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.     

时标上的三阶非线性中立型变时滞动力方程的振动性与渐进性

考虑了时标上一类三阶非线性中立型变时滞动力方程的振动性和渐进性.利用Riccati变换技巧得到了方程所有解振动或渐进趋于零的充分条件.特别地,所得新结果发展了一些已有结论.并给出了例子加以说明本文的主要结果.     

Oscillatory and asymptotic criteria of third order nonlinear delay dynamic equations with damping term on time scales

In this paper, we are concerned with oscillatory and asymptotic behavior of third order nonlinear delay dynamic equations with damping term on time scales. By using a generalized Riccati function and inequality technique, we establish some new oscillatory and asymptotic criteria. The established results on one hand extend some known results in the literature, on the other hand unify continuous and discrete analysis as two special cases of an arbitrary time scale. We also present some applications for the established results.     

Basic Calculus on Time Scales and some of its Applications

The study of dynamic systems on time scales not only unifies continuous and discrete processes, but also helps in revealing diversities in the corresponding results. In this paper we shall develop basic tools of calculus on time scales such as versions of Taylor’s formula, l’Hôspital’s rule, and Kneser’s theorem. Applications of these results in the study of asymptotic and oscillatory behavior of solutions of higher order equations on time scales are addressed. As a further application of Taylor’s formula, Abel-Gontscharoff interpolating polynomial on time scales is constructed and best possible error bounds are offered. We have also included notes at the end of each section which indicate further scope of the calculus developed in this paper.     

带有最大值项的高阶中立型差分方程的振动性

对一类带有最大值项的高阶中立型差分方程的振动性进行了研究,得到一些方程振动的充分条件,推广和改进了已有的一些结果.     

Oscillation of fourth-order delay dynamic equations

This paper is concerned with oscillatory behavior of a class of fourth-order delay dynamic equations on a time scale.In the general time scales case,four oscillation theorems are presented that can be used in cases where known results fail to apply.The results obtained can be applied to an equation which is referred to as Swift-Hohenberg delay equation on a time scale.These criteria improve a number of related contributions to the subject.Some illustrative examples are provided.     

时间测度链上具正负系数的二阶阻尼动力方程的振动准则

研究了时间测度链上一类具正负系数和阻尼项及非线性中立项的二阶变时滞非线性动力方程的振荡性.利用时间测度链上的有关理论及广义Riccati变换,结合大量不等式技巧,建立了该方程若干新的振动准则,这些准则不仅推广和改进了一些已知的结果,而且在时间测度链上统一了二阶非线性时滞阻尼微分方程和二阶非线性时滞阻尼差分方程的振动性质.     

非线性中立型抛物微分方程解的振动性

This paper deals with the oscillatory properties of a class of nonlinear neutral parabolic partial differential equations with several delays. Sufficient criteria for the equation to be oscillatory are obtained by making use of some results of first-order functional differential inequalities. These results fully reveal the essential difference between this type and that without delays.     

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 properties of second-order nonlinear neutral dynamic equations with positive and negative coefficients

This paper is to investigate the oscillatory properties of second-order nonlinear neutral dynamic equation with positive and negative coefficients of the form on an arbitrary time scale $\mathbb{T}$ . We establish several oscillation criteria for the equation by using a generalized Riccati technique. The obtained results extend and improve certain existing results in the literature. Examples are shown to illustrate our main results.     

Warped MPDAE Models including Minimisation Criteria for the Simulation of RF Signals

In radio frequency (RF) applications, electric circuits produce signals including widely separated time scales. A multidimensional representation yields an efficient model by decoupling the time scales. Consequently, a warped multirate partial differential algebraic equation (MPDAE) describes the circuit's behaviour. The appropriate determination of an arising local frequency function is crucial for the efficiency of this approach. Variational calculus implies a necessary condition to a specific solution, which exhibits a minimal amount of oscillations in the whole domain of dependence. We apply a similar strategy to minimise oscillatory performance in some boundary values only. Now variational calculus yields a boundary condition, which can easily be used in numerical methods. We compare the results of both minimisation criteria in a simulation of a warped MPDAE model. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一阶中立型泛函微分方程解的渐近性与振动性

本文研究具有变系数和多偏差的一阶中立型微分方程，讨论了方程的非振动解的渐近性，得到了方程振动的充分判据，其中有些是变号系数的情形．     

Existence and exponential stability of almost‐periodic solutions for neutral BAM neural networks with time‐varying delays in leakage terms on time scales

This paper is concerned with neutral bidirectional associative memory neural networks with time‐varying delays in leakage terms on time scales. Some sufficient conditions on the existence, uniqueness, and global exponential stability of almost‐periodic solutions are established. An example is presented to illustrate the feasibility and effectiveness of the obtained results. Copyright © 2015 John Wiley & Sons, Ltd.     

Oscillation of Second Order Impulsive Differential Equations with Nonpositive Neutral Coefficients

In this work, sufficient conditions are established for a class of nonlinear second order neutral impulsive differential equations to have oscillatory solutions with nonpositive neutral coefficient. Our results extend and complement some of the known results in the literature. Examples are given to illustrate our results.     

Almost periodic solutions for neutral‐type neural networks with the delays in the leakage term on time scales

In this paper, a class of neutral‐type neural networks with delays in the leakage term on time scales are considered. By using the Banach fixed point theorem and the theory of calculus on time scales, some sufficient conditions are obtained for the existence and exponential stability of almost periodic solutions for this class of neural networks. The results of this paper are new and complementary to the previously known results. Finally, an example is presented to illustrate the effectiveness of our results. Copyright © 2016 John Wiley & Sons, Ltd.     

高阶具非线性中立项不稳定型时滞微分方程的振动性和非振动性

在α>1,且0<β<α情性下研究了高阶具非线性中立项不稳定型时滞微分方程[x(t)-pxα(t-τ)](n)=q(t)xβ(t-σ),(t≥t0)的振动和非振动性.利用一些新的技巧,获得了上述方程有界解振动的振动准则和至少存在一个非振动解的非振动准则,所得结果补充和推广了已有文献部分结果.     

Oscillations of higher order differential equations of neutral type

In this paper, sufficient conditions have been obtained for oscillation of solutions of a class of nth order linear neutral delay-differential equations. Some of these results have been used to study oscillatory behaviour of solutions of a class of boundary value problems for neutral hyperbolic partial differential equations.     

时间测度链上一类二阶动力方程的振动性

研究了时间测度链上的一类具有非线性中立项和变时滞的二阶非线性动力方程的振动性.通过引入参数函数和广义的Riccati变换,并借助时间测度链上的有关理论,得到了方程振动的几个充分条件.所得结果推广和改进了现有文献中相应的结果.     

Oscillation of nonlinear neutral delay differential Equations

In this paper, we study the oscillatory behavior of first order nonlinear neutral delay differential equations. Several new sufficient conditions which ensure that all solutions are oscillatory are given. The obtained results extend and improve several known results in the literature. Some examples are considered to illustrate the main results.     

Dynamic Picone's identity and its applications

We prove some Picone-type identities and inequalities for a class of first-order nonlinear dynamic systems and derive various weighted inequalities of Wirtinger type and Hardy type on time scales. As applications we study oscillatory and related properties of these systems including Reid's roundabout theorem on disconjugacy, Sturm's separation and comparison theorems, as well as a variational method in the oscillation theory.