Abstract and Applied Analysis

Volume 2014 (2014), Article ID 321764, 11 pages

http://dx.doi.org/10.1155/2014/321764

Research Article

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

Quanxin Zhang and Xia Song

Department of Mathematics, Binzhou University, Shandong 256603, China

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

Academic Editor: Shurong Sun

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 [1–7]. The research results of oscillation for the second-order linear, nonlinear, or half-linear dynamic equations can be found in [8–23]. 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

so (32) is satisfied. By Theorem 6, (89) is oscillatory on . Similarly, conditions (71) and (72) are satisfied as well. By Theorem 8, we can also obtain that (89) is oscillatory. But the other known results cannot be applied in (89).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to the reviewers for their comments and suggestions. This work was supported by the Natural Science Foundation of Shandong Province of China under Grant no. ZR2013AM003, the Development Program in Science and Technology of Shandong Province of China under Grant no. 2010GWZ20401, and the Science Foundation of Binzhou University under Grant no. BZXYQNLG201009.

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