Abstract and Applied Analysis

New Oscillatory Behavior of Third-Order Nonlinear Delay Dynamic Equations on Time Scales

Li Gao, Quanxin Zhang, and Shouhua Liu

Full-text: Open access

Abstract

A class of third-order nonlinear delay dynamic equations on time scales is studied. By using the generalized Riccati transformation and the inequality technique, four new sufficient conditions which ensure that every solution is oscillatory or converges to zero are established. The results obtained essentially improve earlier ones. Some examples are considered to illustrate the main results.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 914264, 11 pages.

Dates
First available in Project Euclid: 27 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1425049102

Digital Object Identifier
doi:10.1155/2014/914264

Mathematical Reviews number (MathSciNet)
MR3182310

Citation

Gao, Li; Zhang, Quanxin; Liu, Shouhua. New Oscillatory Behavior of Third-Order Nonlinear Delay Dynamic Equations on Time Scales. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 914264, 11 pages. doi:10.1155/2014/914264. https://projecteuclid.org/euclid.aaa/1425049102


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References

  • M. Bohner and S. H. Saker, “Oscillation of second order nonlinear dynamic equations on time scales,” The Rocky Mountain Journal of Mathematics, vol. 34, no. 4, pp. 1239–1254, 2004.
  • L. Erbe, “Oscillation criteria for second order linear equations on a time scale,” The Canadian Applied Mathematics Quarterly, vol. 9, no. 4, pp. 345–375, 2001.
  • L. Erbe, A. Peterson, and P. Rehák, “Comparison theorems for linear dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 275, no. 1, pp. 418–438, 2002.
  • S. Sun, Z. Han, and C. Zhang, “Oscillation of second-order delay dynamic equations on time scales,” Journal of Applied Mathematics and Computing, vol. 30, no. 1-2, pp. 459–468, 2009.
  • Q. Zhang and L. Gao, “Oscillation criteria for second order half-liner delay dynamic equations with damping on time scales,” Scientia Sinica A, vol. 40, no. 7, pp. 673–682, 2010 (Chinese).
  • Q. Zhang, L. Gao, and S. Liu, “Oscillation criteria for second order half-liner delay dynamic equations with damping on time scales(II),” Scientia Sinica A, vol. 41, no. 10, pp. 885–896, 2011 (Chinese).
  • S. R. Grace, R. P. Agarwal, B. Kaymakçalan, and W. Sae-jie, “Oscillation theorems for second order nonlinear dynamic equations,” Journal of Applied Mathematics and Computing, vol. 32, no. 1, pp. 205–218, 2010.
  • R. P. Agarwal, M. Bohner, and S. H. Saker, “Oscillation of second order delay dynamic equations,” The Canadian Applied Mathematics Quarterly, vol. 13, no. 1, pp. 1–17, 2005.
  • L. Erbe, A. Peterson, and S. H. Saker, “Oscillation criteria for second-order nonlinear delay dynamic equations,” Journal of Mathematical Analysis and Applications, vol. 333, no. 1, pp. 505–522, 2007.
  • S. R. Grace, M. Bohner, and R. P. Agarwal, “On the oscillation of second-order half-linear dynamic equations,” Journal of Difference Equations and Applications, vol. 15, no. 5, pp. 451–460, 2009.
  • L. Erbe, A. Peterson, and S. H. Saker, “Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales,” Journal of Computational and Applied Mathematics, vol. 181, no. 1, pp. 92–102, 2005.
  • L. Erbe, A. Peterson, and S. H. Saker, “Hille and Nehari type criteria for third-order dynamic equations,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 112–131, 2007.
  • Z. Han, T. Li, S. Sun, and M. Zhang, “Oscillation behavior of solutions of third-order nonlinear delay dynamic equations on time scales,” Communications of the Korean Mathematical Society, vol. 26, no. 3, pp. 499–513, 2011.
  • L. Gao, Q. Zhang, and S. Liu, “Oscillatory behavior of third-order nonlinear delay dynamic equations on time scales,” Journal of Computational and Applied Mathematics, vol. 256, pp. 104–113, 2014.
  • M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001.
  • M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003.
  • Y. Şahiner, “Oscillation of second-order delay differential equations on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 5–7, pp. e1073–e1080, 2005.
  • G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, UK, 2nd edition, 1998.
  • A. Tuna and S. Kutukcu, “Some integral inequalities on time scales,” Applied Mathematics and Mechanics, vol. 29, no. 1, pp. 23–29, 2008. \endinput