Kamenev-type criteria for nonlinear second-order delay dynamic equations

Year 2018, Volume: 47 Issue: 2, 339 - 345, 01.04.2018

M. Tamer Şenel , Nadide Utku M. M. A. El-sheikh Tongxing Li

https://izlik.org/JA96SA43KH

Abstract

We study oscillation of certain second-order nonlinear delay dynamic equations on arbitrary time scales. Employing a class of kernel functions,
new Kamenev-type oscillation criteria are presented that differ from the known ones. These criteria improve some related results for second-order differential equations.

Keywords

Oscillation , Second-order dynamic equation , Delay dynamic equation , Time scale

References

• Agarwal, R. P., Bohner, M., and Li, T. Oscillatory behavior of second-order half-linear damped dynamic equations, Appl. Math. Comput. 254, 408–418, 2015.
• Agarwal, R. P., Bohner, M., Li, T., and Zhang, C. Oscillation criteria for second-order dynamic equations on time scales, Appl. Math. Lett. 31, 34–40, 2014.
• Agarwal, R. P., Bohner, M., O’Regan, D., and Peterson, A. Dynamic equations on time scales: a survey, J. Comput. Appl. Math. 141, 1–26, 2002.
• Agarwal, R. P., Bohner, M., and Saker, S. H. Oscillation of second order delay dynamic equations, Can. Appl. Math. Q. 13, 1–17, 2005.
• Agarwal, R. P., Grace, S. R., and O’Regan, D. Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations (Kluwer Academic Publishers, Dordrecht, 2002).
• Bohner, M., Hassan, T. S., and Li, T. Fite–Hille–Wintner-type oscillation criteria for second-order half-linear dynamic equations with deviating arguments, Indag. Math. (N.S.) 29, 548–560, 2018.
• Bohner, M. and Li, T. Oscillation of second-order p-Laplace dynamic equations with a nonpositive neutral coefficient, Appl. Math. Lett. 37, 72–76, 2014.
• Bohner, M. and Li, T. Kamenev-type criteria for nonlinear damped dynamic equations, Sci. China Math. 58, 1445–1452, 2015.
• Bohner, M. and Peterson, A. Dynamic Equations on Time Scales: An Introduction with Applications (Birkhäuser, Boston, 2001).
• Bohner, M. and Peterson, A. Advances in Dynamic Equations on Time Scales (Birkhäuser, Boston, 2003).
• Braverman, E. and Karpuz, B. Nonoscillation of second-order dynamic equations with several delays, Abstr. Appl. Anal. 2011, 1–34, 2011.
• Candan, T. Oscillation criteria for second-order nonlinear neutral dynamic equations with distributed deviating arguments on time scales, Adv. Difference Equ. 2013, 1–8, 2013.
• Erbe, L., Peterson, A., and Saker, S. H. Oscillation criteria for second-order nonlinear delay dynamic equations, J. Math. Anal. Appl. 333, 505–522, 2007.
• Fu, Y.-L. and Wang, Q.-R. Oscillation criteria for second-order nonlinear damped differential equations, Dynam. Systems Appl. 18, 375–391, 2009.
• Grace, S. R., Bohner, M., and Sun, S. Oscillation of fourth-order dynamic equations, Hacet. J. Math. Stat. 39, 545–553, 2010.
• Grace, S. R. and Zafer, A. Oscillation of fourth-order nonlinear neutral delay dynamic equations, Hacet. J. Math. Stat. 44, 331–339, 2015.
• Hilger, S. Analysis on measure chains–a unified approach to continuous and discrete calculus, Results Math. 18, 18–56, 1990.
• Huang, Y. and Meng, F. Oscillation criteria for forced second-order nonlinear differential equations with damping, J. Comput. Appl. Math. 224, 339–345, 2009.
• Karpuz, B. Li type oscillation theorem for delay dynamic equations, Math. Methods Appl. Sci. 36, 993–1002, 2013.
• Karpuz, B. and Öcalan, Ö. New oscillation tests and some refinements for first-order delay dynamic equations, Turkish J. Math. 40, 850–863, 2016.
• Li, T., Agarwal, R. P., and Bohner, M. Some oscillation results for second-order neutral dynamic equations, Hacet. J. Math. Stat. 41, 715–721, 2012.
• Li, T. and Saker, S. H. A note on oscillation criteria for second-order neutral dynamic equations on isolated time scales, Commun. Nonlinear Sci. Numer. Simul. 19, 4185–4188, 2014.
• Li, T. and Rogovchenko, Yu. V. Oscillation criteria for second-order superlinear Emden– Fowler neutral differential equations, Monatsh. Math. 184, 489–500, 2017.
• Liu, L. and Bai, Y. New oscillation criteria for second-order nonlinear neutral delay differential equations, J. Comput. Appl. Math. 231, 657–663, 2009.
• Long, Q. and Wang, Q.-R. New oscillation criteria of second-order nonlinear differential equations, Appl. Math. Comput. 212, 357–365, 2009.
• Meng, F. and Huang, Y. Interval oscillation criteria for a forced second-order nonlinear differential equation with damping, Appl. Math. Comput. 218, 1857–1861, 2011.
• O’Regan, D. and Hassan, T. S. Oscillation criteria for solutions to nonlinear dynamic equations of higher order, Hacet. J. Math. Stat. 45, 417–427, 2016.
• Sahiner, Y. Oscillation of second-order delay differential equations on time scales, Nonlinear Anal. 63, e1073–e1080, 2005.
• Senel, M. T. Kamenev-type oscillation criteria for the second-order nonlinear dynamic equations with damping on time scales, Abstr. Appl. Anal. 2012, 1–18, 2012.
• Sun, Y. G. New Kamenev-type oscillation criteria for second-order nonlinear differential equations with damping, J. Math. Anal. Appl. 291, 341–351, 2004.
• Utku, N., Li, T., and Senel, M. T. An asymptotic criterion for third-order dynamic equations with positive and negative coefficients, Hacet. J. Math. Stat. 44, 1157–1162, 2015.
• Xu, R. and Meng, F. New Kamenev-type oscillation criteria for second order neutral nonlinear differential equations, Appl. Math. Comput. 188, 1364–1370, 2007.
• Zafer, A. On oscillation and nonoscillation of second-order dynamic equations, Appl. Math. Lett. 22, 136–141, 2009.