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New Oscillation Criteria of Second-Order Nonlinear Delay Dynamic Equations on Time Scales

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DOI: 10.4236/am.2014.521325    3,308 Downloads   3,833 Views  

ABSTRACT

By using the generalized Riccati transformation and the integral averaging technique, the paper establishes some new oscillation criteria for the second-order nonlinear delay dynamic equations on time scales. The results in this paper unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation on time scales. The Theorems in this paper are new even in the continuous and the discrete cases.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Zhang, Q. and Gao, L. (2014) New Oscillation Criteria of Second-Order Nonlinear Delay Dynamic Equations on Time Scales. Applied Mathematics, 5, 3474-3483. doi: 10.4236/am.2014.521325.

References

[1] Agarwal, R.P., Bohner, M. and Saker, S.H. (2005) Oscillation of Second Order Delay Dynamic Equations. Quarterly of Applied Mathematics, 13, 1-18.
[2] Sahiner. Y. (2005) Oscillation of Second Order Delay Differential Equations on Time Scales. Nonlinear Analysis: Theory, Methods & Applications, 63, 1073-1080.
http://dx.doi.org/10.1016/j.na.2005.01.062
[3] Erbe, L., Peterson, A. and Saker, S.H. (2007) Oscillation Criteria for Second Order Nonlinear Delay Dynamic Equations. Journal of Mathematical Analysis and Applications, 333, 505-522.
http://dx.doi.org/10.1016/j.jmaa.2006.10.055
[4] Saker, S.H. (2005) Oscillation Criteria of Second-Order Half-Linear Dynamic Equations on Time Scales. Journal of Computational and Applied Mathematics, 177, 375-387.
http://dx.doi.org/10.1016/j.cam.2004.09.028
[5] Grace, S.R., Bohner, M. and Agarwal, R.P. (2009) On the Oscillation of Second-Order Half-Linear Dynamic Equations. Journal of Difference Equations and Applications, 15, 451-460.
http://dx.doi.org/10.1080/10236190802125371
[6] Bohner, M. and Saker, S.H. (2004) Oscillation of Second Order Nonlinear Dynamic Equations on Time Scales. Rocky Mountain Journal of Mathematics, 34, 1239-1245.
http://dx.doi.org/10.1216/rmjm/1181069797
[7] Erbe, L. (2001) Oscillation Criteria for Second Order Linear Equations on a Time Scale. The Canadian Applied Mathematics Quarterly, 9, 345-375.
[8] Zhang, Q., Song, X. and Gao, L. (2012) On the Oscillation of Second-Order Nonlinear Delay Dynamic Equations on Time Scales. Applied Mathematics & Information Sciences, 30, 219-234.
[9] Zhang, Q., Gao, L. and Wang, L. (2011) Oscillation of Second-Order Nonlinear Delay Dynamic Equations on Time Scales. Computers & Mathematics with Applications, 61, 2342-2348.
http://dx.doi.org/10.1016/j.camwa.2010.10.005
[10] Han, Z., Li, T., Sun, S. and Zhang, C. (2009) Oscillation for Second-Order Nonlinear Delay Dynamic Equations on Time Scales. Advances in Difference Equations, Article ID: 756171, 13 pages.
[11] Bohner, M. and Peterson, A. (2001) Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston.
http://dx.doi.org/10.1007/978-1-4612-0201-1
[12] Bohner, M. and Peterson, A. (2003) Advances in Dynamic Equations on Time Scales. Birkh?user, Boston.
http://dx.doi.org/10.1007/978-0-8176-8230-9
[13] Sun, S., Han, Z. and Zhang, C. (2009) Oscillation of Second Order Delay Dynamic Equations on Time Scales. Journal of Applied Mathematics and Computing, 30, 459-468.
http://dx.doi.org/10.1007/s12190-008-0185-6
[14] Philos, Ch.G. (1989) Oscillation Theorems for Linear Differential Equations of Second Order. Archiv der Mathematik, 53, 482-492.
http://dx.doi.org/10.1007/BF01324723
[15] Saker, S.H. (2003) Oscillation Theorems for Second-Order Nonlinear Delay Difference Equations. Periodica Mathematica Hungarica, 47, 201-213.
http://dx.doi.org/10.1023/B:MAHU.0000010821.30713.be

  
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