Journal of Industrial Mathematics

Volume 2013 (2013), Article ID 421231, 9 pages

http://dx.doi.org/10.1155/2013/421231

Research Article

Asymptotic Dichotomy in a Class of Odd-Order Nonlinear Differential Equations with Impulses

Kunwen Wen and Lijun Pan

Department of Mathematics, Jiaying University, Meizhou, Guangdong 514015, China

Received 26 September 2013; Accepted 2 December 2013

Academic Editor: Ting Chen

Copyright © 2013 Kunwen Wen and Lijun Pan. 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 investigate the oscillatory and asymptotic behavior of a class of odd-order nonlinear differential equations with impulses. We obtain criteria that ensure every solution is either oscillatory or (nonoscillatory and) zero convergent. We provide several examples to show that impulses play an important role in the asymptotic behaviors of these equations.

1. Introduction

Impulsive effect, likewise, exists in a wide variety of evolutionary processes in which states are changed abruptly at certain moments in time, involving such fields as medicine and biology, economics, mechanics, electronics, telecommunications, and so forth. It has been observed that the solutions of quite a few first-or second-order impulsive differential equations are either oscillatory or (nonoscillatory and) zero convergent (see, e.g., [1–10]). For example, Bainov et al. studied the oscillation properties of first-order impulsive differential equations with deviating arguments [3]. Especially in [4], Chen investigated oscillations of second-order nonlinear differential with impulses, and he promposed that the impulses may change the oscillatory behavior of an equation. Based on [4], the authors were devoted to oscillations of impulsive differential equations (see, e.g., [5–10]). Such a dichotomy may yield useful information in real problems. The implications of this dichotomy are applied to the deflection of an elastic beam [11]. Thus, it is of interest to see whether similar dichotomies occur in different types of impulsive differential equations.

One such type consists of impulsive differential equations which are important in the simulation of processes with jump conditions. But papers devoted to the study of asymptotic behaviors of higher order equations with impulses are quite rare. For this reason, Wen et al. studied in [12] the dichotomous properties of the following third-order nonlinear differential equation with impulses:

where , such that . On the other hand, in [13], Chen and Wen investigated the oscillatory and asymptotic behaviors for odd-order nonlinear differential equations with impulses of the form

where is a positive integer and such that . They obtained some interesting results for assuring that every bounded solution of (2) is either oscillatory or nonoscillatory and zero convergent.

In this paper, we will study a class of odd-order nonlinear differential equations with impulses of the form

where is a positive integer, and such that ,

By a solution of (3), we mean a real function defined on such that

(I) for ;(II), and are continuous on ; for and exist, and for any ;(III) satisfies at each point .

A solution of (3) is said to be nonoscillatory if it is eventually positive or eventually negative. Otherwise, it is said to be oscillatory.

We will establish oscillatory and asymptotic results of (3) based on combinations of the following conditions.

(i) and is continuous on is continuous on for , and , where is positive and continuous on and is differentiable in such that for .(ii)For , are continuous in and there exist positive numbers such that

(iii)

Our plan is the following. We first obtain three theorems (Theorems 1–3) to ensure every solution of (3) is either oscillatory or (nonoscillatory and) zero convergent. We will also illustrate our results with several examples. As applications of our results, we state three corollaries (Corollaries 5–7). These corollaries are new even for the special case (2).

2. Main Results

The main results of the paper are as follows.

Theorem 1. Assume that the conditions (i)–(iii) hold. Suppose further that there exists a positive integer such that for , ,

Then every solution of (3) is either oscillatory or (nonoscillatory and) zero convergent.

Theorem 2. Assume that the conditions (i)–(iii) hold and that for any . Suppose further that there exists a positive integer such that for ,

Then every solution of (3) is either oscillatory or (nonoscillatory and) zero convergent.

Theorem 3. Assume that the conditions (i)–(iii) hold and that for any . Suppose further that is bounded,

Then every solution of (3) is either oscillatory or (nonoscillatory and) zero convergent.

Remark 4. When , (3) reduces to (1). Our Theorems 1–3 are Theorem 2.1, Theorem 2.3, Theorem 2.4 in [12], respectively. So our results generalize and contain results in [12].

Next, for (2), we will also be able to obtain some new results. It is easy to see that (2) has the form of (3) by setting and for and . Let

Our Theorems 1–3 can directly lead us to the following corollaries for (2).

Corollary 5. Assume that the conditions (i)–(iii) hold. Suppose further that there exists a positive integer such that for ,

Then every solution of (2) is either oscillatory or (nonoscillatory and) zero convergent.

Corollary 6. Assume that the conditions (i)–(iii) hold. Suppose further that there exists a positive integer such that for ,

Then every solution of (2) is either oscillatory or (nonoscillatory and) zero convergent.

