New Oscillatory Behavior of Third-Order Nonlinear Delay Dynamic Equations on Time Scales

Abstract

A class of third-order nonlinear delay dynamic equations on time scales is studied. By using the generalized Riccati transformation and the inequality technique, four new sufficient conditions which ensure that every solution is oscillatory or converges to zero are established. The results obtained essentially improve earlier ones. Some examples are considered to illustrate the main results.

1. Introduction

In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various equations on time scales, and we refer the reader to the studies by Bohner and Saker [1] and Erbe et al. [2, 3]. And there are some results dealing with oscillatory behavior of second-order delay dynamic equations on time scales [4–10]. However, there are few results dealing with the oscillation of the solutions of third-order delay dynamic equations on time scales, we refer the reader to the papers [11–14].

In this paper, we consider new oscillatory behavior of all solutions of the third-order nonlinear delay dynamic equation where is the ratio of two positive odd integers.

Throughout this paper, we will assume the following hypotheses.) is a time scale (i.e., a nonempty closed subset of the real numbers ) which is unbounded above, and with , we define the time scale interval of the form by .(), , are positive and real-valued rd-continuous functions defined on , and , satisfy () is a strictly increasing and differentiable function, such that () is a continuous function and there exists some positive constant such that for all .

By a solution of (1), we mean a nontrivial function satisfying (1) which has the properties for , and . Our attention is restricted to those solutions of (1) which satisfy for all , where is the space of -continuous functions. A solution of (1) is said to be oscillatory on if it is neither eventually positive nor eventually negative. Otherwise it is called nonoscillatory. The equation itself is called oscillatory if all its solutions are oscillatory.

If ,  , then (1) simplifies to the third-order nonlinear dynamic equation

If, furthermore, , ,  , then (1) reduces to the third-order linear dynamic equation

If, in addition, , then (1) reduces to the nonlinear delay dynamic equation

In 2005, Erbe et al. [11] considered the general third-order nonlinear dynamic equation (4). By employing the generalized Riccati transformation techniques, they established some sufficient conditions which ensure that every solution of (4) is oscillatory or converges to zero. In 2007, Erbe et al. [12] studied the third-order linear dynamic equation (5), and they obtained Hille and Nehari type oscillation criteria for (5). In 2011, Han et al. [13] extended and improved the results of [12], meanwhile obtaining some oscillatory criteria for (6). In 2014, Gao et al. [14] extended some results of [12, 13] to (1). On this basis, we continue to discuss the oscillation of solutions of (1). By using the generalized Riccati transformation and the inequality technique, we obtain some new sufficient conditions which guarantee that every solution of (1) is oscillatory or converges to zero. Our results will improve some results that have been established in [11–14].

Throughout this paper, we will make use of the following product and quotient rules: For and a differentiable function , the Cauchy integral of is defined by The integration by parts formula reads and improper integrals are defined in the usual way by For more details, see [15, 16].

2. Several Lemmas

In this section we present several lemmas that will be needed in the proofs of our results in Section 3.

Lemma 1. Assume that is an eventually positive solution of (1), then there exists such that either or

Proof. Assume that is an eventually positive solution of (1), then there exists such that and for all . From (1); we obtain Hence, is decreasing and therefore eventually of one sign, so is either eventually positive or eventually negative. We assert that for all .
If there exists such that , we get Let , then Integrating (16) from to provides Then there exists such that . Similarly, , we obtain which contradicts with . So ; this implies that or for all . This completes the proof.

Lemma 2 (see [17]). Assume that () and the following conditions hold:(I), where   for some ;(II),  ,    for .
Then, for each , there exists a constant ,   such that

Lemma 3 (see [15]). If is differentiable, then

Lemma 4 (see [12]). Assume that satisfies Then where the Taylor monomials are defined recursively by

Lemma 5 (see [18]). Assume that and are nonnegative real numbers. Then where the equality holds if and only if .

Lemma 6. Assume that is an eventually positive solution of (1) which satisfies case (II) in Lemma 1, if either or Then .

Proof. Assume that is an eventually positive solution of (1) which satisfies case (II) in Lemma 1. Then is decreasing and . If ; it is easy to see that there exists such that for all . From (14),
If (25) holds, then integrating (27) from to   , we get This is contrary to .
If (26) holds, then integrating (1) from to , we get and hence, Again, integrating this inequality from to , we obtain Finally, integrating the last inequality from to , we get Hence by (26), we obtain , which contradicts . Thus, we get . This completes the proof.

Lemma 7 (see [19]). Let . Then for positive rd-continuous functions , one has where and .

3. Main Results

New we state and prove the main results of this paper.

Theorem 8. Assume that ()–(), (26), and hold. Furthermore, suppose that there exists a positive function with , and for all sufficiently large , there exists , such that where . Then every solution of (1) is either oscillatory or converges to zero.

