`Abstract and Applied AnalysisVolumeΒ 2011Β (2011), Article IDΒ 458275, 11 pageshttp://dx.doi.org/10.1155/2011/458275`
Research Article

## Oscillation of Second-Order Sublinear Impulsive Differential Equations

Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey

Received 25 January 2011; Accepted 27 February 2011

Copyright Β© 2011 A. Zafer. 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

Oscillation criteria obtained by Kusano and Onose (1973) and by Belohorec (1969) are extended to second-order sublinear impulsive differential equations of Emden-Fowler type: , ; ; , by considering the cases and , respectively. Examples are inserted to show how impulsive perturbations greatly affect the oscillation behavior of the solutions.

#### 1. Introduction

We deal with second-order sublinear impulsive differential equations of the form where , , and for some and , is a strictly increasing unbounded sequence of positive real numbers,

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 in the sequel that (a), (b) is a sequence of real numbers, (c), , .

By a solution of (1.1) on an interval , we mean a function which is defined on such that and which satisfies (1.1). Because of the requirement every solution of (1.1) is necessarily continuous.

As usual we assume that (1.1) has solutions which are nontrivial for all large . Such a solution of (1.1) is called oscillatory if it has no last zero and nonoscillatory otherwise.

In case there is no impulse, (1.1) reduces to Emden-Fowler equation with delay and without delay

The problem of oscillation of solutions of (1.3) and (1.4) has been considered by many authors. Kusano and Onose [1] see also [2, 3] proved the following necessary and sufficient condition for oscillation of (1.3).

Theorem 1.1. If , then a necessary and sufficient condition for every solution of (1.3) to be oscillatory is that

The condition is required only for the sufficiency part, and no similar criteria is available for changing sign, except in the case . Without imposing a sign condition on , Belohorec [4] obtained the following sufficient condition for oscillation of (1.4).

Theorem 1.2. If for some , then every solution of (1.4) is oscillatory.

Compared to the large body of papers on oscillation of differential equations, there is only little known about the oscillation of impulsive differential equations; see [5β7] for equations with delay and [8β13] for equations without delay. For some applications of such equations, we may refer to [14β18]. The books [19, 20] are good sources for a general theory of impulsive differential equations.

The object of this paper is to extend Theorems 1.1 and 1.2 to impulsive differential equations of the form (1.1). The results show that the impulsive perturbations may greatly change the oscillatory behavior of the solutions. A nonoscillatory solution of (1.3) or (1.4) may become oscillatory under impulsive perturbations.

The following two lemmas are crucial in the proof of our main theorems. The first lemma is contained in [21] and the second one is extracted from [22].

Lemma 1.3. If each is continuous on , then

Lemma 1.4. , let , , and , and let a sequence of positive real numbers. If and then where

#### 2. The Main Results

We first establish a necessary and sufficient condition for oscillation of solutions of (1.1) when .

Theorem 2.1. If then (1.1) has a solution satisfying

Proof. Choose . In view of Lemma 1.3 by integrating (1.1) twice from to , we obtain Set where . Then Let be such that for all . Replacing by in (2.5) and using the increasing character of , we see that From (2.4), we also see that for and . Now, in view of (2.6) and (2.8), an integration of (2.7) from to leads to Applying Lemma 1.4 with we easily see that Since the inequality (2.11) becomes from which, on using (2.1), we have where In view of (2.5), (2.6), and (2.14) we see that
To complete the proof it suffices to show that approaches a nonzero limit as tends to . To see this we integrate (1.1) from to to get Employing (2.16) we have Therefore, exists. Clearly, we can make by requiring that which is always possible by arranging .

Theorem 2.2. Suppose that and are nonnegative. Then every solution of (1.1) is oscillatory if and only if

Proof. Let (2.20) fail to hold. Then, by Theorem 2.1 we see that there is a solution which satisfies (2.2). Clearly, such a solution is nonoscillatory. This proves the necessity.
To show the sufficiency, suppose that (2.20) is valid but there is a nonoscillatory solution of (1.1). We may assume that is eventually positive; the case being eventually negative is similar. Clearly, there exists such that for all . From (1.1), we have that Thus, is decreasing on every interval not containing . From the impulse conditions in (1.1), we also have . Therefore, we deduce that is nondecreasing on .
We may claim that is eventually positive. Because if eventually, then becomes negative for large values of . This is a contradiction.
It is now easy to show that Therefore, Let be such that for . Using (2.23) and the nonincreasing character of , we have and so, by (1.1), Dividing (2.25) by and integrating from to , we obtain which clearly implies that where Since for and , by taking we see from (2.28) that But, (2.24) gives and hence
Finally, (2.27) and (2.32) result in which contradicts (2.20). The proof is complete.

Example 2.3. Consider the impulsive delay differential equation where and .
We see that , , , and , . Since applying Theorem 2.2 we conclude that every solution of (2.34) is oscillatory.
We note that if the equation is not subject to any impulse condition, then, since the equation has a nonoscillatory solution by Theorem 1.1.

Let us now consider (1.1) when . That is, where and are given by (a) and (b).

The following theorem is an extension of Theorem 1.2. Note that no sign condition is imposed on and .

