Research Article | Open Access

# Oscillation of Certain Second-Order Sub-Half-Linear Neutral Impulsive Differential Equations

**Academic Editor:**Garyfalos Papaschinopoulos

#### Abstract

By introducing auxiliary functions, we investigate the oscillation of a class of second-order sub-half-linear neutral impulsive differential equations of the form where Several oscillation criteria for the above equation are established in both the case and the case which generalize and complement some existing results in the literature. Two examples are also given to illustrate the effect of impulses on the oscillatory behavior of solutions to the equation.

#### 1. Introduction

Impulsive differential equations appear as a natural description of observed evolution phenomena of several real-world problems involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics, and frequency modulates systems [1–5]. In recent years, impulsive differential equations have received a lot of attention.

We are here concerned with the following second-order sub-half-linear neutral impulsive differential equation: where , and for some , is a strictly increasing unbounded sequence of real numbers, for , and

Let denote the set of all real-valued functions defined on such that is continuous for all except possibly at where exists and .

We assume throughout this paper that(a), and ;(b), or ;(c), ;(d) is a sequence of nonnegative real numbers;(e), and .

By a solution of (1.1) we mean a function defined on with such that and satisfies (1.1). It is tacitly assumed that such solutions exist. Note the assumption ; we have that each solution of (1.1) is continuous on . As usual, a nontrivial solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros and nonoscillatory otherwise. Equation (1.1) is said to be oscillatory if its every nontrivial solution is oscillatory.

Compared to equations without impulses, little has been known about the oscillation problem for impulsive differential equations due to difficulties caused by impulsive perturbations [6–17].

When , , and , (1.1) reduces to the following sublinear impulsive delay equation: which has received a lot of attention in the literature. However, for the general sub-half-linear neutral equation (1.1) under the impulse condition given in this paper, little has been known about the oscillation of (1.1) to the best of our knowledge, especially for the case when .

The main objective of this paper is to establish oscillation criteria for the sub-half-linear impulsive differential equation (1.1) in both the case and the case . By introducing an auxiliary function and a function defined below, we establish some new oscillation criteria for (1.1) which complement the oscillation theory of impulsive differential equations. Examples are also given to show the effect of impulses on oscillation of solutions of (1.1).

#### 2. Main Results

Theorem 2.1. *Let . If there exists a positive function such that
**
where , then (1.1) is oscillatory. *

*Proof. *Suppose to the contrary that (1.1) has a nonoscillatory solution . Without loss of generality, we may assume that and for . The case being eventually negative can be similarly discussed. From (1.1), we have that
Based on the impulsive condition , we can deduce that is nonincreasing on . We may claim that holds eventually. Otherwise, there exists such that . Noting that is continuous on , we have that
which implies that is eventually negative since . This is a contradiction. Without loss of generality, say for . Choose sufficiently large such that for , and
which is always possible because . Thus, we have
By choosing sufficiently large such that for and using (2.6) and the nonincreasing character of , we have
Since for and is continuous, we have
By (1.1), (2.7), and (2.8), we get
which implies
From (2.1), we get
Multiplying (2.10) by , we get
Integrating (2.12) from to , we have that
which implies that
On the other hand, by the given impulsive condition, we get
where
Note that , for . Consequently, we see from (2.7), (2.8), and (2.15) that
Substituting (2.17) into (2.14) yields
which contradicts (2.2). This completes the proof.

Theorem 2.2. *Let . If there exists a positive function such that (2.1) holds and
**
where is defined as in Theorem 2.1, then every solution of (1.1) is either oscillatory or tends to zero.*

*Proof. *Suppose to the contrary that there is a solution of (1.1) which is neither oscillatory nor tends to zero. Without loss of generality, we may let and for . Thus, is nonincreasing for . As a result, and are eventually of constant sign. Now, we consider the following two cases: (i) eventually; (ii) eventually. For the case (i), similar to the analysis as in the proof of Theorem 2.1, we have eventually and (2.6) holds. Notice that because ; from (1.1) and (2.6), we get
Following the similar arguments as in the proof of Theorem 2.1, we can get a contradiction with (2.19).

