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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 458275, 11 pages
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: Josef Diblík
Copyright © 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.
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.
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), , .
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  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  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 [5–7] for equations with delay and [8–13] for equations without delay. For some applications of such equations, we may refer to [14–18]. 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.
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
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
the inequality (2.11) becomes
from which, on using (2.1), we have
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.
This work was partially supported by METU-B AP (project no: 01-01-2011-003).
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