Oscillation and Asymptotic Behavior for nth-order Nonlinear Neutral Delay Dynamic Equations on Time Scales

Abstract

In this paper, we derive some sufficient conditions for the oscillation and asymptotic behavior of the nth-order nonlinear neutral delay dynamic equationson 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.

Nonlinear Oscillation and Asymptotic Behavior of Second-order Neutral Dynamic Equations on Time Scales

“…In the last years, a lot of authors have investigated the oscillatory and asymptotic behavior of solutions of different classes of dynamic equations on time scales, and we refer the reader to (Chen & Liu, 2008;Chen, 2010a;Chen, 2010b;Chen & Liu, 2010;Grace et al, 2008;Grace et al, 2009;Hassan, 2008;Karpuz, 2009;Saker, 2005;Saker & O'Regan, 2011;Tripathy, 2009;Xu & Xu, 2009 It is easy to see that the cases considered in (Saker, 2005;Hassan, 2008;Grace et al, 2008;Grace et al, 2009) only are some special cases of (1) and that all the results of (Saker, 2005;Hassan, 2008;Grace et al, 2008;Grace et al, 2009) can not be applied to (1) (Saker, 2005) and (Hassan, 2008). Our results improve and extend some of those in (Saker, 2005;Hassan, 2008;Grace et al, 2008;Grace et al, 2009).…”

Hille and Nehari-Type Oscillation Criteria for Third-Order Emden–Fowler Neutral Delay Dynamic Equations

“…And for the oscillation and nonoscillation of the neutral delay dynamic equations, some excellent works have already been established, and we refer the reader to the various articles [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].…”

Oscillation of second-order quasi-linear neutral functional dynamic equations with distributed deviating arguments

“…Recently, there has been much research activity concerning the oscillation and non-oscillation of solutions of various dynamic equations on time scales, e.g., see [2,[6][7][8][9][10][11][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] and the references cited therein. Agarwal et al [2], Saker [18], Tripathy [26] established some oscillation criteria for second-order nonlinear neutral delay dynamic equation…”