Oscillation criteria for forced second order differential equations with mixed nonlinearities. (English) Zbl 1179.34034
Summary: By using Riccati transformation, new oscillation criteria are given for forced second order differential equations with mixed nonlinearities, which improve and generalize results in the literature. An \((\alpha + 1)\)-degree functional is involved for oscillation, which is widely used in variational theories. Examples, including a forced Duffing equation, are given.
MSC:
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
Keywords:
oscillation; forcing term; mixed nonlinearities; Riccati transformation; variational principleReferences:
| [1] | Wong, J. S.W., Oscillation criteria for a second-order linear differential equation, J. Math. Anal. Appl., 231, 235-240 (1999) · Zbl 0922.34029 |
| [2] | Li, W. T.; Cheng, S. S., An oscillation criterion for nonhomogeneous half-linear differential equations, Appl. Math. Lett., 15, 259-263 (2002) · Zbl 1023.34029 |
| [3] | Zheng, Z.; Meng, F., Oscillation criteria for forced second order Quasi-linear differential equations, Math. Comput. Modelling, 45, 215-220 (2007) · Zbl 1143.34309 |
| [4] | Cakmak, D.; Tiryaki, A., Oscillation criteria for certain forced second-order nonlinear differential equations, Appl. Math. Lett., 17, 275-279 (2004) · Zbl 1061.34017 |
| [5] | Wang, Qi-Ru, Oscillation and asymptotics for second-order half-linear differential equations, Appl. Math. Comput., 122, 253-266 (2001) · Zbl 1030.34031 |
| [6] | Wang, Qi-Ru; Yang, Qi-Gui, Interval criteria for oscillation of second-order half-linear differential equations, J. Math. Anal. Appl., 291, 224-236 (2004) · Zbl 1053.34034 |
| [7] | Wang, Qi-Ru, Interval criteria for oscillation of certain second order nonlinear differential equations, Dyn. Contin. Discrete Impuls. Syst., 12, 769-781 (2005) · Zbl 1087.34015 |
| [8] | Wang, Qi-Ru, Interval criteria for oscillation of second-order nonlinear differential equations, J. Comput. Appl. Math., 205, 231-238 (2007) · Zbl 1142.34331 |
| [9] | Shen, Chun-xue; Zhang, Hong-kui, Interval criteria for oscillation of forced second-order half-linear differential equations, Chinese Quart. J. Math., 20, 411-417 (2005) |
| [10] | A. Elbert, A half-linear second order differential equation, in: Colloq. Math. Soc. Janos Bolyai: Qualitative Theory of Differential Equations, Szeged, 1979, pp. 153-180; A. Elbert, A half-linear second order differential equation, in: Colloq. Math. Soc. Janos Bolyai: Qualitative Theory of Differential Equations, Szeged, 1979, pp. 153-180 · Zbl 0511.34006 |
| [11] | Jaros, J.; Kusano, T., A Picone type identity for second order half-linear differential equations, Acta Math. Univ. Comenian., 68, 1, 137-151 (1999) · Zbl 0926.34023 |
| [12] | Jaros, J.; Kusano, T.; Yoshida, N., Generalized Picone’s formula and forced oscillation for in quasilinear differential equations of the second order, Arch. Math. (Bero), 38, 53-59 (2002) · Zbl 1087.34014 |
| [13] | Wintner, A., A criterion of oscillatory stability, Quart. Appl. Math., 7, 115-117 (1949) · Zbl 0032.34801 |
| [14] | Kamenev, I. V., Integral criterion for oscillation of linear differential equations of second order, Mat. Zametki, 23, 249-251 (1978) · Zbl 0386.34032 |
| [15] | Leighton, W., Comparison theorems for linear differential equations of second order, Proc. Amer. Math. Soc., 13, 603-610 (1962) · Zbl 0118.08202 |
| [16] | Kong, Q., Interval criteria for oscillation of second-order linear differential equation, J. Math. Anal. Appl., 258, 244-257 (2001) · Zbl 0987.34024 |
| [17] | Li, H. J.; Yeh, C. C., Sturm comparison theorem for half-linear second order differential equations, Proc. Roy. Edinburgh, A125, 1193-1240 (1995) · Zbl 0873.34020 |
| [18] | Hong, H. L.; Lian, W. C.; Yeh, C. C., The oscillation of half-linear differential equations with an oscillatory coefficient, Math. Comput. Modelling, 24, 77-86 (1996) · Zbl 0924.34027 |
| [19] | Manojlović, J. V., Oscillation criteria for second-order half-linear differential equations, Math. Comput. Modelling, 30, 109-119 (1999) · Zbl 1042.34532 |
| [20] | Hardy, G. H.; Littlewood, J. E.; Polya, G., Inequalities (1988), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0634.26008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
