- Open Access
Oscillation of second-order damped differential equations
© Fu et al.; licensee Springer. 2013
- Received: 8 May 2013
- Accepted: 30 September 2013
- Published: 19 November 2013
We study oscillatory behavior of a class of second-order differential equations with damping under the assumptions that allow applications to retarded and advanced differential equations. New theorems extend and improve the results in the literature. Illustrative examples are given.
- functional differential equation
- damping term
where , , , , q does not vanish eventually, , for some and for all . Throughout, we assume that solutions of (1.1) exist for any . A solution x of (1.1) is termed oscillatory if it has arbitrarily large zeros; otherwise, we call it nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.
In [8, 15], the authors investigated (1.1) under the assumptions that , , and . The natural question now is: Can one extend the results of to functional equation (1.1)? The purpose of this paper is to give an affirmative answer to this question.
for and for ;
- (ii)H has a nonpositive continuous partial derivative with respect to the second variable satisfying, for some locally integrable continuous function h,
Then (1.1) is oscillatory.
We consider each of two cases separately.
which contradicts (2.3).
one has (2.15), which contradicts (2.3). This completes the proof. □
where , then (1.1) is oscillatory.
which contradicts (2.18). This completes the proof. □
Then (1.1) is oscillatory.
which contradicts (2.25) due to the fact that , where is defined as in (2.26).
The rest of the proof is similar to that of the case where (2.7) holds. Then one can get a contradiction to (2.25). This completes the proof. □
On the basis of Theorem 2.3, similar as in the proof of Theorem 2.2, we have the following result immediately.
where , then (1.1) is oscillatory.
Remark 2.1 Efficient oscillation tests can be derived from Theorems 2.1-2.4 with different choices of the functions H, , and . For example, for , Kamenev’s weight function H defined by , where , belongs to the class . The details are left to the reader.
The following three examples illustrate applications of theoretical results in the previous section.
Hence, by Theorem 2.1, equation (3.1) is oscillatory. As a matter of fact, one such solution is .
Hence, by Theorem 2.1, equation (3.2) is oscillatory. As a matter of fact, one such solution is .
Hence, by Theorem 2.3, equation (3.3) is oscillatory.
Remark 3.1 In this paper, we present some new oscillation criteria for the differential equation with a linear damping term (1.1). Our theorems can be applied to the cases where , , or p is an oscillatory function. Furthermore, the main results can be applied to the cases where the deviating argument τ is delayed or advanced. On the other hand, we do not need to require the assumption that for . Hence, the results obtained supplement and improve those reported in [8, 15].
Remark 3.2 Note that when , Theorems 2.1 and 2.2 include [, Theorem 17] and [, Theorem 19], respectively. On the basis of assumption (2.24), Theorems 2.3 and 2.4 include [, Theorem 17] and [, Theorem 19], respectively.
The authors would like to thank the editors and referees for their thoughtful review of this manuscript and their insightful comments used to improve the quality of this paper. This research is supported by the National Key Basic Research Program of P.R. China (2013CB035604), NNSF of P.R. China (Grant Nos. 61034007, 51277116, 51107069, 61304029), and NSF of Xinjiang (Grant No. 201318101-16).
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