Abstract and Applied Analysis

Volume 2012 (2012), Article ID 351709, 23 pages

doi:10.1155/2012/351709

Research Article

Interval Oscillation Criteria of Second Order Mixed Nonlinear 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 19 December 2011; Revised 10 April 2012; Accepted 11 April 2012

Academic Editor: Agacik Zafer

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 mixed nonlinear impulsive differential equations with delay , where , is a nonnegative constant, denotes the impulsive moments sequence, and . Some sufficient conditions for the interval oscillation criteria of the equations are obtained. The results obtained generalize and improve earlier ones. Two examples are considered to illustrate the main results.

1. Introduction

We consider the following second order impulsive differential equations with delay

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

Let be an interval, and 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 without impulsive differential equation, much research has been done on the qualitative behavior of certain impulsive differential equations (see [1, 2]).

In the last decades, there is constant interest in obtaining new sufficient conditions for oscillation or nonoscillation of the solutions of various impulsive differential equations, 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. In 2007, Özbekler and Zafer [10] investigated the following equations:

where , , and are sequences of real numbers. In 2009, they further gave a research [11] for equations of the form

and obtained some interval oscillation results which improved and extended the earlier ones for the equations without impulses.

For the mixed type Emden-Fowler equations

Liu and Xu [12] established some interval oscillation results. Recently, Özbekler and Zafer [13] investigated the more general cases

where .

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 [14] gave the first research in this direction recently. They considered second order delay differential equations with impulses

and established some interval oscillation criteria which developed some known results for the equations without delay or impulses [15–17].

Motivated mainly by [13, 14], in this paper, we study the interval oscillation of the delay impulsive (1.1). By using some inequalities, Riccati transformation and functions (introduced first by Philos [18]), we establish some interval oscillation criteria which generalize and improve some known results. Moreover, examples are considered to illustrate the main results.

2. Main Results

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

()the exponents satisfy that ;() is nondecreasing, ;() and are real constant sequences such that ,.

It is clear that all solutions of (1.1) are oscillatory if there exists a subsequence of such that for all . So, we assume for all in condition ().

In this section, intervals and are considered to establish oscillation criteria. For convenience, we introduce the following notations (see [12]). Let

For two constants with and 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

(),(),().

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 [19] and second is from [20].

Lemma 2.1. For any given n-tuple satisfying , then there exists an n-tuple such that

where .

Lemma 2.2. Suppose and are nonnegative, then

where equality holds if and if .

Let , , and . Put

It follows from Lemma 2.2 that

Theorem 2.3. Assume that for any , there exist , such that and for

If there exist and such that, for , ,

where is maximum value of on and, for , ,

where with and are positive constants satisfying conditions of Lemma 2.1, 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 . Define

It follows, for , that

Now, let

where are positive constants satisfying conditions of Lemma 2.1 and . Employing in (2.11) the arithmetic-geometric mean inequality (see [20])

and in view of (2.3), we have that

where

First, we consider the case .

In this case, we assume impulsive moments in are . Choosing , multiplying both sides of (2.14) by and then integrating it from to , we obtain

where . Using the integration by parts formula in the left-hand side of above inequality and noting the condition , we obtain

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

In view of the impulse condition in (1.1) and the definition of we have, for , , that

From (2.19), we have

Therefore, we get

On the other hand, for ,

Hence is nonincreasing on .

Because there are different integration intervals in (2.21), we will estimate in each interval of as follows.

Case 1. , for .

Subcase 1. If , then . Thus there is no impulsive moment in . For any , we have

Since , is nondecreasing, function is an increasing function and is nonincreasing on , we have

Therefore,

Integrating both sides of the above inequality from to , we obtain

Subcase 2. 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,

Using the same analysis as (2.24) and (2.25), we have

From (2.29) and (2.31) and note that the monotone properties of and , we get

Then,

In view of , we have

On the other hand, similar to the above analysis, we get

Integrating (2.35) from to , where , we have

From (2.34) and (2.36), we obtain

Case 2 (). Since , then . So, there is no impulsive moment in . Similar to (2.26) of Subcase 1, we have

Case 3 (). Since , then . Hence, there is an impulsive moment in . Making a similar analysis of Subcase 2, we obtain

From (2.21), (2.26), (2.37), (2.38), and (2.39) we get

On the other hand, for , , we have

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

This is

Letting , we have

Using similar analysis on , we get

Then from (2.44), (2.45), and , we have

where and .

