Oscillation Theorems for a Class of Nonlinear Second Order Differential Equations with Damping

Affiliation(s)

School of Science, Beijing University of Civil Engineering and Architecture, Beijing, China.

School of Science, Beijing University of Civil Engineering and Architecture, Beijing, China.

ABSTRACT

The oscillatory behavior of solutions of a class of second order nonlinear differential equations with damping is studied and some new sufficient conditions are obtained by using the refined integral averaging technique. Some well known results in the literature are extended. Moreover, two examples are given to illustrate the theoretical analysis.

Cite this paper

X. Wang and G. Song, "Oscillation Theorems for a Class of Nonlinear Second Order Differential Equations with Damping,"*Advances in Pure Mathematics*, Vol. 3 No. 1, 2013, pp. 226-233. doi: 10.4236/apm.2013.31A032.

X. Wang and G. Song, "Oscillation Theorems for a Class of Nonlinear Second Order Differential Equations with Damping,"

References

[1] Q. R. Wang, “Oscillation Criteria for Nonlinear Second Order Damped Differential Equations,” Acta Mathematica Hungarica, Vol. 102, No. 1-2, 2004, pp. 117-139.

[2] R. P. Agarwal and S. R. Grace, “On the Oscillation of Certain Second Order Differential Equations,” Georgian Mathematical Journal, Vol. 7, No. 2, 2000, pp. 201-213.

[3] B. Ayanlar and A. Tiryaki, “Oscillation Theorems for Nonlinear Second Order Differential Equations with Damping,” Acta Mathematica Hungarica, Vol. 89, No. 1-2, 2000, pp. 1-13. doi:10.1023/A:1026716923088

[4] S. R. Grace, “Oscillation Theorems for Nonlinear Differential Equations of Second Order,” Journal of Mathematical Analysis and Applications, Vol. 171, No. 1, 1992, pp. 220-241. doi:10.1016/0022-247X(92)90386-R

[5] I. V. Kamenev, “An Integral Criterion for Oscillation of Linear Differential Equations of Second Order,” Matematicheskie Zametki, Vol. 23, No. 2, 1978, pp. 136-138.

[6] M. Kirane and Y. V. Rogovchenko, “On Oscillation of Nonlinear Second Order Differential Equation with Damping Term,” Applied Mathematics and Computation, Vol. 117, No. 2-3, 2001, pp. 177-192. doi:10.1016/S0096-3003(99)00172-1

[7] M. Kirane and Y. V. Rogovchenko, “Oscillation Results for a Second Order Damped Differential Equation with Nonmonotonous Nonlinearity,” Journal of Mathematical Analysis and Applications, Vol. 250, No. 1, 2000, pp. 118-138. doi:10.1006/jmaa.2000.6975

[8] H. J. Li, “Oscillation Criteria for Second Order Linear Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 194, No. 1, 1995, pp. 217-234. doi:10.1006/jmaa.1995.1295

[9] W. T. Li, “Oscillation of Certain Second-Order Nonlinear Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 217, No. 1, 1998, pp. 1-14. doi:10.1006/jmaa.1997.5680

[10] Ch. G. Philos, “Oscillation Theorems for Linear Differential Equations of Second Order,” Archiv der Mathematik, Vol. 53, No. 5, 1989, pp. 482-492. doi:10.1007/BF01324723

[11] Y. V. Rogovchenko, “Oscillation Theorems for Second-Order Equations with Damping,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 41, No. 7-8, 2000, pp. 1005-1028. doi:10.1016/S0362-546X(98)00324-1

[12] Y. V. Rogovchenko, “On Oscillation of a Second Order Nonlinear Delay Differential Equation,” Funkcialaj Ekvacioj, Vol. 43, 2000, pp. 1-29.

[13] Y. V. Rogovchenko, “Oscillation Criteria for Second Order Nonlinear Perturbed Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 215, No. 2, 1997, pp. 334-357. doi:10.1006/jmaa.1997.5595

[14] Q. R. Wang, “Oscillation and Asymptotics for Second-Order Half-Linear Differential Equations,” Applied Mathematics and Computation, Vol. 122, No. 2, 2001, pp. 253-266. doi:10.1016/S0096-3003(00)00056-4

[15] Q. R. Wang, “Oscillation Criteria for Even Order Nonlinear Damped Differential Equations,” Acta Mathematica Hungarica, Vol. 95, No. 23, 2002, pp. 169-178. doi:10.1023/A:1015676519998

[16] A. Tiryaki and A. Zafer, “Oscillation Criteria for Second order Nonlinear Differential Equations with Damping,” Turkish Journal of Mathematics, Vol. 24, 2000, pp. 185-196.

[17] J. Yan, “Oscillation Theorems for Second Order Linear Differential Equations with Damping,” Proceedings of the American Mathematical Society, Vol. 98, No. 2, 1986, pp. 276-282. doi:10.1090/S0002-9939-1986-0854033-4

[18] P. Hartman, “On Non-Oscillatory Linear Differential Equations of Second Order,” American Journal of Mathematics, Vol. 74, No. 2, 1952, pp. 389-400. doi:10.2307/2372004

[19] A. Wintner, “A Criterion of Oscillatory Stability,” Quarterly of Applied Mathematics, Vol. 7, 1949, pp. 115-117.

[20] Y. V. Rogovchenko and F. Tuncay, “Oscillation Criteria for Second-Order Nonlinear Differential Equations with Damping,” Nonlinear Analysis, Vol. 69, No. 1, 2008, pp. 208-221. doi:10.1016/j.na.2007.05.012

[21] Y. V. Rogcvchenko and F. Tuncay, “Oscillation Theorems for a Class of Second Order Nonlinear Differential Equations with Damping,” Taiwanese Journal of Mathematics, Vol. 13, No. 6B, 2009, pp. 1909-1928.

