Journal of Applied Mathematics

Volume 2012 (2012), Article ID 285051, 22 pages

doi:10.1155/2012/285051

Research Article

Interval Oscillation Criteria for Super-Half-Linear Impulsive Differential Equations with Delay

Zhonghai Guo,¹ Xiaoliang Zhou,² and Wu-Sheng Wang³

¹Department of Mathematics, Xinzhou Teachers University, Shanxi Xinzhou 034000, China

²Department of Mathematics, Guangdong Ocean University, Guangdong Zhanjiang 524088, China

³Department of Mathematics, Hechi University, Guangxi Yizhou 546300, China

Received 25 April 2012; Accepted 2 July 2012

Academic Editor: Alfredo Bellen

Copyright © 2012 Zhonghai Guo et al. 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 study the following second-order super-half-linear impulsive differential equations with delay , , , , where , , is a nonnegative constant, denotes the impulsive moments sequence with , , and . By some classical inequalities, Riccati transformation, and two classes of functions, we give several interval oscillation criteria which generalize and improve some known results. Moreover, we also give two examples to illustrate the effectiveness and nonemptiness of our results.

1. Introduction

We consider the following second-order super-half-linear impulsive differential equations with delay

where , , is a nonnegative constant, denotes the impulsive moments sequence with , , and .

Let be an interval, we define

For given and , we say is a solution of (1.1) with initial value if satisfies (1.1) for and for .

A solution of (1.1) is said to be nonoscillatory if it is eventually positive or eventually negative. Otherwise, this solution is said to be oscillatory.

Impulsive differential equation is an adequate mathematical apparatus for the simulation of processes and phenomena observed in control theory, physics, chemistry, population dynamics, biotechnologies, industrial robotics, economics, and so forth. Because it has more richer theory than its corresponding nonimpulsive differential equation, much research has been done on the qualitative behavior of certain impulsive differential equations (see [1, 2]).

In the last decades, there has been an increasing interest in obtaining sufficient conditions for oscillation and/or nonoscillation of different classes impulsive differential equations with delay (constant or variable), see, for example, [1–9] and the references cited therein.

In recent years, interval oscillation of impulsive differential equations was also arousing the interest of many researchers, see [10–15].

However, for the impulsive equations almost all of interval oscillation results in the existing literature were established only for the case of “without delay,” in other words, for the case of “with delay” the study on the interval oscillation is very scarce. To the best of our knowledge, Huang and Feng [16] gave the first research in this direction recently. They considered the second-order delay differential equations with impulses

and established some interval oscillation criteria which developed some known results for the equation without delay or impulses [13, 17, 18].

Motivated mainly by [16], in this paper, we study the interval oscillation of the delay impulsive equation (1.1). By using classical inequalities, Riccati transformation, and two classes of functions (introduced first by Philos [19]), we establish some interval oscillation criteria which generalize and improve some known results of [13, 16–18]. Moreover, we also give two examples to illustrate the effectiveness and nonemptiness of our results.

2. Main Results

Throughout the paper, we always assume that the following conditions hold:

) and is nondecreasing, ; () is a quotient of odd positive integers , are real constants satisfying , ; (), and there exist some positive constants and such that for all with .

We introduce the following notations at intervals and

For two constants with , and a function , we define an operator by

where if .

In the discussion of the impulse moments of and , we need to consider the following cases for ,

(S1) and ; (S2) and ; (S3) and ; (S4) and , and the cases for ; (1); (2) ; (3) .

Combining (S*) with (*), we can get 12 cases. In order to save space, throughout the paper, we study (1.1) under the case of combination of (S1) with (1) only. The discussions for other cases are similar and omitted.

The following preparatory lemmas will be useful to prove our theorems. The first is derived from [20] and the second from [21].

Lemma 2.1. Let and be positive real numbers with . Then

for all and .

Lemma 2.2. Suppose and are nonnegative, then

where equality holds if and if .

Let , , and . Put

It follows from Lemma 2.2 that

Lemma 2.3. Assume that for any , there exists , , such that and

If is a nonoscillatory solution of (1.1), then there exist the following estimations of :

where , .

Proof. Without loss of generality, we assume that and for . In this case the selected interval of is . From (1.1) and (2.7), we obtain

Hence is nonincreasing on the interval .

Case (a) (if , then ). Thus there is no impulsive moment in . For any , we have

Since , the function is an increasing function and is non-increasing on , we have

From (2.11) and the conditions and is nondecreasing, we have

Thus

Integrating both sides of above inequality from to , we obtain

Case (b) (if , then ). There is an impulsive moment in . For any , we have

Using the impulsive condition of (1.1) and the monotone properties of , and , we get

Since , we have

In addition,

Similar to the analysis of (2.11)–(2.14), we have

From (2.17) and (2.19) and note that the monotone properties of and , we get

In view of (), we have

On the other hand, using similar analysis of (2.11)–(2.19), we get

Integrating (2.22) from to , where , we have

From (2.21) and (2.23), we obtain

Case (c) (). Since , then . So, there is no impulsive moment in . Similar to (2.14) of Case (a), we have

Case (d) (). Since , then . Hence, there is an impulsive moment in . Making a similar analysis of Case (b), we obtain

When , we can choose interval to study (1.1). The proof is similar and will be omitted. Therefore we complete the proof.

Theorem 2.4. Assume that for any , there exists , , such that and (2.7) holds. If there exists () such that, for ,

and for ,

where , then (1.1) is oscillatory.

Proof. Assume, to the contrary, that is a nonoscillatory solution of (1.1). Without loss of generality, we assume that and for . In this case the interval of selected for the following discussion is .

We define

Differentiating and in view of (1.1) we obtain, for ,

Putting , , , , and , by Lemma 2.1, we see that

where .

