OSCILLATION BEHAVIOR OF SOLUTIONS OF THIRD-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS ON TIME SCALES

Title & Authors
OSCILLATION BEHAVIOR OF SOLUTIONS OF THIRD-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS ON TIME SCALES
Han, Zhenlai; Li, Tongxing; Sun, Shurong; Zhang, Meng;

Abstract
By using the Riccati transformation technique, we study the oscillation and asymptotic behavior for the third-order nonlinear delay dynamic equations $\small{(c(t)(p(t)x^{\Delta}(t))^{\Delta})^{\Delta}+q(t)f(x({\tau}(t)))=0}$ on a time scale T, where c(t), p(t) and q(t) are real-valued positive rd-continuous functions defined on $\small{\mathbb{T}}$. We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our oscillation results are essentially new. Some examples are considered to illustrate the main results.
Keywords
oscillation behavior;third order delay dynamic equations;time scales;
Language
English
Cited by
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On the oscillation for third-order nonlinear neutral delay dynamic equations on time scales, Journal of Applied Mathematics and Computing, 2017, 54, 1-2, 243
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Oscillation criteria for certain second-order Emden-Fowler delay functional dynamic equations with damping on time scales, Advances in Difference Equations, 2015, 2015, 1
5.
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6.
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7.
Oscillation criteria for third order neutral Emden–Fowler delay dynamic equations on time scales, Journal of Applied Mathematics and Computing, 2016
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Oscillatory behavior of third-order nonlinear delay dynamic equations on time scales, Journal of Computational and Applied Mathematics, 2014, 256, 104
References
1.
R. P. Agarwal, M. Bohner, D. O'Regan, and A. Peterson, Dynamic equations on time scales: a survey, J. Comput. Appl. Math. 141 (2002), no. 1-2, 1-26.

2.
R. P. Agarwal, M. Bohner, and S. H. Saker, Oscillation of second order delay dynamic equations, Can. Appl. Math. Q. 13 (2005), no. 1, 1-17.

3.
M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhauser, Boston, 2001.

4.
M. Bohner and A. Peterson, Advances in dynamic Equations on Time Scales, Birkhauser, Boston, 2003.

5.
M. Bohner and S. H. Saker, Oscillation of second order nonlinear dynamic equations on time scales, Rocky Mountain J. Math. 34 (2004), no. 4, 1239-1254.

6.
L. H. Erbe, Oscillation results for second order linear equations on a time scale, J. Difference Equ. Appl. 8 (2002), no. 11, 1061-1071.

7.
L. Erbe, A. Peterson, and S. H. Saker, Oscillation criteria for second-order nonlinear delay dynamic equations, J. Math. Anal. Appl. 333 (2007), no. 1, 505-522.

8.
L. Erbe, A. Peterson, and S. H. Saker, Asymptotic behavior of solution of a third-order nonlinear dynamic equation on time scales, J. Comput. Appl. Math. 181 (2005), no. 1, 92-102.

9.
L. Erbe, A. Peterson, and S. H. Saker, Hille and Nehari type criteria for third-order dynamic equations, J. Math. Anal. Appl. 329 (2007), no. 1, 112-131.

10.
Z. Han, T. Li, S. Sun, and F. Cao, Oscillation criteria for third order nonlinear delay dynamic equations on time scales, Ann. Polon. Math. 99 (2010), no. 2, 143-156.

11.
Z. Han, T. Li, S. Sun, C. Zhang, and B. Han, Oscillation criteria for a class of second order neutral delay dynamic equations of Emden-Fowler type, Abstr. Appl. Anal. 2-11 (2011), Art. ID 653689, 26 pp.

12.
Z. Han, S. Sun, and B. Shi, Oscillation criteria for a class of second order Emden-Fowler delay dynamic equations on time scales, J. Math. Anal. Appl. 334 (2007), no. 2, 847-858.

13.
Z. Han, S. Sun, and B. Shi, Oscillation criteria for second-order delay dynamic equations on time scales, Adv. Difference Equ. 2007 (2007), Art. ID. 70730, 16 pp.

14.
T. S. Hassan, Oscillation criteria for half-linear dynamic equations on time scales, J. Math. Anal. Appl. 345 (2008), no. 1, 176-185.

15.
T. S. Hassan, Oscillation of third order nonlinear delay dynamic equations on time scales, Math. Comput. Modelling 49 (2009), no. 7-8, 1573-1586.

16.
S. Hilger, Analysis on measure chains|a unified approach to continuous and discrete calculus, Results Math. 18 (1990), no. 1-2, 18-56.

17.
Y. Sahiner, Oscillation of second-order delay differential equations on time scales, Non-linear Analysis, TMA 63 (2005), 1073-1080.

18.
S. H. Saker, Oscillation criteria of second-order half-linear dynamic equations on time scales, J. Comput. Appl. Math. 177 (2005), no. 2, 375-387.

19.
S. H. Saker, R. P. Agarwal, and D. O'Regan, Oscillation results for second-order non-linear neutral delay dynamic equations on time scales, Appl. Anal. 86 (2007), no. 1, 1-17.

20.
S. Sun, Z. Han, P. Zhao, and C. Zhang, Oscillation for a class of second order Emden-Fowler delay dynamic equations on time scales, Adv. Difference Equ. 2010 (2010), Art. ID 642356, 15 pp.

21.
Z.-H. Yu and Q.-R. Wang, Asymptotic behavior of solutions of third-order nonlinear dynamic equations on time scales, J. Comput. Appl. Math. 225 (2009), no. 2, 531-540.

22.
B. G. Zhang and Z. Shanliang, Oscillation of second order nonlinear delay dynamic equations on time scales, Comput. Math. Appl. 49 (2005), no. 4, 599-609.