Oscillation criteria for solutions to nonlinear dynamic equations of higher order

Abstract

In this paper using some new dynamic inequalities we present some oscillation results for higher order dynamic equation n rn−1(t) φαn−1 h (rn−2(t)(...(r1(t)φα1 [x ∆(t)])∆...) ∆) ∆ io∆ +p (t) φγ (x (g (t))) = 0, on an unbounded time scale T. Some new oscillation criteria are obtained using comparison techniques. Some applications illustrating our results are included.

Keywords

• Asymptotic behavior,oscillation,higher order,dynamic equations,dynamic inequality,time scales

References

1. Adıvar, M., Akın E. and Higgins R. Oscillatory behavior of solutions of third-order delay and advanced dynamic equations, Journal of Inequalities and Applications 2014, 2014:95, 16 pp.
2. Grover, L.K. and Kaur, P. An improved estimator of the finite population mean in simple random sampling, Model Assisted Statistics and Applications 6 (1), 47-55, 2011. [3] Agarwal, R.P., Bohner, M., Li, T. and Zhang, C. Hille and Nehari type criteria for thirdorder delay dynamic equations, Journal of Difference Equations and Applications 19 (10), 1563-1579, 2013.
3. Binggen, Z., Xinzhou, Y. and Xueyan, L. Oscillation criteria of certain delay dynamic equations on time scales, Journal of Difference Equations and Applications 11 (10), 933- 946, 2005.
4. Bohner, M. Some oscillation criteria for first order delay dynamic equations, Far East J. Appl. Math. 18 (3), 289-304, 2005.
5. Bohner, M. and Peterson, A. Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001.
6. Bohner, M. and Peterson, A., editors, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003.
7. Bohner, M., Karpuz, B. and Öcalan, Ö. Iterated oscillation criteria for delay dynamic equations of first order, Advances in Difference Equations Volume 2008, Article ID 458687, 12 pp.
8. Chen, D. Oscillation and asymptotic behavior of solutions of certain third-order nonlinear delay dynamic equations, Theoretical Mathematics & Applications 3, 19-33, 2013.

9. Erbe, L., Baoguo and Peterson, A. Oscillation of nth order superlinear dynamic equations on time scales, Rocky Mountain Journal of Mathematics 41 (2), 471-491, 2011. 427
10. Erbe, L., Hassan, T.S., Peterson, A. and Saker, S.H. Interval oscillation criteria for forced second-order nonlinear delay dynamic equations with oscillatory potential, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 17(4), 533- 542, 2010.
11. Erbe, L., Hassan, T.S. and Peterson, A. Oscillation of second order functional dynamic equations, International Journal of Difference Equations 5(2), 175-193, 2010.
12. Erbe, L., Karpuz, B. and Peterson, A. Kamenev-type oscillation criteria for higher order neutral delay dynamic equations, International Journal Difference Equations 6 (1), 1-16, 2011.
13. Erbe, L., Mert, R., Peterson, A. and Zafer, A. Oscillation of even order nonlinear delay dynamic equations on time scales, Czechoslovak Mathematical Journal 63 (138), 265-279, 2013.
14. Erbe, L. and Hassan, T.S. New oscillation criteria for second order sublinear dynamic equations, Dynamic Systems and Applications 22 (1), 49-63, 2013.
15. Grace, S.R. and Hassan, T.S. Oscillation criteria for higher order nonlinear dynamic equations, Mathematische Nachrichten 287 (14-15), 1659-1673, 2014.
16. Grace, S.R., Agarwal, R.P. and Zafer, A. Oscillation of higher order nonlinear dynamic equations on time scales, Advances in Difference Equations 2012, 2012:67, 18 pp.
17. Han, Z., Li, T., Sun S. and Zhang, M. Oscillation behavior of solutions of third-order nonlinear delay dynamic equations on time scales, Communications of the Korean Mathematical Society 26 (3), 499-513, 2011.
18. Hassan, T.S. Oscillation criteria for second-order nonlinear dynamic equations, Advances in Difference Equations 2012:171, 13 pp, 2012.
19. Hassan, T.S. Interval oscillation for second order nonlinear differential equations with a damping term, Serdica Mathematical Journal 34 (4), 715-732, 2008.
20. Hassan, T.S. Asymptotic behavior of solutions of second order nonlinear dynamic equations, Dynamic Systems and Applications 23 (2-3), 179-187, 2014.
21. Hilger, S. Analysis on measure chains — a unified approach to continuous and discrete calculus, Results in Mathematics 18 (1-2), 18-56, 1990.
22. Karpuz, B. Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coefficients, Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 34, 14 pp.
23. Kiguradze, I. T. On oscillatory solutions of some ordinary differential equations, Soviet Mathematics Doklady 144, 33-36, 1962.
24. Mert, R. Oscillation of higher order neutral dynamic equations on time scales, Advances in Difference Equations 2012, 2012:68, 11 pp.
25. Saker, S.H. Oscillation criteria of second-order half-linear dynamic equations on time scales, Journal of Computational and Applied Mathematics 177 (2), 375-387, 2005.
26. Sun, Y. and Hassan, T.S. Comparison criteria for odd order forced nonlinear functional neutral dynamic equations, Applied Mathematics and Computation, 251, 387–395, 2015.
27. Sun, T., Yu, W. and Xi, H. Oscillatory behavior and comparison for higher order nonlinear dynamic equations on time scales, Journal of Applied Mathematics & Informatics 30 (1-2), 289-304, 2012.
28. Zhang, B. G. and Deng, X. Oscillation of delay differential equations on time scales, Mathematical and Computer Modeling 36 (11-13), 1307–1318, 2002.