Abstract
In this paper, we derive some sufficient conditions for the oscillation and asymptotic behavior of the nth-order nonlinear neutral delay dynamic equations
on time scales, where α>0 is a constant, γ>0 is a quotient of odd positive integers and λ=±1. Our results in this paper not only extend and improve some known results but also present a valuable unified approach for the investigation of oscillation and asymptotic behavior of nth-order nonlinear neutral delay differential equations and nth-order nonlinear neutral delay difference equations. Examples are provided to show the importance of our main results.
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Hilger, S.: Analysis on measure chains—a unified approach to continuous and discrete calculus. Result. Math. 18, 18–56 (1990)
Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)
Spedding, V.: Taming Nature’s Numbers, pp. 28–31. New Scientist, London (2003)
Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)
Došlý, O., Hilgerb, S.: A necessary and sufficient condition for oscillation of the Sturm–Liouville dynamic equation on time scales. J. Comput. Appl. Math. 141, 147–158 (2002)
Saker, S.H., Agarwal, R.P., O’Regan, D.: Oscillation of second-order damped dynamic equations on time scales. J. Math. Anal. Appl. 330, 1317–1337 (2007)
Bohner, M., Saker, S.H.: Oscillation criteria for perturbed nonlinear dynamic equations. Math. Comput. Model. 40, 249–260 (2004)
Erbe, L., Peterson, A., Saker, S.H.: Hille and Nehari type criteria for third-order dynamic equations. J. Math. Anal. Appl. 329, 112–131 (2007)
Erbe, L., Peterson, A., Saker, S.H.: Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales. J. Comput. Appl. Math. 181, 92–102 (2005)
Zhang, B.G., Shanliang, Z.: Oscillation of second-order nonlinear delay dynamic equations on time scales. Comput. Math. Appl. 49, 599–609 (2005)
Bohner, M., Erbe, L., Peterson, A.: Oscillation for nonlinear second order dynamic equations on a time scale. J. Math. Anal. Appl. 301, 491–507 (2005)
Şahiner, Y.: Oscillation of second-order delay differential equations on time scales. Nonlinear Anal. 63, 1073–1080 (2005)
Han, Z., Sun, S., Shi, B.: Oscillation criteria for a class of second-order Emden–Fowler delay dynamic equations on time scales. J. Math. Anal. Appl. 334, 847–858 (2007)
Agarwal, R.P., Grace, S.R., O’Regan, D.: Oscillation criteria for certain nth order differential equations with deviating arguments. J. Math. Anal. Appl. 262, 601–622 (2001)
Agarwal, R.P., Grace, S.R., O’Regan, D.: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht (2000)
Wong, P.J.Y., Agarwal, R.P.: Oscillation theorems and existence criteria of asymptotically monotone solutions for second order differential equations. Dyn. Syst. Appl. 4, 477–496 (1995)
Wong, P.J.Y., Agarwal, R.P.: Oscillatory behaviour of solutions of certain second order nonlinear differential equations. J. Math. Anal. Appl. 198, 337–354 (1996)
Xu, Z., Xia, Y.: Integral averaging technique and oscillation of certain even order delay differential equations. J. Math. Anal. Appl. 292, 238–246 (2004)
Xu, R., Meng, F.: Some new oscillation criteria for second order quasi-linear neutral delay differential equations. Appl. Math. Comput. 182, 797–803 (2006)
Chern, J.L., Lian, W.Ch., Yeh, C.C.: Oscillation criteria for second order half-linear differential equations with functional arguments. Publ. Math. (Debr.) 48, 209–216 (1996)
Agarwal, R.P., Shieh, S.L., Yeh, C.C.: Oscillation criteria for second-order retarded differential equations. Math. Comput. Model. 26, 1–11 (1997)
Džurina, J., Stavroulakis, I.P.: Oscillation criteria for second-order delay differential equations. Appl. Math. Comput. 140, 445–453 (2003)
Sun, Y.G., Meng, F.W.: Note on the paper of Dzurina and Stavroulakis. Appl. Math. Comput. 174, 1634–1641 (2006)
Jeroš, J., Kusano, T.: Oscillation properties of first order nonlinear functional differential equations of neutral type. Diff. Integral Equ. 4, 425–436 (1991)
Meng, F., Xu, R.: Oscillation criteria for certain even order quasi-linear neutral differential equations with deviating arguments. Appl. Math. Comput. 190, 458–464 (2007)
Agarwal, R.P., O’Regan, D., Saker, S.H.: Oscillation criteria for second-order nonlinear neutral delay dynamic equations. J. Math. Anal. Appl. 300, 203–217 (2004)
Saker, S.H.: Oscillation of second-order nonlinear neutral delay dynamic equations. J. Comput. Appl. Math. 187, 123–141 (2006)
Wu, H., Zhuang, R., Mathsen, R.M.: Oscillation criteria for second-order nonlinear neutral variable delay dynamic equations. Appl. Math. Comput. 178, 321–331 (2006)
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This work was supported by the Scientific Research Foundation of Education Department of Hunan Province of People’s Republic of China (No. 06C242).
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Chen, DX. Oscillation and Asymptotic Behavior for nth-order Nonlinear Neutral Delay Dynamic Equations on Time Scales. Acta Appl Math 109, 703–719 (2010). https://doi.org/10.1007/s10440-008-9341-0
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DOI: https://doi.org/10.1007/s10440-008-9341-0