Oscillation of a class of second-order linear impulsive differential equations
Advances in Difference Equations volume 2012, Article number: 205 (2012)
In this paper, we investigate the oscillation of a class of second-order linear impulsive differential equations of the form
By using the equivalence transformation and the associated Riccati techniques, some interesting results are obtained.
Impulsive differential equations are recognized as adequate mathematical models for studying evolution processes that are subject to abrupt changes in their states at certain moments. Many applications in physics, biology, control theory, economics, applied sciences and engineering exhibit impulse effects (see [1–4]). In recent years, the study of the oscillation of all solutions of impulsive differential equations have been the subject of many research works. See, for example, [5–11] and the references cited therein.
In this article, we consider the second-order linear impulsive differential equation of the form
where , , and , and are two known sequences of positive real numbers, λ is a real number, and
Let be an interval and .
A function is said to be a solution of Eq. (1.1) if
satisfies for and ,
, for each , and and are left continuous for each , .
Definition 1.1 A nontrivial solution of Eq. (1.1) is said to be nonoscillatory if the solution is eventually positive or eventually negative. Otherwise, it is said to be oscillatory. Eq. (1.1) is said to be oscillatory if all solutions are oscillatory.
If , then Eq. (1.1) reduces to the second-order linear differential equation with impulses
In  Luo et al. and  Guo et al. gave some excellent results on the oscillation and nonoscillation of Eq. (1.2) by using associated Riccati techniques and an equivalence transformation. Moreover, Luo et al. showed that the oscillation of Eq. (1.2) can be caused by impulsive perturbations, though the corresponding equation without impulses admits a nonoscillatory solution.
If and , then Eq. (1.1) reduces to the impulsive Langevin equation of the form
The Langevin equation (first formulated by Langevin in 1908) is found to be an effective tool to describe the evolution of physical phenomena in fluctuating environments. For more details of the Langevin equation without impulses and its applications, we refer the reader to .
If and for all , then Eq. (1.1) reduces to the self-adjoint second-order differential equation
2 Main results
Now we are in the position to establish the main result.
Lemma 2.1 If the second-order differential equation
is oscillatory, then Eq. (1.1) is oscillatory, where , .
Proof For the sake of contradiction, suppose that Eq. (1.1) has an eventually positive solution . Then there exists a constant such that for .
can be obtained. Therefore,
On the other hand, we have
Let , then we get
Now, we define
Then, for , we get that
Therefore, is continuous on .
Then, for , and from (2.2), we get that
The left-hand and the right-hand derivatives of at are given by
Hence, for , we have
Then, for and
From (2.5), we obtain
This implies that is an eventually positive solution of Eq. (2.1) which is a contradiction. A similar argument can be used to prove that Eq. (2.1) cannot have an eventually negative solution. Therefore, from Definition 1.1, the solution of Eq. (2.1) is oscillatory. The proof is complete. □
Lemma 2.2 (Leighton type oscillation criteria)
Assume that the functions and .
Proof Let be a nonoscillatory solution of the Eq. (2.6). Without loss of generality, we assume that there exists a such that for . We define
Then the equation
has a solution on . It is easy to see that the solution of Eq. (2.7) satisfies the following equation:
Since , then there exists such that
for all t in . Hence, from (2.8), it follows that
Integrating (2.9) from to ∞, we obtain
which is a contradiction. Thus, the solution is oscillatory. The proof is complete. □
Theorem 2.3 Assume that
where , . Then Eq. (1.1) is oscillatory.
If , , then , and (1.1) becomes
Theorem 2.4 Eq. (2.12) is oscillatory if and only if
Without loss of generality, we suppose that is an eventually positive solution of (2.13) such that for . Set
Then, for , we have , and for ,
Moreover, for , we have
Now we have for
We get that is an eventually positive solution of (2.12), a contradiction, and so the proof is complete. □
Corollary 2.5 Assume that
Then Eq. (2.12) is oscillatory.
3 Some examples
In this section, we illustrate our results with two examples.
Example 3.1 Consider the following impulsive Langevin equation:
Set , , and . If for some integer , then we get
where denotes the greatest integer function.
By Theorem 2.3, Eq. (3.1) is oscillatory.
Example 3.2 Consider the equation
where is a known sequence of positive real numbers. It is easy to see that
By Corollary 2.5, Eq. (3.2) is oscillatory.
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This research work is financially supported by the Office of the Higher Education Commission of Thailand, and King Mongkut’s University of Technology North Bangkok, Thailand.
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
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Tariboon, J., Thiramanus, P. Oscillation of a class of second-order linear impulsive differential equations. Adv Differ Equ 2012, 205 (2012). https://doi.org/10.1186/1687-1847-2012-205