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.
Keywordsoscillation 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.
2 Main results
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.
3 Applications and discussion
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).
- Agarwal RP, Bohner M, Li WT Monographs and Textbooks in Pure and Applied Mathematics 267. In Nonoscillation and Oscillation: Theory for Functional Differential Equations. Dekker, New York; 2004.View ArticleGoogle Scholar
- Agarwal RP, Grace SR, O’Regan D: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht; 2000.MATHView ArticleGoogle Scholar
- Aktaş MF, Çakmak D, Tiryaki A: On the qualitative behaviors of solutions of third-order nonlinear functional differential equations. Appl. Math. Lett. 2011, 24: 1849-1855. 10.1016/j.aml.2011.05.004MATHMathSciNetView ArticleGoogle Scholar
- Aktaş MF, Tiryaki A, Zafer A: Oscillation criteria for third-order nonlinear functional differential equations. Appl. Math. Lett. 2010, 23: 756-762. 10.1016/j.aml.2010.03.003MATHMathSciNetView ArticleGoogle Scholar
- Baculíková B, Džurina J, Rogovchenko YV: Oscillation of third order trinomial delay differential equations. Appl. Math. Comput. 2012, 218: 7023-7033. 10.1016/j.amc.2011.12.049MATHMathSciNetView ArticleGoogle Scholar
- Candan T: Oscillation criteria for second-order nonlinear neutral dynamic equations with distributed deviating arguments on time scales. Adv. Differ. Equ. 2013, 2013: 1-8. 10.1186/1687-1847-2013-1MathSciNetView ArticleGoogle Scholar
- Candan T: Oscillation of second-order nonlinear neutral dynamic equations on time scales with distributed deviating arguments. Comput. Math. Appl. 2011, 62: 4118-4125. 10.1016/j.camwa.2011.09.062MATHMathSciNetView ArticleGoogle Scholar
- Bohner M, Saker SH: Oscillation of damped second order nonlinear delay differential equations of Emden-Fowler type. Adv. Dyn. Syst. Appl. 2006, 1: 163-182.MATHMathSciNetGoogle Scholar
- Džurina J, Komariková R: Asymptotic properties of third-order delay trinomial differential equations. Abstr. Appl. Anal. 2011., 2011: Article ID 730128Google Scholar
- Grace SR, Lalli BS, Yeh CC: Oscillation theorems for nonlinear second order differential equations with a nonlinear damping term. SIAM J. Math. Anal. 1984, 15: 1082-1093. 10.1137/0515084MATHMathSciNetView ArticleGoogle Scholar
- Kirane M, Rogovchenko YV: Oscillation results for a second order damped differential equation with nonmonotonous nonlinearity. J. Math. Anal. Appl. 2000, 250: 118-138. 10.1006/jmaa.2000.6975MATHMathSciNetView ArticleGoogle Scholar
- Kirane M, Rogovchenko YV: On oscillation of nonlinear second order differential equation with damping term. Appl. Math. Comput. 2001, 117: 177-192. 10.1016/S0096-3003(99)00172-1MATHMathSciNetView ArticleGoogle Scholar
- Li, T, Rogovchenko, YV, Tang, S: Oscillation of second-order nonlinear differential equations with damping. Math. Slovaca (2012, in press)Google Scholar
- Li WT, Agarwal RP: Interval oscillation criteria for second order nonlinear differential equations with damping. Comput. Math. Appl. 2000, 40: 217-230. 10.1016/S0898-1221(00)00155-3MATHMathSciNetView ArticleGoogle Scholar
- Liu S, Zhang Q, Yu Y: Oscillation of even-order half-linear functional differential equations with damping. Comput. Math. Appl. 2011, 61: 2191-2196. 10.1016/j.camwa.2010.09.011MATHMathSciNetView ArticleGoogle Scholar
- Rogovchenko SP, Rogovchenko YV: Oscillation of differential equations with damping. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 2003, 10: 447-461.MATHMathSciNetGoogle Scholar
- Rogovchenko SP, Rogovchenko YV: Oscillation theorems for differential equations with a nonlinear damping term. J. Math. Anal. Appl. 2003, 279: 121-134. 10.1016/S0022-247X(02)00623-6MATHMathSciNetView ArticleGoogle Scholar
- Rogovchenko YV: Oscillation criteria for second order nonlinear perturbed differential equations. J. Math. Anal. Appl. 1997, 215: 334-357. 10.1006/jmaa.1997.5595MATHMathSciNetView ArticleGoogle Scholar
- Rogovchenko YV: Oscillation theorems for second-order equations with damping. Nonlinear Anal. 2000, 41: 1005-1028. 10.1016/S0362-546X(98)00324-1MATHMathSciNetView ArticleGoogle Scholar
- Rogovchenko YV, Tuncay F: Oscillation criteria for second-order nonlinear differential equations with damping. Nonlinear Anal. 2008, 69: 208-221. 10.1016/j.na.2007.05.012MATHMathSciNetView ArticleGoogle Scholar
- Rogovchenko YV, Tuncay F: Oscillation theorems for a class of second order nonlinear differential equations with damping. Taiwan. J. Math. 2009, 13: 1909-1928.MATHMathSciNetGoogle Scholar
- Şenel MT, Temtek P: On behaviour of solutions for third order nonlinear ordinary differential equations with damping terms. J. Comput. Anal. Appl. 2009, 11: 346-355.MATHMathSciNetGoogle Scholar
- Tiryaki A, Aktaş MF: Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping. J. Math. Anal. Appl. 2007, 325: 54-68. 10.1016/j.jmaa.2006.01.001MATHMathSciNetView ArticleGoogle Scholar
- Tiryaki A, Zafer A: Interval oscillation of a general class of second-order nonlinear differential equations with nonlinear damping. Nonlinear Anal. 2005, 60: 49-63.MATHMathSciNetView ArticleGoogle Scholar
- Yan J: Oscillation theorems for second order linear differential equations with damping. Proc. Am. Math. Soc. 1986, 98: 276-282. 10.1090/S0002-9939-1986-0854033-4MATHView ArticleGoogle Scholar
- Yeh CC: Oscillation theorems for nonlinear second order differential equations with damping term. Proc. Am. Math. Soc. 1982, 84: 397-402. 10.1090/S0002-9939-1982-0640240-9MATHView ArticleGoogle Scholar
- Kamenev IV: An integral criterion for oscillation of linear differential equations. Mat. Zametki 1978, 23: 249-251.MATHMathSciNetGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.