Research Article | Open Access
Oscillation Criteria of Third-Order Nonlinear Impulsive Differential Equations with Delay
This paper deals with the oscillation of third-order nonlinear impulsive equations with delay. The results in this paper improve and extend some results for the equations without impulses. Some examples are given to illustrate the main results.
In this paper, we are concerned with oscillation of the third-order nonlinear impulsive equations with delay where is the delay, is the sequence of impulsive moments which satisfies and , .
Throughout this paper, we will assume that the following assumptions are satisfied: (H1) is continuous in , for ; (H2) for , where is continuous in , , for all ; (H3) are positive constants.
Our attention is restricted to those solutions of (1) which exist on half line and satisfy for any . For the general theory of impulsive differential equations with/without delay, we refer the readers to monographs or papers [1–4]. A solution of (1) is said to be oscillatory if it has arbitrarily large zeros; otherwise, it is nonoscillatory. It is well known that there is a drastic difference in the behavior of solutions between differential equations with impulses and those without impulses. Some differential equations are nonoscillatory, but they may become oscillatory if some proper impulse controls are added to them, see  and Example 13 in Section 4. In the recent years, the oscillation theory and asymptotic behavior of impulsive differential equations and their applications have been and still are receiving intensive attention. For contribution, we refer to the recent survey paper by Agarwal et al.  and the references cited therein. But to the best of the authors' knowledge, it seems that little has been done for oscillation of third-order impulsive differential equations .
Our aim in this paper is to establish some new sufficient conditions which ensure that the solutions of (1) oscillate or converge to a finite limit as tends to infinity. In particular, we extend some results in [9, 11] to impulsive delay differential equations. The results in this paper are more general compared by those obtained by Mao and Wan  and improve some of the results in  (see Example 13 in Section 4). The new results will be proved by making use of the techniques used in [9, 11].
The paper is organized as follows. In Section 2, we prove some lemmas which play important roles in the proof of the main results. In Section 3, some new sufficient conditions which guarantee that the solution of (1) oscillates or converges to a finite limit are established. In Section 4, two examples are given to illustrate the main results.
2. Preliminary Results
In this section, we state and prove some lemmas which we will need in the proofs of the main results. First of all, we introduce the following notations: and are the sets of real numbers and positive integer numbers, respectively, is defined by
The following lemma is from Lakshmikantham et al. [3, Page 32, Theorem 1.4.1].
Lemma 1. Assume that(i) is the impulse moments sequence with , ; (ii), and for , , it holds that where , , and are real constants. Then,
Lemma 2. Suppose that is an eventually positive solution of (1), and Then, it holds that one of the following two cases for sufficiently large : (i), and , (ii), and , with and .
Proof. Assume that is an eventually positive solution of (1). We may assume that there exists such that and for . First, we assert that for any . Suppose not, there exists some such that . By
we have monotonically decreasing in , . Thus, , , . Consider the impulsive differential inequalities
By Lemma 1, we have
There are two cases of the sign of .
Case 1. If there exists some such that , since , then and . By induction it easily show , and hence for . So, we obtain the following impulsive differential inequalities: which follows from Lemma 1 that
From (11) and applying Lemma 1, noting that for and for any , we have Thus, by (5) we have for sufficiently large which is a contradiction.
Case 2. If for any , noting that for , we have . By induction, we get that for any , . So the following impulsive differential inequalities hold: According to Lemma 1, we get Hence, the condition (6) implies that when is sufficiently large, which contradicts to for again. In terms of the above discussion, we see that for any with sufficiently large . Consequently, noting that for any , we have .
Next, if there exists a such that , then , . Therefore, by induction, we have for , . So case (i) is satisfied. Otherwise, if for all , then . Thus, for , using , we have ; hence, case (ii) is satisfied. This completes the proof.
Remark 3. Suppose that is an eventually negative solution of (1). If (5) and (6) hold, one can prove it holds that one of the following two cases in a similar way as Lemma 2: (i), and , , (ii), and , , with and .
Lemma 4. Let be a piecewise continuous function on , which is continuous at and is left continuous at . If (1) for ; (2) is monotone nonincreasing (monotone nondecreasing) on for large enough; (3) converges, then .
Proof. First, we claim that is convergence. In fact, since is decreasing on , and is defined in Lemma 2), then Obviously, by induction, we can get Since is bounded, we conclude that is bounded, which follows that there exists such that Hence, is convergence since is convergence, which follows from Lemma 4 that exists. The proof is complete.
