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

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Abstract

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

$$\begin{array}{rcl}&&\left\{a(t)\Psi(x(t))\left[|(x(t)+p(t)x(\tau(t)))^{\Delta ^{n-1}}|^{\alpha-1}(x(t)+p(t)x(\tau(t)))^{\Delta^{n-1}}\right]^{\gamma}\right\}^{\Delta}\\[12pt]&&{}\quad +\lambda F(t,x(\delta(t)))=0,\end{array}$$

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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Authors and Affiliations

  1. Department of Mathematics and Physics, Hunan Institute of Engineering, Xiangtan, 411104, Hunan, People’s Republic of China

    Da-Xue Chen

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  1. Da-Xue Chen
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Correspondence to Da-Xue Chen.

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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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