Oscillation of a class of second-order linear impulsive differential equations
© Tariboon and Thiramanus; licensee Springer 2012
Received: 15 August 2012
Accepted: 13 November 2012
Published: 27 November 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.
Keywordsoscillation nonoscillation impulsive differential equation
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.
Let be an interval and .
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.
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.
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 .
2 Main results
Now we are in the position to establish the main result.
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 .
Therefore, is continuous on .
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 .
which is a contradiction. Thus, the solution is oscillatory. The proof is complete. □
where , . Then Eq. (1.1) is oscillatory.
We get that is an eventually positive solution of (2.12), a contradiction, and so the proof is complete. □
Then Eq. (2.12) is oscillatory.
3 Some examples
In this section, we illustrate our results with two examples.
where denotes the greatest integer function.
By Theorem 2.3, Eq. (3.1) is oscillatory.
By Corollary 2.5, Eq. (3.2) is oscillatory.
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.
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