For the case (ii), assume that for . It must now hold that for . Let us consider two cases: (a) is unbounded; (b) is bounded. If is unbounded, then we have
On the other hand, there exists a sequence satisfying , , and . Let be sufficiently large such that and . Then, we have which contradicts (2.21). If is bounded, then we can prove that . In fact,
which implies that since . This is a contradiction. The proof of Theorem 2.2 is complete.

When there is no impulse, (1.1) reduces to The following oscillation results for (2.23) are immediate.

Corollary 2.3. *Let . If there exists a positive function such that (2.1) holds and
**
where is the same as in Theorem 2.1, then (2.23) is oscillatory. *

Corollary 2.4. *Let . If there exists a positive function such that (2.1) holds and
**
where is the same as in Theorem 2.1, then every solution of (2.23) is either oscillatory or tends to zero. *

Next, we introduce the function defined in [18] to further study oscillation of (1.1). Say that defined on belongs to the function class if , , , and for .

Theorem 2.5. *Let . If there exist a positive function and such that (2.1) holds and
**
where is defined as in Theorem 2.1, then (1.1) is oscillatory.*

*Proof. *Suppose to the contrary that (1.1) has a nonoscillatory solution . Without loss of generality, we may assume that and for . Similar to the proof of Theorem 2.1, we have that (2.12) holds. Multiplying on both sides of (2.12) and integrating it from to , we get
which implies that
Therefore,
Proceeding as in the proof of Theorem 2.1, we get a contradiction with (2.26). This completes the proof.

For the case , we have the following oscillation result. Since the proof is similar to that of Theorem 2.2, we omit it here.

Theorem 2.6. *Let . If there exist a positive function and such that (2.1) holds and
**
where is defined as in Theorem 2.1, then every solution of (1.1) is either oscillatory or tends to zero. *

The following two corollaries for (2.23) are immediate.

Corollary 2.7. *Let . If there exist a positive function and such that (2.1) holds and
**
where is defined as in Theorem 2.1, then (2.23) is oscillatory.*

Corollary 2.8. *Let . If there exist a positive function and such that (2.1) holds and
**
where is defined as in Theorem 2.1, then every solution of (2.23) is either oscillatory or tends to zero.*

#### 3. Examples

We now present two examples to illustrate the effect of impulses on oscillation of solutions of (1.1).

*Example 3.1. *Consider the following impulsive delay differential equation:
where , is a constant, , and . We see that , , , , , . Let . A straightforward computation yields . Therefore, when , it is not difficult to verify that (2.1) and (2.2) hold. Thus, (3.1) is oscillatory by Theorem 2.1. However, when there is no impulse in (3.1), Corollary 2.3 cannot guarantee the oscillation of (3.1) since condition (2.24) is invalid for this case. Therefore, the impulsive perturbations may greatly affect the oscillation of (3.1). If , then we have that every solution of (3.1) is either oscillatory or tends to zero by Theorem 2.2. Such behavior of solutions of (3.1) is determined by the impulsive perturbations to a great extent, since Corollary 2.4 fails to apply for this case.

*Example 3.2. *Consider the following impulsive delay differential equation:
where , is a constant, , and . We see that , , , , , . Let . It is not difficult to verify that (2.1) and (2.2) hold if , which implies that (3.2) is oscillatory by Theorem 2.1. We also can verify that and (2.1) and (2.19) hold if . Thus, by Theorem 2.2, every solution of (3.2) is either oscillatory or tends to zero. However, Corollaries (1.1) and (2.1) do not apply for this case. Therefore, the impulsive perturbations play a key role in the oscillation problem of (3.2).

