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Abstract and Applied Analysis

VolumeΒ 2011Β (2011), Article IDΒ 458275, 11 pages

http://dx.doi.org/10.1155/2011/458275

## 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

Academic Editor: JosefΒ DiblΓk

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 [5β7] for equations with delay and [8β13] for equations without delay. For some applications of such equations, we may refer to [14β18]. 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).

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