New Oscillation Results for Forced Second Order Differential Equations with Mixed Nonlinearities

Affiliation(s)

Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpa?a University, Tokat, Turkey.

Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpa?a University, Tokat, Turkey.

ABSTRACT

Some new oscillation criteria are given for forced second order differential equations with mixed nonlinearities by using the generalized variational principle and Riccati technique. Our results generalize and extend some known oscillation results in the literature.

Some new oscillation criteria are given for forced second order differential equations with mixed nonlinearities by using the generalized variational principle and Riccati technique. Our results generalize and extend some known oscillation results in the literature.

Cite this paper

E. Tunç and A. Kaymaz, "New Oscillation Results for Forced Second Order Differential Equations with Mixed Nonlinearities,"*Applied Mathematics*, Vol. 3 No. 2, 2012, pp. 147-153. doi: 10.4236/am.2012.32023.

E. Tunç and A. Kaymaz, "New Oscillation Results for Forced Second Order Differential Equations with Mixed Nonlinearities,"

References

[1] J. S. W. Wong, “Oscillation Criteria for a Forced SecondOrder Linear Differential Equation,” Journal of Mathematical Analysis and Applications, Vol. 231, No. 1, 1999, pp. 235-240. doi:10.1006/jmaa.1998.6259

[2] W. T. Li and S. S. Cheng, “An Oscillation Criterion for Nonhomogeneous Half-Linear Differential Equations,” Applied Mathematics Letters, Vol. 15, No. 3, 2002, pp. 259-263. doi:10.1016/S0893-9659(01)00127-6

[3] J. V. Manojlovic, “Oscillation Criteria for Second-Order Half-Linear Differential Equations,” Mathematical and Computer Modelling, Vol. 30, No. 5-6, 1999, pp. 109119. doi:10.1016/S0895-7177(99)00151-X

[4] Q. R. Wang, “Oscillation and Asymptotics for SecondOrder Half-Linear Differential Equations,” Applied Mathematics and Computation, Vol. 122, No. 2, 2001, pp. 253266. doi:10.1016/S0096-3003(00)00056-4

[5] Q. R. Wang and Q. G. Yang “Interval Criteria for Oscillation of Second-Order Half-Linear Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 291, No. 1, 2004, pp. 224-236. doi:10.1016/j.jmaa.2003.10.028

[6] J. Jaros and T. Kusano, “A Picone Type Identity for Second Order Half-Linear Differential Equations,” Acta Mathematica Universitatis Comenianae, Vol. 68, No. 1, 1999, pp. 137-151.

[7] A. Elbert, “A Half-Linear Second Order Differential Equation,” Colloquia Mathematica Societatis Janos Bolyai: Qualitative Theory of Differential Equations, Szeged, 1979, pp. 153-180.

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

[9] I. V. Kamenev, “An Integral Criterion for Oscillation of Linear Differential Equations of Second Order,” Matematicheskie Zametki Vol. 23, No. 2, 1978, pp. 249-251.

[10] W. Leighton, “Comparison Theorems for Linear Differential Equations of Second Order,” Proceedings of the American Mathematical Society, Vol. 13, 1962, pp. 603610. doi:10.1090/S0002-9939-1962-0140759-0

[11] Q. Kong, “Interval Criteria for Oscillation of Second-Order Linear Ordinary Differential Equation,” Journal of Mathematical Analysis and Applications, Vol. 229, No. 1, 1999, pp. 258-270. doi:10.1006/jmaa.1998.6159

[12] H. J. Li and C. C. Yeh, “Sturm Comparison Theorem for Half-Linear Second Order Differential Equations,” Proceedings of the Royal Society of Edinburgh, Vol. A125, 1995, pp. 1193-1240. doi:10.1017/S0308210500030468

[13] O. Do?ly and P. ?ehák, “Half-Linear Differential Equations,” North-Holland Mathematics Studies, Vol. 202, Elsevier Science, Amsterdam, 2005.

[14] R. P. Agarwal, S. R. Grace and D. O’Regan, “Oscillation Theory for Second Order Linear, Half-Linear, Superlinear Dynamic Equations,” Kluver, Dordrecht, 2002.