3. Main Results
In this section, we establish some sufficient conditions which guarantee that every solution of (1) either oscillates or has a finite limit. Occasionally, we will make the additional assumption where here it is understood that Now we are ready to state and prove the main results in this paper. The results will be proved by making use of the technique in .
Theorem 6. Assume that (5), (6), and (19) hold, and is a solution of (1). Furthermore, assume that , , and for . If there exists a positive differentiable function such that where , and Then is oscillatory.
Proof. Let be a nonoscillatory solution of (1), without loss of generality, we may assume that eventually ( eventually can be achieved in the similar way). By Lemma 2, either case (i) or case (ii) in Lemma 2 holds. Assume that satisfies case (i), then , for , ( is defined in Lemma 2). Define the Riccati transformation by
Thus, for , , and
If , namely, , is decreasing in and , respectively. In view of the following we have Thus,
If , that is, , then
Similarly, we have Thus, we obtain
On the other hand,
Observing that , we have
Applying Lemma 1, it follows from (30), (31), and (32) that which yields for all large . This is contrary to , and so, case (i) in Lemma 2 is not possible.
If satisfies the case (ii) in Lemma 2, that is, , and , , which proves that the solution is positive and decreasing. Integrating (1) from to (), we obtain Noting and , then it holds that which leads to and hence Integrating the above inequality again from to (), one has Using and , we have Now, we integrate the last inequality from to () to obtain Since and is decreasing, then for , , . Thus, we get and then, which contradicts the condition (19). The proof is complete.
Proof. By the proof of Theorem 6, we know the case (i) in Lemma 2 is not possible, too, since the condition (19) is not required to prove it. So it suffices to show if there is a solution satisfying case (ii) in Lemma 2, that is, if with and . then exists. This is obtained by applying Lemma 5 which leads to exists. The proof is complete.
Proof. By the proof of Theorem 7, exists, and we define it by . We now show . If not, then . So, . Hence, there exists such that for . Then and note that since , which imply that Thus, in virtue of (44) it holds that and contradicts for large enough, the proof is complete.
Remark 9. Theorem 6 and Corollary 8 extend the results in [11, Theorem 3.1] and [9, Corollary 1], respectively. In fact, when for which implies that the impulses in (1) disappear. In such a case, (5) and (6) hold naturally, and (21) and (44) are reduced to which are similar to those in [11, Theorem 3.1] and [9, Corollary 1], respectively.
Next, we present some new oscillation results for (1), by using an integral averaging condition of Kamenev’s type.
Theorem 10. Assume (5), (6), and (19) hold. Furthermore, , and , . If there exists a positive differentiable function such that where is defined by (21) and . Then every solution of (1) is oscillatory.
Proof. We choose large enough such that Lemma 2 holds. By Lemma 2 there are two possible cases. First, if the case (i) holds, proceeding as in the proof of Theorem 6, we will end up with (32). By (30), we have
If , for , we obtain An integration by parts of the right-hand side leads to Take into account (31), (32), , and , we have If , similarly we also get So, it yields which follows that
Hence, which is a contradiction of (48).
If case (ii) holds, then as a manner with case (ii) in Theorem 6, it is not possible, too. The proof is complete.
Remark 12. Corollary 11 is an extension of [11, Theorem 3.2] into impulsive case. Especially, let in (48), it reduces to naturally, which can be considered as the extension of Kamenev-type oscillation criteria for third-order impulsive differential equations with delay (see [8, 14, 15]).
Example 13. Consider the third-order impulsive differential equation with delay where , are constants, for any .
When for any , the impulses in (59) disappear, by [16, Theorem 4], (59) is nonoscillatory if and . However, we may change its oscillation by proper impulsive control. In fact, let and and in (59); choose , , and ; a simple calculation leads to Then, let Obviously, (5), (6), and (19) hold, and Thus, (21) is also satisfied. By Theorem 6, every solution of (59) is oscillatory.
Example 15. Consider the third-order impulsive differential equation with delay where , , , .
The author is very grateful to Professor H. Saker who presents the references [8–12, 17] and gives many helpful suggestions, which leads to an improvement of this paper. The work is supported in part by the NSF of Guangdong province (S2012010010034).
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