Oscillation of Third-order Delay Dynamic Equations on Time Scales

This paper is concerned with the oscillatory behavior of a class of third-order noonlinear variable delay neutral functional dynamic equations on time scale. By using the generalized Riccati transformation and inequality technique, we establish some new oscilla- tion criteria for the equations. Our results extend and improve some known results, but also unify the oscillation of third-order nonlinear variable delay functional differential equations and functional difference equations with a nonlinear neutral term. Some examples are given to illustrate the importance of our results.     

Oscillatory Behavior of Third-order Nonlinear Differential Equations with a Sublinear Neutral Term

The authors present some new criteria for oscillation and asymptotic behavior of solutions of thirdorder nonlinear differential equations with a sublinear neutral term of the form ■where z(t) = x(t)+∫_a~b p(t, ξ)x~γ(τ(t, ξ)) dξ, 0 < γ ≤ 1. Under the conditions∫_t0~∞ r-1/α(t)dt = ∞ or∫_t0~∞ r-1/α(t)dt <∞. The results obtained here extend, improve and complement to some known results in the literature. Examples are provided to illustrate the theorems.     

An Oscillation Criterion for Nonlinear Third-Order Differential Equations

Sufficient conditions for oscillation of a certain class of nonlinear third-order differential equations are found.     

Approximate methods based on order reduction for the periodic solutions of nonlinear third-order ordinary differential equations

Four approximate methods based on order reduction, the introduction of a book-keeping parameter and power series expansions for the solution and the frequency of oscillation are used to analyze three autonomous, nonlinear, third-order ordinary differential equations which have analytical periodic solutions. The first method introduces the velocity in both sides of the equation if this (linear) term is not present. The second one is based on the first one but employs a new independent variable, whereas, in the third and fourth techniques, a term equal to the velocity times the square of the unknown frequency of oscillation is introduced in both sides of the equation. The third method uses the original independent variable, whereas the fourth one employs a new independent variable which depends linearly on the unknown frequency of oscillation. It is shown that the second method provides accurate solutions only for initial velocities close to unity, whereas the third one is found to yield very accurate results for the first and second equations considered here and only for large initial velocities for the third one. The fourth technique provides as accurate results as or more accurate results than parameter-perturbation techniques which deal with the third-order equations directly and are based on the expansion of certain constants that appear in the differential equations in terms of a book-keeping parameter.     

Oscillation and asymptotic behavior of third-order nonlinear neutral delay differential equations with distributed deviating arguments

This paper concerns the oscillation and asymptotic behavior of a class of third-order nonlinear neutral delay differential equations with distributed deviating arguments. By employing a generalized Riccati transformation and integral averaging technique, we establish some sufficient conditions to ensure that all solutions of the considered equations are either oscillatory or converge to zero, which extend and improve some known results in the literature.     

时标上一类三阶中立型动力方程的振动准则

利用广义Riccati代换给出了时标上一类三阶中立型动力方程的振动准则,其结果不仅得到了几个振动解存在的条件,也拓展了一些已有的结论,在最后还给出了几个例子来验证结论.     

Oscillatory criteria for third-order nonlinear difference equation with impulses

In this paper, we study the oscillation of a third-order nonlinear difference equation with impulses. Some sufficient conditions for the oscillatory behavior of solutions of third-order impulsive nonlinear difference equation are obtained. Some known results in the literature are generalized and improved.     

A Study on Oscillatory and Asymptotic Nature of Impulsive Neutral Differential Equations of Order Three

In this paper, we consider a class of third-order neutral impulsive differential equations. An equivalent class of neutral differential equations is obtained by using a suitable substitution. Some new oscillation results are proved. Moreover, we discuss the asymptotic behavior of the solution. The results presented here are illustrated via examples.     

带强迫项与阻尼项的二阶非线性微分方程的区间振动性

研究了一类二阶非线性微分方程的振动性.利用广义黎卡提变换和积分平均技巧建立了方程(1)的新的振动准则,推广了文献中已有的一些结果.     

On nonexistence of Kneser solutions of third-order neutral delay differential equations

The aim of this paper is to complement existing oscillation results for third-order neutral delay differential equations by establishing sufficient conditions for nonexistence of so-called Kneser solutions. Combining newly obtained results with existing ones, we attain oscillation of all solutions of the studied equations.     

一类二阶非线性泛函微分方程的振动性定理

研究了一类较为广泛的二阶非线性泛函微分方程解的振动性质.在一定条件下,利用广义黎卡提变换、积分平均技巧和分类讨论的方法建立了两个新的振动准则,改进和推广了已知的一些结果.     

Oscillation of second-order nonlinear differential equations with nonlinear damping

This paper is concerned with the oscillation of a class of general type second-order differential equations with nonlinear damping terms. Several new oscillation criteria are established for such a class of differential equations under quite general assumptions. Examples are also given to illustrate the results.     

非线性偶阶中立型时滞偏微分方程的振动性

建立了一类非线性偶阶中立型时滞偏微分方程的若干新的振动准则,所得的结果推广和改进了文献中的一些振动结果。     

一类二阶非线性泛函微分方程的振动性质

研究了一类较为广泛的二阶非线性泛函微分方程解的振动性质.在一定条件下,利用广义黎卡提变换和积分平均技巧建立了方程(1)的两个新的振动准则,推广和改进了已知的一些结果.     

二阶非线性微分方程的振动准则

本文给出关于二阶非线性常微分方程和时滞微分方程的一些新的振动准则．     

非线性二阶阻尼微分方程的振动定理

考虑带有非线性阻尼项的一类二阶微分方程的振动性质.在一般的假设下我们建立了这类方程的若干新的振动准则.所得结果推广和改进了文献中的某些结果.     

Hille and Nehari type criteria for third-order dynamic equations

In this paper, we extend the oscillation criteria that have been established by Hille [E. Hille, Non-oscillation theorems, Trans. Amer. Math. Soc. 64 (1948) 234-252] and Nehari [Z. Nehari, Oscillation criteria for second-order linear differential equations, Trans. Amer. Math. Soc. 85 (1957) 428-445] for second-order differential equations to third-order dynamic equations on an arbitrary time scale T, which is unbounded above. Our results are essentially new even for third-order differential and difference equations, i.e., when T=R and T=N. We consider several examples to illustrate our results.     

非线性脉冲中立型时滞抛物方程解的振动性质

研究一类非线性脉冲中立型时滞抛物方程,借助于一阶脉冲中立型微分不等式,获得了该类方程在Robin,Dirichlet边值条件下所有解振动的若干新的充分性判据.所得结果充分反映了脉冲和时滞在振动中的影响作用.     

三阶非线性中立型变时滞动力方程的渐近性与振动性(英文)

考虑了时标上三阶非线性中立型变时滞动力方程的渐近性和振动性.利用广义Ric-cati技巧与完全平方技巧,获得了方程所有解渐近和振动的准则.所得结果推广了已有三阶动力方程的结果,给出了一些例子加以说明本文的主要结论.     

On oscillation of nonlinear second-order differential equations with damping term

The paper is concerned with oscillation of a novel class of nonlinear differential equations with a damping term. First it is demonstrated how known oscillation results for another intensively studied class of equations can be translated to the one in question, and vice versa. Advantages and drawbacks of such translation are carefully examined. Then an oscillation criterion for the new class of equations is established. The principal result of the paper is compared to those reported in the literature, and an illustrative example to which known oscillation criteria fail to apply is provided.