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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 351709, 23 pages
Interval Oscillation Criteria of Second Order Mixed Nonlinear Impulsive Differential Equations with Delay
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.
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.
We consider the following second order impulsive differential equations with delay where , is a nonnegative constant, denotes the impulsive moments sequence, and , for all .
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  investigated the following equations: where , , and are sequences of real numbers. In 2009, they further gave a research  for equations of the form and obtained some interval oscillation results which improved and extended the earlier ones for the equations without impulses.
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  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 ), 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 ). 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.
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
where are positive constants satisfying conditions of Lemma 2.1 and . Employing in (2.11) the arithmetic-geometric mean inequality (see )
and in view of (2.3), we have that
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 , 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 . Moreover, the above substitution can lead to some divergent integrals, for example, the integrals in (2.22), (2.24) in . 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.
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.
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: