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

Abstract

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.

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

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

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.

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.
There are 33 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

M. Tamer Şenel

Nadide Utku This is me

M. M. A. El-sheikh This is me

Tongxing Li

Publication Date April 1, 2018
Published in Issue Year 2018 Volume: 47 Issue: 2

Cite

APA Şenel, M. T., Utku, N., El-sheikh, M. M. A., Li, T. (2018). Kamenev-type criteria for nonlinear second-order delay dynamic equations. Hacettepe Journal of Mathematics and Statistics, 47(2), 339-345.
AMA Şenel MT, Utku N, El-sheikh MMA, Li T. Kamenev-type criteria for nonlinear second-order delay dynamic equations. Hacettepe Journal of Mathematics and Statistics. April 2018;47(2):339-345.
Chicago Şenel, M. Tamer, Nadide Utku, M. M. A. El-sheikh, and Tongxing Li. “Kamenev-Type Criteria for Nonlinear Second-Order Delay Dynamic Equations”. Hacettepe Journal of Mathematics and Statistics 47, no. 2 (April 2018): 339-45.
EndNote Şenel MT, Utku N, El-sheikh MMA, Li T (April 1, 2018) Kamenev-type criteria for nonlinear second-order delay dynamic equations. Hacettepe Journal of Mathematics and Statistics 47 2 339–345.
IEEE M. T. Şenel, N. Utku, M. M. A. El-sheikh, and T. Li, “Kamenev-type criteria for nonlinear second-order delay dynamic equations”, Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 2, pp. 339–345, 2018.
ISNAD Şenel, M. Tamer et al. “Kamenev-Type Criteria for Nonlinear Second-Order Delay Dynamic Equations”. Hacettepe Journal of Mathematics and Statistics 47/2 (April 2018), 339-345.
JAMA Şenel MT, Utku N, El-sheikh MMA, Li T. Kamenev-type criteria for nonlinear second-order delay dynamic equations. Hacettepe Journal of Mathematics and Statistics. 2018;47:339–345.
MLA Şenel, M. Tamer et al. “Kamenev-Type Criteria for Nonlinear Second-Order Delay Dynamic Equations”. Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 2, 2018, pp. 339-45.
Vancouver Şenel MT, Utku N, El-sheikh MMA, Li T. Kamenev-type criteria for nonlinear second-order delay dynamic equations. Hacettepe Journal of Mathematics and Statistics. 2018;47(2):339-45.