[22] X. J. Wang and G. H. Song, “Oscillation Criteria for a Second-Order Nonlinear Damped Differential Equation,” International Journal of Information and Systems Sciences, Vol. 7, No. 1, 2011, pp. 73-82.

[1] Q. R. Wang, “Oscillation Criteria for Nonlinear Second Order Damped Differential Equations,” Acta Mathematica Hungarica, Vol. 102, No. 1-2, 2004, pp. 117-139.

[2] R. P. Agarwal and S. R. Grace, “On the Oscillation of Certain Second Order Differential Equations,” Georgian Mathematical Journal, Vol. 7, No. 2, 2000, pp. 201-213.

[3] B. Ayanlar and A. Tiryaki, “Oscillation Theorems for Nonlinear Second Order Differential Equations with Damping,” Acta Mathematica Hungarica, Vol. 89, No. 1-2, 2000, pp. 1-13. doi:10.1023/A:1026716923088

[4] S. R. Grace, “Oscillation Theorems for Nonlinear Differential Equations of Second Order,” Journal of Mathematical Analysis and Applications, Vol. 171, No. 1, 1992, pp. 220-241. doi:10.1016/0022-247X(92)90386-R

[5] I. V. Kamenev, “An Integral Criterion for Oscillation of Linear Differential Equations of Second Order,” Matematicheskie Zametki, Vol. 23, No. 2, 1978, pp. 136-138.

[6] M. Kirane and Y. V. Rogovchenko, “On Oscillation of Nonlinear Second Order Differential Equation with Damping Term,” Applied Mathematics and Computation, Vol. 117, No. 2-3, 2001, pp. 177-192. doi:10.1016/S0096-3003(99)00172-1

[7] M. Kirane and Y. V. Rogovchenko, “Oscillation Results for a Second Order Damped Differential Equation with Nonmonotonous Nonlinearity,” Journal of Mathematical Analysis and Applications, Vol. 250, No. 1, 2000, pp. 118-138. doi:10.1006/jmaa.2000.6975

[8] H. J. Li, “Oscillation Criteria for Second Order Linear Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 194, No. 1, 1995, pp. 217-234. doi:10.1006/jmaa.1995.1295

[9] W. T. Li, “Oscillation of Certain Second-Order Nonlinear Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 217, No. 1, 1998, pp. 1-14. doi:10.1006/jmaa.1997.5680

[10] Ch. G. Philos, “Oscillation Theorems for Linear Differential Equations of Second Order,” Archiv der Mathematik, Vol. 53, No. 5, 1989, pp. 482-492. doi:10.1007/BF01324723

[11] Y. V. Rogovchenko, “Oscillation Theorems for Second-Order Equations with Damping,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 41, No. 7-8, 2000, pp. 1005-1028. doi:10.1016/S0362-546X(98)00324-1

[12] Y. V. Rogovchenko, “On Oscillation of a Second Order Nonlinear Delay Differential Equation,” Funkcialaj Ekvacioj, Vol. 43, 2000, pp. 1-29.

[13] Y. V. Rogovchenko, “Oscillation Criteria for Second Order Nonlinear Perturbed Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 215, No. 2, 1997, pp. 334-357. doi:10.1006/jmaa.1997.5595

[14] Q. R. Wang, “Oscillation and Asymptotics for Second-Order Half-Linear Differential Equations,” Applied Mathematics and Computation, Vol. 122, No. 2, 2001, pp. 253-266. doi:10.1016/S0096-3003(00)00056-4

[15] Q. R. Wang, “Oscillation Criteria for Even Order Nonlinear Damped Differential Equations,” Acta Mathematica Hungarica, Vol. 95, No. 23, 2002, pp. 169-178. doi:10.1023/A:1015676519998

[16] A. Tiryaki and A. Zafer, “Oscillation Criteria for Second order Nonlinear Differential Equations with Damping,” Turkish Journal of Mathematics, Vol. 24, 2000, pp. 185-196.

[17] J. Yan, “Oscillation Theorems for Second Order Linear Differential Equations with Damping,” Proceedings of the American Mathematical Society, Vol. 98, No. 2, 1986, pp. 276-282. doi:10.1090/S0002-9939-1986-0854033-4

[18] P. Hartman, “On Non-Oscillatory Linear Differential Equations of Second Order,” American Journal of Mathematics, Vol. 74, No. 2, 1952, pp. 389-400. doi:10.2307/2372004

[19] A. Wintner, “A Criterion of Oscillatory Stability,” Quarterly of Applied Mathematics, Vol. 7, 1949, pp. 115-117.

[20] Y. V. Rogovchenko and F. Tuncay, “Oscillation Criteria for Second-Order Nonlinear Differential Equations with Damping,” Nonlinear Analysis, Vol. 69, No. 1, 2008, pp. 208-221. doi:10.1016/j.na.2007.05.012

[21] Y. V. Rogcvchenko and F. Tuncay, “Oscillation Theorems for a Class of Second Order Nonlinear Differential Equations with Damping,” Taiwanese Journal of Mathematics, Vol. 13, No. 6B, 2009, pp. 1909-1928.

[22] X. J. Wang and G. H. Song, “Oscillation Criteria for a Second-Order Nonlinear Damped Differential Equation,” International Journal of Information and Systems Sciences, Vol. 7, No. 1, 2011, pp. 73-82.