[15] J. Shao, “A New Oscillation Criterion for Forced SecondOrder Quasi-Linear Differential Equations,” Discrete Dynamics in Nature and Society, Vol. 2011, Hindawi Publishing Corporation, New York, pp. 1-8.

[16] Z. Zheng and F. Meng, “Oscillation Criteria for Forced Second Order Quasi-Linear Differential Equations,” Mathematical and Computer Modelling, Vol. 45, No. 1-2, 2007, pp. 215-220. doi:10.1016/j.mcm.2006.05.005

[17] Z. Zheng, X. Wang and H. Han, “Oscillation Criteria for Forced Second Order Differential Equations with Mixed Nonlinearities,” Applied Mathematics Letters, Vol. 22, No. 7, 2009, pp. 1096-1101. doi:10.1016/j.aml.2009.01.018

[18] J. Jaros, T. Kusano and N. Yoshida, “Generalized Picone’s Formula and Forced Oscillation in Quasi-Linear Differential Equations of the Second Order,” Archivum Mathematicum, Vol. 38, No. 1, 2002, pp. 53-59.

[19] J. Shao and F. Meng, “Generalized Variational Principles on Oscillation for Nonlinear Nonhomogeneous Differential Equations,” Abstract and Applied Analysis, Vol. 2011, 2011, pp. 1-10.

[20] Q. Yang, “Interval Oscillation Criteria for a Forced Second Order Nonlinear Ordinary Differential Equations with Oscillatory Potential,” Applied Mathematics and Computation, Vol. 135, No. 1, 2003, pp. 49-64.

[21] D. ?akmak and A. Tiryaki, “Oscillation Criteria for Certain Forced Second-Order Nonlinear Differential Equations,” Applied Mathematics Letters, Vol. 17, No. 3, 2004, pp. 275-279. doi:10.1016/S0893-9659(04)90063-8

[22] E. Tun?, “A Note on the Oscillation of Second-Order Differential Equations with Damping,” Journal of Computational Analysis and Applications, Vol. 12, No. 2, 2010, pp. 444-453.

[23] E. Tun?, “Interval Oscillation Criteria for Certain Forced Second-Order Differential Equations,” Carpathian Journal of Mathematics, Vol. 28, No. 1, 2012, in Press.

[24] V. Komkov, “A Generalization of Leighton’s Variational Theorem,” Applicable Analysis: An International Journal, Vol. 2, No. 4, 1972, pp. 377-383. doi:10.1080/00036817208839051

[25] G. H. Hardy, J. E. Littlewood and G. Polya, “Inequalities,” 2nd Edition, Cambridge University Press, Cambridge, 1988.

[1] J. S. W. Wong, “Oscillation Criteria for a Forced SecondOrder Linear Differential Equation,” Journal of Mathematical Analysis and Applications, Vol. 231, No. 1, 1999, pp. 235-240. doi:10.1006/jmaa.1998.6259

[2] W. T. Li and S. S. Cheng, “An Oscillation Criterion for Nonhomogeneous Half-Linear Differential Equations,” Applied Mathematics Letters, Vol. 15, No. 3, 2002, pp. 259-263. doi:10.1016/S0893-9659(01)00127-6

[3] J. V. Manojlovic, “Oscillation Criteria for Second-Order Half-Linear Differential Equations,” Mathematical and Computer Modelling, Vol. 30, No. 5-6, 1999, pp. 109119. doi:10.1016/S0895-7177(99)00151-X

[4] Q. R. Wang, “Oscillation and Asymptotics for SecondOrder Half-Linear Differential Equations,” Applied Mathematics and Computation, Vol. 122, No. 2, 2001, pp. 253266. doi:10.1016/S0096-3003(00)00056-4

[5] Q. R. Wang and Q. G. Yang “Interval Criteria for Oscillation of Second-Order Half-Linear Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 291, No. 1, 2004, pp. 224-236. doi:10.1016/j.jmaa.2003.10.028

[6] J. Jaros and T. Kusano, “A Picone Type Identity for Second Order Half-Linear Differential Equations,” Acta Mathematica Universitatis Comenianae, Vol. 68, No. 1, 1999, pp. 137-151.