#### Acknowledgments

The author thanks the reviewers for their valuable comments and suggestions on this paper. This paper was supported by the Natural Science Foundation of Shandong Province under the Grant ZR2010AL002.

#### References

- G. Ballinger and X. Liu, “Permanence of population growth models with impulsive effects,”
*Mathematical and Computer Modelling*, vol. 26, no. 12, pp. 59–72, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Z. Lu, X. Chi, and L. Chen, “Impulsive control strategies in biological control of pesticide,”
*Theoretical Population Biology*, vol. 64, no. 1, pp. 39–47, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Sun, F. Qiao, and Q. Wu, “Impulsive control of a financial model,”
*Physics Letters A*, vol. 335, no. 4, pp. 282–288, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Tang and L. Chen, “Global attractivity in a “food-limited” population model with impulsive effects,”
*Journal of Mathematical Analysis and Applications*, vol. 292, no. 1, pp. 211–221, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y. Zhang, Z. Xiu, and L. Chen, “Dynamics complexity of a two-prey one-predator system with impulsive effect,”
*Chaos, Solitons and Fractals*, vol. 26, pp. 131–139, 2005. View at: Publisher Site | Google Scholar - R. P. Agarwal, F. Karakoç, and A. Zafer, “A survey on oscillation of impulsive ordinary differential equations,”
*Advances in Difference Equations*, vol. 2010, Article ID 354841, 52 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y. S. Chen and W. Z. Feng, “Oscillations of second order nonlinear ode with impulses,”
*Journal of Mathematical Analysis and Applications*, vol. 210, no. 1, pp. 150–169, 1997. View at: Publisher Site | Google Scholar - L. H. Erbe, Q. Kong, and B. G. Zhang,
*Oscillation theory for functional-differential equations*, vol. 190, Marcel Dekker, New York, NY, USA, 1995. - K. Gopalsamy and B. G. Zhang, “On delay differential equations with impulses,”
*Journal of Mathematical Analysis and Applications*, vol. 139, no. 1, pp. 110–122, 1989. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Luo, “Second-order quasilinear oscillation with impulses,”
*Computers & Mathematics with Applications*, vol. 46, no. 2-3, pp. 279–291, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH - A. Özbekler and A. Zafer, “Sturmian comparison theory for linear and half-linear impulsive differential equations,”
*Nonlinear Analysis*, vol. 63, no. 5–7, pp. 289–297, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - A. Özbekler and A. Zafer, “Forced oscillation of super-half-linear impulsive differential equations,”
*Computers & Mathematics with Applications*, vol. 54, no. 6, pp. 785–792, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - A. Özbekler and A. Zafer, “Interval criteria for the forced oscillation of super-half-linear differential equations under impulse effects,”
*Mathematical and Computer Modelling*, vol. 50, no. 1-2, pp. 59–65, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Shen, “Qualitative properties of solutions of second-order linear ODE with impulses,”
*Mathematical and Computer Modelling*, vol. 40, no. 3-4, pp. 337–344, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X. Liu and Z. Xu, “Oscillation of a forced super-linear second order differential equation with impulses,”
*Computers & Mathematics with Applications*, vol. 53, no. 11, pp. 1740–1749, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X. Liu and Z. Xu, “Oscillation criteria for a forced mixed type Emden-Fowler equation with impulses,”
*Applied Mathematics and Computation*, vol. 215, no. 1, pp. 283–291, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - E. M. Bonotto, L. P. Gimenes, and M. Federson, “Oscillation for a second-order neutral differential equation with impulses,”
*Applied Mathematics and Computation*, vol. 215, no. 1, pp. 1–15, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Ch. G. Philos, “Oscillation theorems for linear differential equations of second order,”
*Archiv der Mathematik*, vol. 53, no. 5, pp. 482–492, 1989. View at: Publisher Site | Google Scholar | Zentralblatt MATH

#### Copyright

Copyright © 2011 Yuangong Sun. 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.