## International Journal of Differential Equations

### Oscillation Theorems for Second-Order Damped Nonlinear Differential Equations

#### Abstract

We present new oscillation criteria for the differential equation of the form ${[r(t)U(t)]}^{\prime }+p(t){k}_{2}(x(t), {x}^{\prime }(t)){|x(t)|}^{\nu }$$U(t)+q(t)\varphi (x({g}_{1}(t)),{x}^{\prime }({g}_{2}(t)))f(x(t))=0$, where $U(t)={k}_{1}(x(t),{x}^{\prime }(t)){|{x}^{\prime }(t)|}^{\alpha -1}{x}^{\prime }(t)$, $\alpha \le \beta , \nu =(\beta -\alpha )/(\alpha +1)$. Our research is different from most known ones in the sense that H function is not employed in our results, though Riccati's substitution and its generalized forms are used. Our criteria which are established under quite general assumptions are an extension for previous results. In particular, by taking $\beta =\alpha$, the above-mentioned equation can be reduced into the various types of equations concerned by people currently.

#### Article information

Source
Int. J. Differ. Equ., Volume 2009 (2009), Article ID 714357, 15 pages.

Dates
Revised: 28 January 2009
Accepted: 23 March 2009
First available in Project Euclid: 26 January 2017

https://projecteuclid.org/euclid.ijde/1485399812

Digital Object Identifier
doi:10.1155/2009/714357

Mathematical Reviews number (MathSciNet)
MR2525714

Zentralblatt MATH identifier
1207.34083

#### Citation

Qin, Hui-Zeng; Ren, Yongsheng. Oscillation Theorems for Second-Order Damped Nonlinear Differential Equations. Int. J. Differ. Equ. 2009 (2009), Article ID 714357, 15 pages. doi:10.1155/2009/714357. https://projecteuclid.org/euclid.ijde/1485399812

#### References

• A. Tiryaki and A. Zafer, “Oscillation criteria for second order nonlinear differential equations with damping,” Turkish Journal of Mathematics, vol. 24, no. 2, pp. 185–196, 2000.MR1796670
• O. G. Mustafa, S. P. Rogovchenko, and Y. V. Rogovchenko, “On oscillation of nonlinear second-order differential equations with damping term,” Journal of Mathematical Analysis and Applications, vol. 298, no. 2, pp. 604–620, 2004.MR2086978
• F. Lu and F. Meng, “Oscillation theorems for superlinear second-order damped differential equations,” Applied Mathematics and Computation, vol. 189, no. 1, pp. 796–804, 2007.MR2330256
• E. M. Elabbasy, T. S. Hassan, and S. H. Saker, “Oscillation of second-order nonlinear differential equations with a damping term,” Electronic Journal of Differential Equations, vol. 2005, no. 76, pp. 1–13, 2005.MR2162237
• W.-T. Li, “Interval oscillation criteria for second order nonlinear differential equations with damping,” Taiwanese Journal of Mathematics, vol. 7, no. 3, pp. 461–475, 2003.MR1998768
• N. Yamaoka, “Oscillation criteria for second-order damped nonlinear differential equations with $p$-Laplacian,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 932–948, 2007.MR2270061
• J. V. Manojlović, “Oscillation criteria for sublinear differential equations with damping,” Acta Mathematica Hungarica, vol. 104, no. 1-2, pp. 153–169, 2004.MR2071250
• Z. Zheng, “Oscillation criteria for nonlinear second order differential equations with damping,” Acta Mathematica Hungarica, vol. 110, no. 3, pp. 241–252, 2006.MR2204062
• Y. G. Sun, “Oscillation of second order functional differential equations with damping,” Applied Mathematics and Computation, vol. 178, no. 2, pp. 519–526, 2006.MR2248512
• Q. Yang, “Oscillation of self-adjoint linear matrix Hamiltonian systems,” Journal of Mathematical Analysis and Applications, vol. 296, no. 1, pp. 110–130, 2004.MR2070496
• Q. Yang, “On the oscillation of certain nonlinear neutral partial differential equations,” Applied Mathematics Letters, vol. 20, no. 8, pp. 900–907, 2007.MR2323129
• O. G. Mustafa, S. P. Rogovchenko, and Y. V. Rogovchenko, “On oscillation of nonlinear čommentComment on ref. [8?]: This reference is a repetition of [2?]. Please check. second-order differential equations with damping term,” Journal of Mathematical Analysis and Applications, vol. 298, no. 2, pp. 604–620, 2004.MR2086978