[7] A. Elbert, “A Half-Linear Second Order Differential Equation,” Colloquia Mathematica Societatis Janos Bolyai: Qualitative Theory of Differential Equations, Szeged, 1979, pp. 153-180.

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

[9] I. V. Kamenev, “An Integral Criterion for Oscillation of Linear Differential Equations of Second Order,” Matematicheskie Zametki Vol. 23, No. 2, 1978, pp. 249-251.

[10] W. Leighton, “Comparison Theorems for Linear Differential Equations of Second Order,” Proceedings of the American Mathematical Society, Vol. 13, 1962, pp. 603610. doi:10.1090/S0002-9939-1962-0140759-0

[11] Q. Kong, “Interval Criteria for Oscillation of Second-Order Linear Ordinary Differential Equation,” Journal of Mathematical Analysis and Applications, Vol. 229, No. 1, 1999, pp. 258-270. doi:10.1006/jmaa.1998.6159

[12] H. J. Li and C. C. Yeh, “Sturm Comparison Theorem for Half-Linear Second Order Differential Equations,” Proceedings of the Royal Society of Edinburgh, Vol. A125, 1995, pp. 1193-1240. doi:10.1017/S0308210500030468

[13] O. Do?ly and P. ?ehák, “Half-Linear Differential Equations,” North-Holland Mathematics Studies, Vol. 202, Elsevier Science, Amsterdam, 2005.

[14] R. P. Agarwal, S. R. Grace and D. O’Regan, “Oscillation Theory for Second Order Linear, Half-Linear, Superlinear Dynamic Equations,” Kluver, Dordrecht, 2002.

[15] J. Shao, “A New Oscillation Criterion for Forced SecondOrder Quasi-Linear Differential Equations,” Discrete Dynamics in Nature and Society, Vol. 2011, Hindawi Publishing Corporation, New York, pp. 1-8.

[16] Z. Zheng and F. Meng, “Oscillation Criteria for Forced Second Order Quasi-Linear Differential Equations,” Mathematical and Computer Modelling, Vol. 45, No. 1-2, 2007, pp. 215-220. doi:10.1016/j.mcm.2006.05.005

[17] Z. Zheng, X. Wang and H. Han, “Oscillation Criteria for Forced Second Order Differential Equations with Mixed Nonlinearities,” Applied Mathematics Letters, Vol. 22, No. 7, 2009, pp. 1096-1101. doi:10.1016/j.aml.2009.01.018

[18] J. Jaros, T. Kusano and N. Yoshida, “Generalized Picone’s Formula and Forced Oscillation in Quasi-Linear Differential Equations of the Second Order,” Archivum Mathematicum, Vol. 38, No. 1, 2002, pp. 53-59.

[19] J. Shao and F. Meng, “Generalized Variational Principles on Oscillation for Nonlinear Nonhomogeneous Differential Equations,” Abstract and Applied Analysis, Vol. 2011, 2011, pp. 1-10.

[20] Q. Yang, “Interval Oscillation Criteria for a Forced Second Order Nonlinear Ordinary Differential Equations with Oscillatory Potential,” Applied Mathematics and Computation, Vol. 135, No. 1, 2003, pp. 49-64.

[21] D. ?akmak and A. Tiryaki, “Oscillation Criteria for Certain Forced Second-Order Nonlinear Differential Equations,” Applied Mathematics Letters, Vol. 17, No. 3, 2004, pp. 275-279. doi:10.1016/S0893-9659(04)90063-8

[22] E. Tun?, “A Note on the Oscillation of Second-Order Differential Equations with Damping,” Journal of Computational Analysis and Applications, Vol. 12, No. 2, 2010, pp. 444-453.

[23] E. Tun?, “Interval Oscillation Criteria for Certain Forced Second-Order Differential Equations,” Carpathian Journal of Mathematics, Vol. 28, No. 1, 2012, in Press.

[24] V. Komkov, “A Generalization of Leighton’s Variational Theorem,” Applicable Analysis: An International Journal, Vol. 2, No. 4, 1972, pp. 377-383. doi:10.1080/00036817208839051

[25] G. H. Hardy, J. E. Littlewood and G. Polya, “Inequalities,” 2nd Edition, Cambridge University Press, Cambridge, 1988.