First, we consider the case .

In this case, we assume impulsive moments in are . Choosing a , multiplying both sides of (2.31) by and then integrating it from to , we obtain

where . Using the integration by parts on the left side of above inequality and noting the condition , we obtain

Letting , , and using (2.6), for the integrand function in above inequality we have that

Meanwhile, for , , we have

Hence

Therefore, we get

On the other hand, for , , we have

In view of , and note that the monotone properties of , , and , we obtain

This is

Let , it follows

Similar analysis on , we can get

Then from (2.41), (2.42), and , we have

where and . From (2.37) and (2.43) and applying Lemma 2.3, we obtain

This contradicts (2.27).

Next we consider the case . By the condition (1) we know there is no impulsive moment in . Multiplying both sides of (2.31) by and integrating it from to , we obtain

Using the integration by parts on the left-hand side and noting the condition , we obtain

It follows that

Letting , , and and applying the inequality (2.6), we get

Using same way as Case (a) we get

From (2.48) and (2.49) we obtain

This contradicts our assumption (2.28).

When , we can choose interval to study (1.1). The proof is similar and will be omitted. Therefore we complete the proof.

Remark 2.5. When , , and the delay term , (1.1) reduces to that studied by Liu and Xu [13]. Therefore our Theorem 2.4 generalizes Theorem 2.1 of [13].

Remark 2.6. When , and , (1.1) reduces to the (1.3) studied by Huang and Feng [16]. Therefore our Theorem 2.4 extends Theorem 2.1 of [16].

Remark 2.7. When , for all ., the impulses in (1.1) disappear, Theorem 2.4 reduces to the main results of [17, 18].

In the following we will establish a Kemenev type interval oscillation criteria for (1.1) by the ideas of Philos [19] and Kong [22].

Let , , then a pair function is said to belong to a function set , defined by , if there exists satisfying the following conditions:

(), , for ; () , .

We assume there exist () which satisfy for any . Noticing whether or not there are impulsive moments of in and , we should consider the following four cases, namely,

(S1); (S2) ; (S3); (S4) .

Moreover, in the discussion of the impulse moments of , it is necessary to consider the following two cases:

(1);    (2) .

In the following theorem, we only consider the case of combination of (S1) with (1). For the other cases, similar conclusions can be given and their proofs will be omitted here.

For convenience in the expressing blow, we define

where ,  () and .

Theorem 2.8. Assume that for any , there exists , , such that and if there exists a pair of such that

then (1.1) is oscillatory.

Proof. Assume, to the contrary, that is a nonoscillatory solution of (1.1). Without loss of generality, we assume that and for . In this case the interval of selected for the following discussion is . Using the same proof as in Theorem 2.4, we can get (2.31). Multiplying both sides of (2.31) by and integrating it from to , we have

where . Noticing impulsive moments are in and using the integration by parts on the left-hand side of above inequality, we obtain

Substituting (2.54) into (2.53), we have

Letting , , and using (2.6) to the right side of above inequality, we have

Because there are different integration intervals in (2.56), we need to divide the integration interval into several subintervals for estimating the function . Using Lemma 2.2, (2.6), we get estimation for the left-hand side of above inequality as follows,

From (2.56) and (2.57), we have

Multiplying both sides of (2.31) by and using similar analysis to the above, we can obtain

Dividing (2.58) and (2.59) by and , respectively, and adding them, we get

On the other hand, similar to (2.43), we have

From (2.60) and (2.61), we can obtain a contradiction to the condition (2.52).

When , we can choose interval to study (1.1). The proof is similar and will be omitted. Therefore we complete the proof.

Remark 2.9. Let , and , the conditions (), () can be changed into

(), , for ; (), .

We know that () and () are the main assumptions used in [13, 16] to obtain Kemenev type oscillation criteria. Therefore, Theorem 2.8 is a generalization of Theorem 2.3 in [13] and Theorem 2.5 in [16].

3. Examples

In this section, we give two examples to illustrate the effectiveness and nonemptiness of our results.

Example 3.1. Consider the following equation

where , ,  ., , , , and are positive constants.

For any , we can choose large such that

There are impulsive moments in and in . From and for all , we know that condition is satisfied. Moreover, we also see the conditions (S1) and (2.7) are satisfied.

Let . It is easy to get that . In view of as , by a simple calculation, the left side of (2.27) is the following

On the other hand, we have

Thus condition (2.27) is satisfied for if

Similarly, for we can get the following condition

which ensures (2.27). Hence, by Theorem 2.4, if (3.5) and (3.6) hold, (3.1) is oscillatory. It is easy to see that (3.5) and (3.6) may be satisfied when or is large enough. Particularly, let , for and , condition (3.5) and (3.6) become a simple form

Example 3.2. Consider the following equation

where : , , , (), , , , , (>0) is a constant. Clearly, . In addition, let

For any , we choose large enough such that and let , , and . Then , , and on and satisfy (2.7). Let , then . By simple calculation, we get

Then the left side of the inequality (2.52) is

Because , and , it is easy to get that the right side of the inequality (2.52) for is

Thus (2.52) is satisfied for if

When , with the similar argument above we get that the left side of inequality (2.52) is

and the right side of the inequality (2.52) is

Therefore (2.52) is satisfied for if

Hence, by Theorem 2.8, (3.8) is oscillatory if (3.13) and (3.16) hold. Particularly, let , for , condition (3.13) and (3.16) become a simple form

Acknowledgments

The authors thank the anonymous reviewers for their detailed and insightful comments and suggestions for improvement of paper. This work was supported by the NNSF of China (11161018), and the NSF of Guangdong Province (10452408801004217).

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