This paper concerns the oscillation problem of second-order nonlinear damped ODE with functional terms.We give some new interval oscillation criteria which is not only based on constructing a lower solution of a Riccati type equation but also based on constructing an upper solution for corresponding Riccati type equation. We use a recently developed pointwise comparison principle between those lower and upper solutions to obtain our results. Some illustrative examples are also provided to demonstrate our results.
@article{bwmeta1.element.doi-10_1515_math-2015-0023,
author = {S\"uleyman \"O\v grek\c ci},
title = {New interval oscillation criteria for second-order functional differential equations with nonlinear damping},
journal = {Open Mathematics},
volume = {13},
year = {2015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_math-2015-0023}
}
Süleyman Öǧrekçi. New interval oscillation criteria for second-order functional differential equations with nonlinear damping. Open Mathematics, Tome 13 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_math-2015-0023/
[1] M.A. El-Sayed, An oscillation criteria for a forced second-order linear differential equations, Proc. Amer. Math. Soc., 118 (1993), 813–817.
[2] G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, second ed., Cambridge University Press, Cambridge, 1988.
[3] Y. Huang and F. Meng, Oscillation criteria for forced second-order nonlinear differential equations with damping, Journal of Computational and Applied Mathematics 224 (2009) 339-345. | Zbl 1167.34325
[4] F. Jiang, F. Meng, New oscillation criteria for a class of second-order nonlinear forced differential equations, J. Math. Anal. Appl., 336 (2007), 1476–1485. | Zbl 1128.34018
[5] Q. Kong, Interval criteria for oscillation of second-order linear ordinary differential equations, J. Math. Anal. Appl., 229 (1999), 258–270. | Zbl 0924.34026
[6] W.T. Li, Interval oscillation criteria for second-order quasi-linear nonhomogeneous differential equations with damping, Appl. Math. Comput., 147 (2004), 753–763. | Zbl 1054.34054
[7] Q. Long, Q.R. Wang, New oscillation criteria of second-order nonlinear differential equations, Appl. Math. Comput., 212 (2009), 357–365. | Zbl 1172.34322
[8] F. Meng and Y. Huang, Interval oscillation criteria for a forced second-order nonlinear differential equations with damping, Applied Mathematics and Computation 218 (2011) 1857-1861. | Zbl 1235.34104
[9] M. Pasic, Fite-Wintner-Leighton type oscillation criteria for second-order differential equations with nonlinear damping, Abstract and Applied Analysis, Volume 2013 (2013), Article ID 852180, 10 pages.
[10] S.P. Rogovchenko and Yu.V. Rogovchenko, Oscillation of second order differential equations with damping, Dynam. Contin. Discrete Impuls. Syst. Ser. A 10 (2003) 447–461. | Zbl 1048.34071
[11] S.P. Rogovchenko, Y.V. Rogovchenko, Oscillation theorems for differential equations with a nonlinear damping term, J. Math. Anal. Appl., 279 (2003), 121–134. | Zbl 1027.34040
[12] N.Shang and H. Qin, Comments on the paper: “Oscillation of second-order nonlinear ODEwith damping” [Applied Mathematics and Computation 199 (2008) 644–652], Applied Mathematics and Computation 218 (2011) 2979–2980. [WoS]
[13] A. Tiryaki, B. Ayanlar, Oscillation theorems for certain nonlinear differential equations of second order, Comput. Math. Appl., 47 (2004), 149–159. | Zbl 1057.34019
[14] A. Tiryaki, A. Zafer, Oscillation of second-order nonlinear differential equations with nonlinear damping, Math. Comput. Modelling, 39 (2004), 197–208. | Zbl 1049.34040
[15] A. Tiryaki, Y. Ba¸sçı, I. Güleç, Interval criteria for oscillation of second-order functional differential equations, Computers and Mathematics with Applications 50 (2005) 1487-1498. | Zbl 1091.34036
[16] YG Sun, C.H. Ou, J.S.W. Wong, Interval oscillation theorems for a second-order linear differential equation, Computers and Mathematics with Applications 48 (2004) 1693-1699. | Zbl 1069.34049
[17] YG Sun, J.S.W. Wong, Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities, J. Math. Anal. Appl. 334 (2007) 549–560. | Zbl 1125.34024
[18] YG Sun, Oscillation of second order functional differential equations with damping, Applied Mathematics and Computation 178 (2006) 519–526. | Zbl 1105.34326
[19] Q.R. Wang, X.M. Wu, S.M. Zhu, Oscillation criteria for second-order nonlinear damped differential equations, Comput. Math. Appl., 46 (2003), 1253–1262. | Zbl 1059.34039
[20] J.S.W. Wong, Oscillation criteria for a forced second-order linear differential equations, J. Math. Anal. Appl., 231 (1999), 235–240. | Zbl 0922.34029
[21] L. Xing, Z. Zheng, New oscillation criteria for forced second order halflinear differential equations with damping, Appl. Math. Comput., 198 (2008), 481–486. | Zbl 1144.34330
[22] Q. Yang, Interval oscillation criteria for a forced second-order nonlinear ordinary differential equations with oscillatory potential, Appl. Math. Comput., 135 (2003), 49–64. | Zbl 1030.34034
[23] Q. Yang, R.M. Mathsen, Interval oscillation criteria for second order nonlinear delay differential equations, Rocky Mountain Journal of Mathematics, 34 (2004) 1539-1563. [WoS] | Zbl 1062.34036
[24] A. Zhao, Y. Wang, J. Yan, Oscillation criteria for second-order nonlinear differential equations with nonlinear damping, Comput. Math. Appl., 56 (2008), 542–555. | Zbl 1155.34320