Oscillation Theorems for Second-order Nonlinear Neutral Dynamic Equations with Variable Delays and Damping

In this paper, we study the oscillation of a class of second-order nonlinear neutral damped variable delay dynamic equations on time scales. By using a generalized Riccati transformation technique, we obtain some sufficient conditions for the oscillation of the equations. The results of this paper improve and extend some known results. We also illustrate our main results with some examples.

• Da-Xue Chen
• Guang-Hui Liu

• Oscillation theorem
• second-order nonlinear neutral dynamic equation
• variable delay
• damping
• Riccati transformation.

[1] A. Zafer, On oscillation and nonoscillation of second-order dynamic
equations, Appl. Math. Lett. 22 (2009) 136-141.
[2] S. Hilger, Ein Maβkettenkalk┬¿ul mit Anwendung auf Zentrumsmannigfaltigkeiten
(PhD thesis), Universit¨at W¨urzburg, W¨urzburg, 1988.
[3] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An
Introduction with Applications, Birkh¨auser, Boston, 2001.
[4] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time
Scales, Birkh¨auser, Boston, 2003.
[5] R. P. Agarwal, M. Bohner, D. O-Regan and A. Peterson, Dynamic
equations on time scales: a survey, J. Comput. Appl. Math. 141 (2002)
1-26.
[6] V. Spedding, Taming Nature-s Numbers, New Scientist, 179 (2003) 28-
31.
[7] L. Erbe, T. S. Hassan and A. Peterson, Oscillation criteria for nonlinear
damped dynamic equations on time scales, Appl. Math. Comput. 203
(2008) 343-357.
[8] T. S. Hassan, Oscillation criteria for half-linear dynamic equations on
time scales, J. Math. Anal. Appl. 345 (2008) 176-185.
[9] L. Ou, Atkinson-s super-linear oscillation theorem for matrix dynamic
equations on a time scale, J. Math. Anal. Appl. 299 (2004) 615-629.
[10] A. D. Medico and Q. Kong, Kamenev-type and interval oscillation
criteria for second-order linear differential equations on a measure chain,
J. Math. Anal. Appl. 294 (2004) 621-643.
[11] M. Bohner and S. H. Saker, Oscillation criteria for perturbed nonlinear
dynamic equations, Math. Comput. Modelling 40 (2004) 249-260.
[12] D. -X. Chen, Oscillation and asymptotic behavior for nth-order nonlinear
neutral delay dynamic equations on time scales, Acta Appl. Math.
109 (2010) 703-719.
[13] D. -X. Chen and J. -C. Liu, Asymptotic behavior and oscillation of
solutions of third-order nonlinear neutral delay dynamic equations on
time scales, Can. Appl. Math. Q. 16 (2008) 19-43.
[14] O. Doˇsl'y and S. Hilger, A necessary and sufficient condition for
oscillation of the Sturm-Liouville dynamic equation on time scales, J.
Comput. Appl. Math. 141 (2002) 147-158.
[15] B. G. Zhang and Z. Shanliang, Oscillation of second-order nonlinear
delay dynamic equations on time scales, Comput. Math. Appl. 49 (2005)
599-609.
[16] Y. S┬© ahiner, Oscillation of second-order delay differential equations on
time scales, Nonlinear Anal. 63 (2005) 1073-1080.
[17] L. Erbe, A. Peterson and S. H. Saker, Hille and Nehari type criteria for
third-order dynamic equations, J. Math. Anal. Appl. 329 (2007) 112-
131.
[18] D. -X. Chen, Oscillation of second-order Emden-Fowler neutral delay
dynamic equations on time scales, Math. Comput. Modelling 51 (2010)
1221-1229.
[19] G. H. Hardy, J. E. Littlewood and G. P'olya, Inequalities (second ed.),
Cambridge University Press, Cambridge, 1988.
[20] L. Erbe, A. Peterson and S. H. Saker, Asymptotic behavior of solutions
of a third-order nonlinear dynamic equation on time scales, J. Comput.
Appl. Math. 181 (2005) 92-102.
[21] R. P. Agarwal, D. O-Regan and S. H. Saker, Oscillation criteria for
second-order nonlinear neutral delay dynamic equations, J. Math. Anal.
Appl. 300 (2004) 203-217.
[22] S. H. Saker, Oscillation of second-order nonlinear neutral delay dynamic
equations on time scales, J. Comput. Appl. Math. 187 (2006) 123-141.
[23] H. -W. Wu, R. -K. Zhuang and R. M. Mathsen, Oscillation criteria for
second-order nonlinear neutral variable delay dynamic equations, Appl.
Math. Comput. 178 (2006) 321-331.