Theorem 2.4. If for some , then every solution of (2.38) is oscillatory.

Proof. Assume on the contrary that (2.38) has a nonoscillatory solution such that for all for some . The proof is similar when is eventually negative. We set It is not difficult to see that and hence From (2.41), we have and so In view of (2.42), by a straightforward integration of (2.44), we have which combined with (2.44) leads to Finally, by using (2.39) in the last inequality, we see that there is a such that which, however, implies that as , a contradiction with . The proof is complete.

Example 2.5. Consider the impulsive differential equation where and .
We have that , , and , . Taking we see from (2.38) that Since the conditions of Theorem 2.4 are satisfied, every solution of (2.48) is oscillatory.
Note that if the impulses are absent, then, since the equation is oscillatory by Theorem 1.2.

#### Acknowledgment

This work was partially supported by METU-B AP (project no: 01-01-2011-003).

#### References

1. T. Kusano and H. Onose, βNonlinear oscillation of a sublinear delay equation of arbitrary order,β Proceedings of the American Mathematical Society, vol. 40, pp. 219β224, 1973.
2. H. E. Gollwitzer, βOn nonlinear oscillations for a second order delay equation,β Journal of Mathematical Analysis and Applications, vol. 26, pp. 385β389, 1969.
3. V. N. Sevelo and O. N. Odaric, βCertain questions on the theory of the oscillation (non-oscillation) of the solutions of second order differential equations with retarded argument,β Ukrainskii Matematicheskii Zhurnal, vol. 23, pp. 508β516, 1971 (Russian).
4. S. Belohorec, βTwo remarks on the properties of solutions of a nonlinear differential equation,β Acta Facultatis Rerum Naturalium Universitatis Comenianae/Mathematica, vol. 22, pp. 19β26, 1969.
5. D. D. Bainov, Yu. I. Domshlak, and P. S. Simeonov, βSturmian comparison theory for impulsive differential inequalities and equations,β Archiv der Mathematik, vol. 67, no. 1, pp. 35β49, 1996.
6. K. Gopalsamy and B. G. Zhang, βOn delay differential equations with impulses,β Journal of Mathematical Analysis and Applications, vol. 139, no. 1, pp. 110β122, 1989.
7. J. Yan, βOscillation properties of a second-order impulsive delay differential equation,β Computers & Mathematics with Applications, vol. 47, no. 2-3, pp. 253β258, 2004.
8. C. Yong-shao and F. Wei-zhen, βOscillations of second order nonlinear ODE with impulses,β Journal of Mathematical Analysis and Applications, vol. 210, no. 1, pp. 150β169, 1997.
9. Z. He and W. Ge, βOscillations of second-order nonlinear impulsive ordinary differential equations,β Journal of Computational and Applied Mathematics, vol. 158, no. 2, pp. 397β406, 2003.
10. C. Huang, βOscillation and nonoscillation for second order linear impulsive differential equations,β Journal of Mathematical Analysis and Applications, vol. 214, no. 2, pp. 378β394, 1997.
11. J. Luo, βSecond-order quasilinear oscillation with impulses,β Computers & Mathematics with Applications, vol. 46, no. 2-3, pp. 279β291, 2003.
12. A. Γzbekler and A. Zafer, βSturmian comparison theory for linear and half-linear impulsive differential equations,β Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 5β7, pp. e289βe297, 2005.
13. A. Γzbekler and A. Zafer, βPicone's formula for linear non-selfadjoint impulsive differential equations,β Journal of Mathematical Analysis and Applications, vol. 319, no. 2, pp. 410β423, 2006.
14. G. Ballinger and X. Liu, βPermanence of population growth models with impulsive effects,β Mathematical and Computer Modelling, vol. 26, no. 12, pp. 59β72, 1997.
15. Z. Lu, X. Chi, and L. Chen, βImpulsive control strategies in biological control of pesticide,β Theoretical Population Biology, vol. 64, no. 1, pp. 39β47, 2003.
16. J. Sun, F. Qiao, and Q. Wu, βImpulsive control of a financial model,β Physics Letters A, vol. 335, no. 4, pp. 282β288, 2005.
17. S. Tang and L. Chen, βGlobal attractivity in a βfood-limitedβ population model with impulsive effects,β Journal of Mathematical Analysis and Applications, vol. 292, no. 1, pp. 211β221, 2004.
18. S. Tang, Y. Xiao, and D. Clancy, βNew modelling approach concerning integrated disease control and cost-effectivity,β Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 3, pp. 439β471, 2005.
19. V. Lakshmikantham, D. D. BaΔ­nov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989.
20. A. M. SamoΔ­lenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, River Edge, NJ, USA, 1995.
21. M. Akhmetov and R. Sejilova, βThe control of the boundary value problem for linear impulsive integro-differential systems,β Journal of Mathematical Analysis and Applications, vol. 236, no. 2, pp. 312β326, 1999.
22. D. Bainov and V. Covachev, Impulsive Differential Equations with a Small Parameter, vol. 24 of Series on Advances in Mathematics for Applied Sciences, World Scientific, River Edge, NJ, USA, 1994.