- Open Access
Oscillation criteria for a certain second-order nonlinear perturbed differential equations
© Temtek and Tiryaki; licensee Springer. 2013
- Received: 15 February 2013
- Accepted: 20 September 2013
- Published: 11 November 2013
In this paper, a class of second-order nonlinear perturbed differential equation and its special cases are studied. By using the generalized Riccati transformation and well-known techniques, some new oscillation criteria are established. The results obtained essentially generalize and improve some known results and can be applied to the well-known half-linear and damped half-linear-type equations.
- Differentiable Function
- Oscillatory Behavior
- Rocket Engine
- Oscillation Theory
- Nonoscillatory Solution
where , , , , and α is a positive real number. Throughout the paper, according to the results, we shall impose the following conditions:
(H1) Let , and there exists a constant such that and for ,
(H2) , and there exists a continuous function such that for ,
(H3) , and there exists a continuous function such that for , ,
(H4) , and there exists a continuous function such that for , ,
(H5) and for every .
By a solution of (1), we mean a function , , which has the property and satisfies (1) on . We consider only those solutions of (1), which satisfy for all . We assume that (1) possesses such a solution. A nontrivial solution of (1) is said to be oscillatory if it has a sequence of zeros tending to infinity, otherwise, it is called nonoscillatory. An equation is said to be oscillatory if all its solutions are oscillatory.
In the last century, oscillation theory of differential equations has developed quickly and played an important role in qualitative theory of differential equations and theory of boundary value problem. The study of oscillation theory plays an important role in physical science and technology; for example in the oscillation of building or machine, electromagnetic vibration in radio technology and optical science, self-exited vibration in control system, sound vibration, beam vibration in synchrotron accelerator, the vibration sparked for burning rocket engine, the complicated oscillation in chemical reaction, and also in the research of a lossless high-speed computer network and physical sciences [1, 2]. All of this different phenomena can be unified into oscillation theory through an oscillation equation. There are many books on oscillation theory, we choose to refer to [3–6].
This problem has received the attention of many authors. Many criteria have been found, some of which involve the average behavior of the integral of the alternating coefficient. Among numerous papers dealing with this subject, we refer in particular to [7–35].
The first attempt for Equation (1) was due to Graef et al. , who investigated the case of (1) with and . In 1996, Wong and Agarwal studied the oscillatory behavior of (1) with and the existence of a positive monotone solution of the damped equation given in . Note that their paper contains a lot of new results and has been the motivation for the work for many others. It is the motivation for two recent papers and this work. In , Zhang and Wang studied the oscillation of Equation (1) with . We should note that Wong’s  result for corresponds to the special case with in two main results in . On the other hand, in another recent paper, Remili  studied the oscillation results for Equation (1) with , . Two results of Remili, are also similar to Wong’s results with addition of a suitable weighted function.
In this paper, motivated by the ideas in , we obtain several new oscillation criteria for Equation (1) and its special cases by using generalized Riccati transformation and well known techniques. The results obtained essentially generalize and improve some known results and can be applied to the well-known half-linear equation and damped half-linear-type equations.
In this section, we prove our main results.
then Equation (1) is oscillatory.
Noting condition (2), and the fact that , this implies that the left-hand side of this inequality, that is, is lower bounded. But the right-hand side of it tend towards mines infinite, so contradiction exists. The proof is complete. □
all conditions of Theorem 2.1 are satisfied. Hence it is oscillatory.
Corollary 2.1 Let the conditions of Theorem 2.1 be satisfied except that condition (3) is replaced by (12) and (13). Then Equation (1) is oscillatory.
Now, supposing that this condition is not an establishment, we discuss the oscillatory behavior of (1). We have the following result.
then Equation (1) is oscillatory.
hence (16) takes the form for all large T.
Proof Let be a nonoscillatory solution on the interval I of the differential Equation (1). We suppose, as in Theorem 2.1, that is positive on I. We consider the following three cases for the behavior of .
This contradicts condition (17).
which contradicts the fact that oscillates.
for all as it was shown in . The remaining part of the proof is similar to that of Theorem 2.1. □
Remark 2.2 When and , Theorem 2.1 and Theorem 2.2 reduce to Theorem 1 and 2 in , respectively.
then Equation (1) is oscillatory.
The rest of the proof can be made as in the proof of Theorem 2.1. □
then any solution of Equation (1) such that is bounded is oscillatory.
By (25) and (H5), the right side of (27) tends to −∞ as . However, the left side of (27) is nonnegative.
In view of (29), the left side of (30) is infinite, whereas the right side of (30) is finite by (H5).
The rest of the proof is as in the proof of Theorem 2.1. Hence we omit it. □
Note that it can be easily seen from the proof of Theorem 2.4 when , it is not necessary to assume that is bounded. In this case, conclusion of Theorem 2.4 leads to the following results.
then Equation (1) is oscillatory.
Remark 2.3 If we take in our results, then condition (H1) may include weaker conditions. In fact, if we replace condition (H1) with the condition , then all oscillation criteria above are valid with . Hence when and , Theorem 2.1, Theorem 2.2 and Corollary 2.2 reduce to Theorem 2.1 and Theorem 2.2 and Theorem 2.9 in , respectively. When , and , Theorem 2.2 and Corollary 2.2 give Theorem 1 and Theorem 2 in , respectively. When and , Theorem 2.2 and Corollary 2.2 reduce Theorem 1 and Theorem 2 in , respectively.
then Equation (1) with is oscillatory.
for . Proceeding similarly as in Theorem 2.1, using condition (33), we obtain a contradiction. □
As an immediate consequence of Theorem 2.5, we have the following interesting criteria for the oscillation of (1).
Then the conclusion Theorem 2.5 holds.
It is clear that condition (38) is a necessary condition for (34) to hold.
In case (34) failed to be satisfied that the following theorem may be applicable.
then Equation (1) with is oscillatory.
Proof Proof is a similar to the proof of Theorem 2.2. □
Remark 2.5 More recently, Ouyang and et al.  gave some oscillation criteria for Equation (32) under the main condition . However, their results impose sign conditions on function and . In our results, we assume that hypothesis (H1) is stronger than . But our results do not depend on signs of the functions and . Note that Theorem 3.6 given in , which also does not depend on the signs of and , is different from Theorem 2.5 above, because it contains suitable averaging functions.
The authors would like to express their thanks to the anonymous referee for his(her) valuable suggestions and comments on the manuscript of this paper.
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