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality, we may assume that there exists sufficiently large , such that and for all . In the case when is eventually negative, the proof is similar. By Lemma 1, we see that satisfies either case (I) or case (II).
If case (I) holds, then ,  . Define the function by Then . By the product rule (7) and the quotient rule (8), we have Let , from case (I) in Lemma 1, we get ,  . In view of that and , it is not difficult to see that . Thus, by Lemma 2, for every , there exists with , such that for all ; this implies that By Lemma 3, we get , that is, Using (38) and (39), Let for all ; it is easy to see that ,  ,  ,  . Thus, by Lemma 4, there exists such that for all . Then, we get From , we obtain . By (41), we get Therefore, from (38) and (42), there exists with such that Using (43), we get that is, Now, set where ,   and . Using the equality (24), we obtain From (47), we obtain Integrating (48) from to , we get consequently, This is contrary to (34).
If case (II) holds, from (26), we get . This completes the proof.

Theorem 9. Assume that ()–(), (26), and hold. Furthermore, suppose that there exist functions , where such that and has a nonpositive continuous -partial derivative on with respect to the second variable and satisfies, for all sufficiently large , that there exists , such that where and are defined in Theorem 8. . Then every solution of (1) is either oscillatory or converges to zero.

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality, we may assume that there exists sufficiently large , such that and for all . In the case when is eventually negative, the proof is similar. By Lemma 1, we see that satisfies either case (I) or case (II). If case (I) holds, we proceed as in the proof of Theorem 8 and get (45). In (45), replace by and multiply both sides by and integrate with respect to from to , ; we get Integrating by parts using (51) and (52), we obtain and so Now, set where , and . Using inequality (24), we obtain Combining (56) and (58), we get which contradicts (53).
If case (II) holds, from (26), we get . This completes the proof.

If (53) is not held, then we get the following result.

Theorem 10. Assume that ()–(), (26), and hold. Furthermore, suppose that there exist functions , where , such that (51) holds, has a nonpositive continuous -partial derivative on with respect to the second variable and satisfies (52). Assume that and a real rd-continuous function such that for , where and are defined in Theorem 8, . Then every solution of (1) is either oscillatory or converges to zero.

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality, we may assume that there exists sufficiently large , such that and for all . In the case when is eventually negative, the proof is similar. By Lemma 1, we see that satisfies either case (I) or case (II).
If case (I) holds, proceeding as in the proof of Theorem 9, we get that (56) and (58) hold. Then we conclude that From (63), we obtain By (56), we get We denote meanwhile noting that (63), we obtain Now we assert that holds. Suppose to the contrary that by (60), there exists a constant such that from (71); there exists for arbitrary real number such that By (10), we obtain From (72), there exists , we get for , so that . Since is arbitrary, we obtain Selecting a sequence : with satisfying then there exists a constant such that for sufficiently large positive integer . From (75), we can easily see (77) implies that From (77) and (78), we obtain that is, for sufficiently large positive integer , which together with (79) implies On the other hand, by Lemma 7, we obtain The above inequality shows that Hence, (82) implies This contradicts (61). Therefore (70) holds. Noting for , by using (70), we obtain This contradicts (62). This completes the proof.

If case (II) holds, from (26), we get . This completes the proof.

Theorem 11. Assume that ()–(), (26), (52), (60), (62), and hold, where ,  , and are defined in Theorem 10. Furthermore suppose that there is a real rd-continuous function such that for , where is defined in Theorem 8, . Then every solution of (1) is either oscillatory or converges to zero.

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality, we may assume that there exists sufficiently large , such that and for all . In the case when is eventually negative, the proof is similar. By Lemma 1, we see that satisfies either case (I) or case (II).
If case (I) holds, proceeding as in the proof of Theorem 9, we get that (56) and (58) hold. We conclude that From (88), we obtain Using (87) and (92), we get Thus, there exists a sequence : with such that We define and also, as in the proof of Theorem 10. From (56) and (91), we obtain For the above sequence , we get Similar to the proof of Theorem 10, we get (70). The rest proofs are the same as the Theorem 10, here omitted. This completes the proof.

Remark 12. From Theorems 9, 10, and 11, we can obtain different sufficient conditions for the oscillation of (1) with different choices of the functions and .

Remark 13. The theorems in this paper are new even for the cases of and .

Example 14. Consider the third-order nonlinear delay dynamic equation Here , , , , , and .
Conditions ()–() are clearly satisfied, and () holds with .  , and so (26) holds.
Let , then . Since so that Then by Theorem 8, every solution of (97) is either oscillatory or converges to zero. But the other known results cannot be applied in (97).

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. This research is supported by Shandong Provincial Natural Science Foundation (ZR2011AL001 and ZR2013AM003) and the Science Foundation of Binzhou University (BZXYQNLG201009).