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Forced oscillation of second-order differential equations with mixed nonlinearities
Journal of Inequalities and Applications volume 2014, Article number: 520 (2014)
Abstract
We study oscillatory behavior of a class of second-order forced differential equations with mixed nonlinearities. Some new oscillation theorems are presented that improve and complement those related results given in the literature. An example is provided to illustrate the main results.
MSC:34K11.
1 Introduction
This paper is concerned with the oscillation of solutions to a class of second-order forced differential equations with mixed nonlinearities
where , is a natural number, () are constants, , , , , (), . We also assume that there exists a function such that (), , , and .
We consider only those solutions x of equation (1.1) which satisfy condition for all . We assume that (1.1) possesses such solutions. As usual, a solution of (1.1) is called oscillatory if it has arbitrarily large zeros on the interval ; otherwise, it is termed nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.
Functional differential equations arise in many applied problems in natural sciences, technology, and automatic control; see, for instance, Hale [1]. In mechanical and engineering problems, questions related to the existence of oscillatory and nonoscillatory solutions play an important role. As a result, many theoretical studies have been undertaken during the past few years. We refer the reader to [2–12] and the references cited therein.
In what follows, we briefly comment on the related results that motivate our study. Li and Cheng [9] studied a differential equation
Zheng et al. [11] considered the equation
Equation (1.1) was studied by Zhong et al. [12] who established the following oscillation theorem.
Theorem 1.1 (see [[12], Theorem 3.1])
Assume that
and there exists a function such that
where
and
Then equation (1.1) is oscillatory.
The purpose of this paper is to refine Theorem 1.1 in some cases and analyze the oscillatory behavior of solutions to (1.1) in the case when the integral in (1.2) is finite. This paper proceeds as follows: in Section 2, we present our main results; in Section 3, an example is provided to illustrate the results obtained.
2 Oscillation criteria
In what follows, all functional inequalities are tacitly assumed to hold eventually, that is, for all t large enough. Before stating the main results, we begin with the following lemma.
Lemma 2.1 (Bernoulli’s inequality)
For and ,
Theorem 2.2 Assume that condition (1.2) is satisfied, and let
If there exists a function such that, for all constants ,
where is defined as in (1.4), then equation (1.1) is oscillatory.
Proof Assume that (1.1) has a nonoscillatory solution x. Without loss of generality, we can assume that x is an eventually positive solution, i.e., there exists a such that for . Equation (1.1) yields
With a proof similar to that of [[12], Theorem 3.1], we conclude that
For , define a function
Then for . Differentiating (2.5), by virtue of (2.3) and (2.4), we have , and so
Let . It follows from Lemma 2.1 that
Hence, we deduce that
By virtue of (2.4), there exists a constant such that
Thus, by (2.8), we obtain
Substitution of (2.9) into (2.6) implies that
Integrating the latter inequality from to t, we conclude that
which contradicts (2.2). This completes the proof. □
On the basis of Theorem 2.2, we can obtain the following results due to condition (2.1).
Corollary 2.3 Assume that conditions (1.2) and (2.1) are satisfied. If there exists a function such that
where is defined as in (1.4), then equation (1.1) is oscillatory.
Using in Corollary 2.3, we can get the following criterion.
Corollary 2.4 Assume that conditions (1.2) and (2.1) are satisfied. If
then equation (1.1) is oscillatory.
In what follows, we derive some oscillation criteria for (1.1) in the case where
Theorem 2.5 Assume that conditions (2.1) and (2.10) are satisfied, and let (). Suppose also that there exists a function such that (2.2) holds. If
holds for all constants , where
then equation (1.1) is oscillatory.
Proof Assume that (1.1) has a nonoscillatory solution x. As above, we may assume that there is a such that for . By virtue of (1.1), we have (2.3). Then there exist two possible cases, i.e., or
Assume first that . Then we obtain (2.4). Proceeding as in the proof of Theorem 2.2, we can obtain a contradiction to (2.2). Suppose now that (2.13) holds. Define a new function ω by
Then for and
On the other hand, we have (2.7), and so
due to (). By (2.13), there exists a constant such that . Hence, by virtue of (2.16), we conclude that
It follows now from (2.15), (2.17), and that
Using the condition , we have, for ,
Integrating the latter inequality from t to l, we deduce that
Passing to the limit as , we have
which yields
i.e.,
Multiplying (2.18) by and integrating the resulting inequality from to t, we obtain
Hence, we derive from (2.19) that
which contradicts (2.11). The proof is complete. □
Theorem 2.6 Assume that conditions (2.1) and (2.10) are satisfied, and let (). Suppose further that there exists a function such that (2.2) holds. If, for all constants ,
where δ is as in (2.12), then equation (1.1) is oscillatory.
Proof Assume again that there exists a such that for . From (1.1), we have (2.3). Then there exist two possible cases, i.e., or (2.13). Suppose that . Following the same lines as in Theorem 2.2, we can obtain a contradiction to (2.2). Assume now that (2.13) is satisfied. Define the function ω by (2.14). We have for and (2.15). On the other hand, it has been established in Theorems 2.2 and 2.5 that (2.7) and (2.19) hold. By virtue of (2.19),
It follows from the latter inequality, (), and (2.7) that
and
Since , there exists a constant such that . Hence, by (2.22), we conclude that
Using (2.15), (2.21), and (2.23), we obtain
The remainder of the proof is similar to that of Theorem 2.5 and hence is omitted. This completes the proof. □
Remark 2.7 From the proof of Theorems 2.5 and 2.6, one can obtain oscillation results for equation (1.1) with delayed and advanced arguments. The details are left to the reader.
3 Example
The following example illustrates possible applications of the theoretical results presented in this paper.
Example 3.1 For , consider a second-order differential equation
where is a constant. Let , , , , , , , , , and . Then and
Hence, by Corollary 2.4, equation (3.1) is oscillatory for any .
Let Q be defined as in Theorem 1.1. Then
Using the latter inequality and in (1.3), we observe that Theorem 1.1 cannot ensure oscillation of (3.1) on the interval . Therefore, Corollary 2.4 improves Theorem 1.1.
Remark 3.2 In this paper, several new oscillation criteria for equation (1.1) are obtained by using the Riccati substitution and Bernoulli’s inequality. Employing inequalities different from those exploited in [12], we improve Theorem 1.1; see Example 3.1. Furthermore, Theorems 2.5 and 2.6 complement those by Zhong et al. [12] since our results can be applied to the case where (2.10) holds.
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Acknowledgements
The authors express their sincere gratitude to the editors and anonymous referees for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results and accentuate important details. This research is supported by the AMEP of Linyi University, P.R. China.
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Wang, Y., Li, T. & Thandapani, E. Forced oscillation of second-order differential equations with mixed nonlinearities. J Inequal Appl 2014, 520 (2014). https://doi.org/10.1186/1029-242X-2014-520
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DOI: https://doi.org/10.1186/1029-242X-2014-520
Keywords
- oscillation
- forced differential equation
- second-order
- mixed nonlinearities