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Abstract and Applied Analysis
Volume 2013, Article ID 340574, 17 pages
http://dx.doi.org/10.1155/2013/340574
Research Article

Oscillatory and Asymptotic Properties on a Class of Third Nonlinear Dynamic Equations with Damping Term on Time Scales

School of Science, Shandong University of Technology, Zhangzhou Road 12, Zibo 255049, China

Received 4 March 2013; Revised 10 May 2013; Accepted 10 May 2013

Academic Editor: Narcisa C. Apreutesei

Copyright © 2013 Qinghua Feng and Huizeng Qin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. M. Bohner, “Some oscillation criteria for first order delay dynamic equations,” Far East Journal of Applied Mathematics, vol. 18, no. 3, pp. 289–304, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. B. G. Zhang and X. Deng, “Oscillation of delay differential equations on time scales,” Mathematical and Computer Modelling, vol. 36, no. 11–13, pp. 1307–1318, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. Z.-Q. Zhu and Q.-R. Wang, “Existence of nonoscillatory solutions to neutral dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 751–762, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. E. Akin-Bohner, M. Bohner, and S. H. Saker, “Oscillation criteria for a certain class of second order Emden-Fowler dynamic equations,” Electronic Transactions on Numerical Analysis, vol. 27, pp. 1–12, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. S. R. Grace, R. P. Agarwal, M. Bohner, and D. O'Regan, “Oscillation of second-order strongly superlinear and strongly sublinear dynamic equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 8, pp. 3463–3471, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. 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. View at Publisher · View at Google Scholar · View at Scopus
  11. S. H. Saker, “Oscillation of nonlinear dynamic equations on time scales,” Applied Mathematics and Computation, vol. 148, no. 1, pp. 81–91, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. B. Jia, L. Erbe, and A. Peterson, “New comparison and oscillation theorems for second-order half-linear dynamic equations on time scales,” Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2744–2756, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. Z. Han, S. Sun, and B. Shi, “Oscillation criteria for a class of second-order Emden-Fowler delay dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 334, no. 2, pp. 847–858, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. M. Bohner and S. H. Saker, “Oscillation criteria for perturbed nonlinear dynamic equations,” Mathematical and Computer Modelling, vol. 40, no. 3-4, pp. 249–260, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. L. Erbe, A. Peterson, and S. H. Saker, “Oscillation criteria for second-order nonlinear dynamic equations on time scales,” Journal of the London Mathematical Society, vol. 67, no. 3, pp. 701–714, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. L. Erbe, A. Peterson, and S. H. Saker, “Kamenev-type oscillation criteria for second-order linear delay dynamic equations,” Dynamic Systems and Applications, vol. 15, no. 1, pp. 65–78, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. H. Saker, “Oscillation criteria of second-order half-linear dynamic equations on time scales,” Journal of Computational and Applied Mathematics, vol. 177, no. 2, pp. 375–387, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. H. Saker, “Oscillation of second-order nonlinear neutral delay dynamic equations on time scales,” Journal of Computational and Applied Mathematics, vol. 187, no. 2, pp. 123–141, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. Huang and W. Feng, “Forced oscillation of second order nonlinear dynamic equations on time scales,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2008, no. 36, pp. 1–13, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. L. Erbe, A. Peterson, and S. H. Saker, “Oscillation and asymptotic behavior of a third-order nonlinear dynamic equation,” The Canadian Applied Mathematics Quarterly, vol. 14, no. 2, pp. 129–147, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. L. Erbe, T. S. Hassan, and A. Peterson, “Oscillation of third order functional dynamic equations with mixed arguments on time scales,” Journal of Applied Mathematics and Computing, vol. 34, no. 1-2, pp. 353–371, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. T. S. Hassan, “Oscillation of third order nonlinear delay dynamic equations on time scales,” Mathematical and Computer Modelling, vol. 49, no. 7-8, pp. 1573–1586, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. B. Karpuz, “Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coefficients,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2009, no. 34, pp. 1–14, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. B. Karpuz, “Asymptotic behaviour of bounded solutions of a class of higher-order neutral dynamic equations,” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2174–2183, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. Z.-H. Yu and Q.-R. Wang, “Asymptotic behavior of solutions of third-order nonlinear dynamic equations on time scales,” Journal of Computational and Applied Mathematics, vol. 225, no. 2, pp. 531–540, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. Z. Han, T. Li, S. Sun, and F. Cao, “Oscillation criteria for third order nonlinear delay dynamic equations on time scales,” Annales Polonici Mathematici, vol. 99, no. 2, pp. 143–156, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. Z. Han, T. Li, S. Sun, and C. Zhang, “Oscillation behavior of third-order neutral Emden-Fowler delay dynamic equations on time scales,” Advances in Difference Equations, vol. 2010, Article ID 586312, 23 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. L. Erbe, T. S. Hassan, and A. Peterson, “Oscillation of third order nonlinear functional dynamic equations on time scales,” Differential Equations and Dynamical Systems, vol. 18, no. 1-2, pp. 199–227, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. S. R. Grace, J. R. Graef, and M. A. El-Beltagy, “On the oscillation of third order neutral delay dynamic equations on time scales,” Computers & Mathematics with Applications, vol. 63, no. 4, pp. 775–782, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. Y. Sun, Z. Han, Y. Sun, and Y. Pan, “Oscillation theorems for certain third order nonlinear delay dynamic equations on time scales,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2011, no. 75, pp. 1–14, 2011. View at Google Scholar · View at MathSciNet
  34. S. H. Saker, “Oscillation of third-order functional dynamic equations on time scales,” Science China. Mathematics, vol. 54, no. 12, pp. 2597–2614, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. S. H. Saker, “On oscillation of a certain class of third-order nonlinear functional dynamic equations on time scales,” Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, vol. 54, no. 4, pp. 365–389, 2011. View at Google Scholar · View at MathSciNet
  36. R. Agarwal, M. Bohner, and A. Peterson, “Inequalities on time scales: a survey,” Mathematical Inequalities & Applications, vol. 4, no. 4, pp. 535–557, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK, 1988. View at MathSciNet