Corollary 7. Assume that the conditions (i)–(iii) hold. Suppose further that , is bounded,

Then every solution of (2) is either oscillatory or (nonoscillatory and) zero convergent.

Remark 8. We note that the above corollaries for ensuring every solution of (2) to be either oscillatory or tend to zero with fixed sign eventually is also new for (2).

Example 9. Consider the equation

where for , and . It is not difficult to see that conditions (i)–(iii) are satisfied. Furthermore,

Thus by Theorem 1, every solution of (14) is either oscillatory or (nonoscillatory and) zero convergent.

Example 10. Consider the equation

where for , and

where

Here, we do not assume that is bounded, monotonic, or differentiable. It is not difficult to see that conditions (i)–(iii) are satisfied. Furthermore,

Thus by Theorem 3, every solution of (16) is either oscillatory or (nonoscillatory and) zero convergent.

Example 11. Consider the equation

where for ,;. It is not difficult to see that conditions (i)–(iii) are satisfied. Furthermore,

Thus, by Theorem 2, every solution of (20) is either oscillatory or (nonoscillatory and) zero convergent. But the ordinary differential equation

has a solution which tends to as . This example shows that impulses play an important role in oscillatory and asymptotic behaviors of equations.

3. Proofs

To prove our Theorems, we need the following Lemmas.

Lemma 12 (Lakshmikantham et al. [1]). Assume that

(H₀) and is left-continuous at for (H₁)for and ,

where , and are real constants. Then for ,

Remark 13. If the inequalities in (23) are reversed, then the inequality in (24) should be reversed as well.

Lemma 14. Suppose that conditions (i)–(iii) hold and is a solution of (3). One has the following statements:

(a)if there exists some such that and for , then there exists some such that for ;(b)if there exists and some such that and for , then there exists some such that for .

Proof. First of all, we will prove the result of (a) to be true. Without loss of generality, we may assume that and for . We assert that there exists some such that for . If this is not true, then for any , we have . Since is increasing on intervals of the form , we see that for . Since is increasing on intervals of the form , we see that for ,

that is,

In particular,

Similarly, for , we have

By induction, we know that

From condition (ii), we have

It follows from (29), (30), and Lemma 12, that for ,

Note that , , and the second equality of condition (iii) hold. Thus, we get for all sufficiently large , which is contrary to for . So there exists some such that and . Since is increasing on intervals of the form for each , then for , we have

The assertion (a) is thus proved.

Next, we will prove the result of (b) to be true. Without loss of generality, we may assume that and for . We assert there exists some such that for . If this is not true, then for any , we have . Since is increasing on intervals of the form , we see that for . By , and , we have that is nondecreasing on . For , we have

In particular,

Similarly, for , we have

By induction, we know that

From condition (ii), we have

Set . Then from (36) and (37), we see that

It follows from Lemma 12 that

that is,

Note that , and the first equality of condition (iii) holds. Thus, we get for all sufficiently large . The relation leads to a contradiction. So there exists some such that and . Then

Since , then for every positive integer , we have that is increasing on . For , we have

In particular,

Similarly, for , we have

By induction, for , we have

Summing up the above discussion, we know that there exists some such that

The proof of Lemma 14 is complete.

Remark 15. We may prove in similar manners the following statements.

()If we replace the condition (a) in Lemma 14 “ and for ” with “ and for ,” then under conditions (i)–(iii), there exists some such that for .()If we replace the condition (b) in Lemma 14 “ and for ” with “ and for ,” then under conditions (i)–(iii), there exists some such that for .

Lemma 16. Suppose conditions (i)–(iii) hold. Let be a solution of (3).

(a)If there exists some such that and for , then for all sufficiently large .(b)If there exists and some such that and for , then for all sufficiently large .

Proof. First of all, we will prove the result of (a) to be true. We assert that for any . If this is not true, then there exists some such that . Since and are strictly decreasing on for and for , we have

Similarly, for , we have

We can easily prove that, for any positive integer and , we have

Hence, for . By the result (a) of Remark 13, for sufficiently large , we have . Using the result (b) of Remark 13 repeatedly, for all sufficiently large , we get , which is contrary to for . Hence we have for any . So we get for all sufficiently large .

Next, we will prove the result of (b) to be true. We assert that for any . If this is not true, then there exists some such that . Since is strictly monotony decreasing on for and for , we have

Similarly, for , we have

We can easily prove that for any positive integer and , we have

Hence, for . By the result () of Remark 8, for sufficiently large , we have . Similarly, by using the result () of Remark 8 again, we can conclude that for all sufficiently large , . That is contrary with for . Hence, we have for any . So we get for all sufficiently large .

The proof of Lemma 16 is complete.

