .

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality we may assume that there exists a , such that and for all . So and there exists a such that for . Define the function by
We proceed as in the proof of Theorem 6 to obtain (44) and (47), so that
Hence, (72) implies
From the above inequality and (71), we have
Therefore, there exists a sequence with such that
Definitions of and are as in Theorem 6. From (44), and noting (77), we have
For the above sequence ,
We proceed by reduction to absurdity to obtain (54). The rest of the proof is similar to that of Theorem 6 and hence is omitted. This completes the proof.

If (4) is not satisfied, that is, if the condition (5) holds, we can obtain the following result.

Theorem 9. Assume that (H1)–(H4), (5), and (28)–(32) hold. Suppose that , , , and are defined in Theorem 6. Assume that holds. Then every solution of (1) is either oscillatory or converges to zero on .

Proof. As the proof of Theorem 6, assume that (1) has a nonoscillatory solution on . Without loss of generality we may assume that there exists , such that and for all . So and there exists such that for . In the proof of Lemma 5, we find that is either eventually positive or eventually negative. Thus, we shall distinguish the following two cases:(I) for ;(II) for .
Case (I). When is eventually positive, the proof is similar to that of the proof of Theorem 6, and we can obtain that (1) is oscillatory.
Case (II). When is eventually negative, is decreasing and exists. Therefore, there exists , such that for . Define the function . Equations (1) and (85) yield
The inequality (86) is the assumed inequality of [8, Theorem 6.1] (see also [27, Lemma 1]). All assumptions of [8, Theorem 6.1] are satisfied as well. Hence the conclusion of [8, Theorem 6.1] holds, that is, for all , and thus for all . Assuming and using (83) in (88), we can get , and this is a contradiction to the fact that for . Thus , that is . Then, it follows from that holds. This completes the proof.

Using the same method as in the proofs of Theorems 8 and 9, we can easily obtain the following result.

Theorem 10. Assume that (H1)–(H3), (5), (28)-(29), (31), (71)-(72), and (83) hold. Suppose that , , , and are defined in Theorem 8. Then every solution of (1) is either oscillatory or converges to zero on .

Remark 11. The theorems in this paper are new even for the cases of and .
Example. Consider a second-order half-linear delay 2-difference equation where . Here, we have
Then is unbounded above, , and . Conditions (H1)–(H3) are clearly satisfied, and (H4) holds with . Next, we have
Hence (4) is satisfied. Now let , then
Since
let ; then condition (28) holds. We have so condition (28) holds. Let as , then for all , and for all . Hence
We get and thus condition (30) holds. Let , then that is, condition (31) holds. Since then
Moreover, (97) implies
Thus, when is enough large, we have