From (2.40) and (2.46), we obtain

This contradicts (2.8).

Next we consider the case . From the condition (1) we know that there is no impulsive moment in . Multiplying both sides of (2.47) by and integrating it from to , we obtain

Similar to the proof of (2.21), we have

Using same way as Subcase 1, we get

From (2.49) and (2.50) we obtain

This contradicts condition (2.9).

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

Remark 2.4. In article [14], the authors obtained the following inequalities:

See [14, equation ],

See [14, equation ].

Dividing [14, equation ] by [14, equation ], they obtained

See [14, equation ]

This is an error. Moreover, similar errors appeared many times in the later arguments, for example, in inequalities (2.15), (2.19), and (2.20) in [14]. Moreover, the above substitution can lead to some divergent integrals, for example, the integrals in (2.22), (2.24) in [14]. Therefore, the conditions of their Theorems 2.1–2.5 must be defective. In the proof of our Theorem 2.3, this error is remedied.

Remark 2.5. When , that is, the delay disappears, (1.1) reduces to (1.7) studied by Özbekler and Zafer [13]. In this case, our result with is Theorem 2.1 of [13].

Remark 2.6. When , that is, the delay disappears in (1.1) and , our result reduces to Theorem 2.1 of [12].

Remark 2.7. When for all and , that is, both impulses and delay disappear in (1.1), our result with and reduces to Theorem 1 of [21].

In the following we will establish a Kong-type interval oscillation criteria for (1.1) by the ideas of Philos [18] and Kong [22].

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

;.

We assume that there exist such that for any . Noticing whether or not there are impulsive moments of in and , we should consider the following four cases, namely, (S5) ; (S6) ; (S7) and (S8) . Moreover, in the discussion of the impulse moments of , it is necessary to consider the following two cases: () and () . In the following theorem, we only consider the case of combination of (S5) with (). For the other cases, similar conclusions can be given and the proofs will be omitted here.

For convenience in the expression below, we define, for ,

where , and with and are positive constants satisfying conditions of Lemma 2.1.

Theorem 2.8. Assume (2.7) holds. 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 . Similar to the proof of Theorem 2.3, we can get (2.14) and (2.19). Multiplying both sides of (2.14) by and integrating it from to , we have

where , with and are positive constants satisfying conditions of Lemma 2.1.

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

Substituting (2.58) into (2.57), we obtain

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

Similar to the proof of Theorem 2.3, we need to divide the integration interval into several subintervals for estimating the function . Using the methods of (2.26), (2.37), (2.38), and (2.39) we estimate the left-hand side of above inequality as follows:

From (2.60) and (2.61), we have

On the other hand, multiplying both sides of (2.14) by and using similar analysis to the above, we can obtain

Dividing (2.62) and (2.63) by , and respectively, and adding them, we get

Using the same methods as (2.46), we have

From (2.64), (2.65), we can obtain a contradiction to the condition (2.56).

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

Remark 2.9. When , that is, the delay disappears and in (1.1), our result Theorem 2.8 reduces to Theorem 2.2 of [12].

3. Examples

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

Example 3.1. Consider the following delay differential equation with impulse:

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.

We can choose such that Lemma 2.1 holds. Let and . It is easy to verify that . By a simple calculation, the left side of (2.8) is the following:

On the other hand, we have

Thus if

the condition (2.8) is satisfied in . Similarly, we can show that for the condition (2.8) is satisfied if

Hence, by Theorem 2.3, (3.1) is oscillatory, if (3.4) and (3.5) hold. Particularly, let , for all , condition (3.4) and (3.5) become

Example 3.2. Consider the following equation:

where are positive constants; , , , and . In addition, let

For any , we choose large enough such that , and let , , and . It is easy to see that condition (2.7) in Theorem 2.8 is satisfied. Letting , we get . By simple calculation, we have

Then the left-hand side of the inequality (2.56) is

Because , and , it is easy to get that the right-hand side of the inequality (2.56) for is

Thus (2.56) is satisfied with if

When , with the same argument as above we get that the left-hand side of inequality (2.56) is

and the right-hand side of the inequality (2.56) is

Therefore, (2.51) is satisfied with if

Hence, by Theorem 2.8, (3.7) is oscillatory if

Particularly, when , for all , condition (3.16) becomes

Acknowledgments

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

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