Lemma 17. Suppose conditions (i)–(iii) hold. Let be a solution of (3). Suppose that and for . Then there exists some and such that for ,

Proof. Let for . By (3) and condition (i), we have

By the result (a) of Lemma 16, there exists some such that for . Without loss of generality, we may let for . Then is strictly increasing on . If for any , then for . If there exists some such that , since is strictly monotony increasing and , then for . Thus, there exists some such that for . So one of the following statements holds:

If (55) holds, then by the result (b) of Lemma 14, for all sufficiently large . Using the result (b) of Lemma 14 repeatedly, for all sufficiently large , we can conclude that

If (56) holds, by Lemma 16, we have for all sufficiently large . Similarly, there exists some such that one of the following statements holds:

Repeating the discussion above, we can eventually get that there exist some and such that for ,

The proof of Lemma 17 is complete.

Lemma 18 (see [13]). Suppose for is continuous at , left-continuous at , and exists for . Further, assume that

(H₂)there exists , such that for ;(H₃) is nonincreasing (resp., nondecreasing) on for ;(H₄) is convergent.

Then exists and (resp. ).

We now turn to the proof of Theorem 1. Without loss of generality, we may assume that . If (3) has a nonoscillatory solution , we may assume that for . By Lemma 17, there exists a and such that for , (53) holds. Next, we rewrite (53) as

If (60) holds, then we see that the conditions (H₂) and (H₃) of Lemma 18 are satisfied. Furthermore, note that and , then one has

Since and  , one obtains for any ,

By (62) and (63), we know is bounded. Thus, there exists such that . It follows from condition (ii) that

From (64) and note that is convergent, we know that is convergent. Therefore, the condition (H₄) of Lemma 18 is also satisfied. By Lemma 18, we have . We assert that . If , then there exists , such that for any . Note that ; hence we may see that for . Let . Then for . By conditions (i), (ii), and (3), one has for ,

From (65) and Lemma 12, one gets for ,

It is easy to see from (7) and (66) that for sufficiently large . This is contrary to for . Thus, ; that is, .

If (61) holds, then let . Then for . By (3) and condition (i), one has for ,

From the conditions (i) and (ii) and note that , we know that

From (67), (68), and Lemma 12, one gets for ,

It is easy to see from (7) and (69) that for sufficiently large . This is contrary to for . Thus, every solution of (3) is oscillatory. The proof of Theorem 1 is complete.

We now give the proof of Theorem 2. Without loss of generality, we may assume that . If (3) has a nonoscillatory solution , we may assume that for . By Lemma 17, there exists a and such that for , (53) holds. Next, we may rewrite (53) as (60) and (61).

If (60) holds, then note that . Since is decreasing on for and each , one has for ,

So, since is decreasing and bounded on , we know that is convergent as . Let . Then . We assert that . If , then there exists such that for . Since , one has . By (3) and condition (i), one has for ,

From condition (ii) and noting that , one has

By (71), (72), and Lemma 12, one has for ,

It is easy to see from (8) and (73) that for sufficiently large . This is contrary to for . Thus , that is, ;

If (61) holds, then let . We see that for . By (3) and the condition (i), one gets for

From the conditions (i) and (ii), we know that

From (74), (75), and Lemma 12, one gets for ,

It is easy to see from (8) and (76) that for sufficiently large . This is contrary to for . Thus, every solution of (3) is oscillatory. The proof of Theorem 2 is complete.

Finally, we give the proof of Theorem 3. Without loss of generality, we may assume that . If (3) has a nonoscillatory solution , we may assume that for . By Lemma 17, there exists a and such that for , (53) hold. Next, we rewrite (53) as (60) and (61).

If (60) holds, then we see that the conditions (H₂) and (H₃) of Lemma 18 are satisfied. Furthermore, since for , we see that there exists some such that for any ,

Since is bounded, in view of (77), we know that is bounded. Thus, there exists , such that . It follows from the condition (ii) that

By (78), we know that is convergent. Therefore, the condition (H₄) of Lemma 18 is also satisfied. By Lemma 18, we know that . We assert that . If , then there exists , such that for . Since , we see that for . By , the boundedness of and the condition (ii), we know that is bounded. Note that for . So we may also see that there exists such that . Therefore, one has

By (3) and the condition (i), one has

Integrating (80) from to , it follows from (79) and the condition (ii) that

Since is convergent, it is easy to see from (9) and (81) that for sufficiently large . This is contrary to for . Thus ; that is, .

If (61) holds, then let . We see that for . Similar to the proof of (76), we also obtain

It is easy to see from (9) and (82) that for sufficiently large . This is contrary to for . Thus, every solution of (3) is oscillatory. The proof of Theorem 3 is complete.

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