一类二阶非线性变时滞微分方程的振动准则

研究二阶非线性变时滞微分方程x″(t)+p(t)f(x(g(t)))=0,对振动因子p(t)变符号的情况讨论了方程的振动性,通过两个已有引理得到了方程振动的两个充分条件.所得结论推广了原有的二阶非线性微分方程与变时滞微分方程当系数不变号时的振动性结论,完善了具变符号振动因子的二阶非线性变时滞微分方程的研究.     

Numerical approach for solving fractional relaxation–oscillation equation

In this study, we will obtain the approximate solutions of relaxation–oscillation equation by developing the Taylor matrix method. A relaxation oscillator is a kind of oscillator based on a behavior of physical system’s return to equilibrium after being disturbed. The relaxation–oscillation equation is the primary equation of relaxation and oscillation processes. The relaxation–oscillation equation is a fractional differential equation with initial conditions. For this propose, generalized Taylor matrix method is introduced. This method is based on first taking the truncated fractional Taylor expansions of the functions in the relaxation–oscillation equation and then substituting their matrix forms into the equation. Hence, the result matrix equation can be solved and the unknown fractional Taylor coefficients can be found approximately. The reliability and efficiency of the proposed approach are demonstrated in the numerical examples with aid of symbolic algebra program, Maple.     

Oscillation and other properties of linear impulsive and nonimpulsive delay equations

Many of the oscillation results for linear impulsive equations were justified by the following scheme. First, the equivalence of the oscillation of the impulsive equation and some specially constructed nonimpulsive equation was established; further, on the base of well-known results for the nonimpulsive case, the oscillation of the impulsive equation was analyzed. In the present paper, we prove the “oscillation equivalence” result for a linear impulsive equation with a distributed delay and discuss the possibility to expand this approach to the other properties of impulsive equations, for example, stability and asymptotic behavior. In addition, a linear impulsive equation of the second order is considered.     

Oscillation criteria of a class of fourth order differential equations

In the present paper, we consider the oscillation of a class of self‐adjoint fourth order differential equation. Some oscillation and non‐oscillation criteria on a perturbation equation of Euler differential equation are given. In particular, we find the exact value of a constant in some oscillation criteria, which has answered the open problem presented by Do?lý in 2004. Copyright © 2011 John Wiley & Sons, Ltd.     

Oscillation behavior of solution of impulsive fractional differential equation

In this paper, we study the oscillation of impulsive Caputo fractional differential equation. Sufficient conditions for the asymptotic and oscillation of the equation are obtained by using the inequality principle and Bihari Lemma. An example is given to illustrate the results. This is the first time to study the oscillation of impulsive fractional differential equation with Caputo derivative.     

一类具变号系数的二阶非线性变时滞微分方程的振动性

研究一类二阶非线性变时滞微分方程x″(t)+p(t)f(x(g(t)))=0的振动性,对振动因子p(t)可变符号的情况,通过两个引理,得出了方程振动的两个充分性定理.所得结论推广了二阶非线性变时滞微分方程当系数不变号时的振动性结论.     

具有可变时滞的非线性非自治差分方程的振动性及其应用

本文获得了一类具有可变时滞的非线性非自治差分方程的振动准则,建立了这类差分方程的几个线性化振动性定理,并得到了具有可变时滞的离散型非自治广义Logistic方程关于其正平衡点振动的一系列充分条件.     

高阶中立型时滞差分方程解的振动性

研究了一类具有变系数的高阶中立型时滞差分方程解的振动性，给出了其有界解振动的两个充分条件。     

具有正负系数的二阶脉冲中立型方程的振动性

考虑一类具有正负系数的多时滞二阶线性脉冲中立型微分方程,证明了该方程解的振动性等价于一类非脉冲中立型方程解的振动性,得到了方程所有解存在的充分条件.     

Oscillation and stability of nonlinear neutral impulsive delay differential equations

In this paper, oscillation and stability of nonlinear neutral impulsive delay differential equation are studied. The main result of this paper is that oscillation and stability of nonlinear impulsive neutral delay differential equation are equivalent to oscillation and stability of corresponding nonimpulsive neutral delay differential equations. At last, two examples are given to illustrate the importance of this study.     

一类二阶非线性差分方程的振动性定理

应用分析方法,研究了一类二阶非线性差分方程解的振动性质,获得了一个新的振动性定理,推广和改进了已知的一些结果.     

二阶中立型变时滞差分方程解的振动性

研究了二阶中立型变时滞差分方程Δ2(xn+pxn-l)+qnf(xσ(n))=0解的振动性,获得了该类方程全部非平凡解振动的三个定理.所得结果将二阶中立型差分方程已有的振动性的相应结论推广到了二阶中立型变时滞差分方程.     

里卡蒂方法研究带泛函参数的非线性脉冲时滞双曲方程的振动性

本文研究了带泛函参数的非线性脉冲时滞双曲方程的振动性问题.利用积分平均法和里卡蒂方法得到了这类方程解的振动性的一个充分条件,对非线性时滞双曲方程解的震动性进行了推广,能更好地利用一些现有的脉冲时滞常微分方程解的振动性的结论.     

具有正负周期系数的差分方程的振动性与线性化振动性

本文得到具有正负周期系数的差分方程振动的一个充分必要条件,并利用此充要条件获得了一个线性化振动性结果。     

某类系数变号的二阶非线性变时滞微分方程的振动性

研究了二阶非线性变时滞微分方程x″(t)+p(t)f(x(g(t)))=0的振动性,对振动因子p(t)变号的情况,给出了两个重要的引理,并得到方程振动的一个充分性定理.所得结论推广了二阶非线性变时滞微分方程当系数不变号时原有的振动性结论.     

具有阻尼项的二阶半线性偏微分方程解的振动性

研究了一类具有阻尼项的二阶半线性偏微分方程div(A(x)‖▽u(x)‖p-2▽u(x))+〈■(x),‖▽u(x)‖p-2▽u(x)〉+C(x)u(x)p-2u(x)=0,p>1运用偏Riccati变换和H函数方法,获得了该方程解的振动性的若干充分条件.     

具有连续变量的非线性差分方程的强迫振动

本文讨论了一类具有连续变量的非线性差分方程的振动问题 ,给出了方程存在最终正解和最终负解的充要条件及强迫项对解振动的影响 ,得到了几个充分条件 .     

含连续时滞和阻尼项的二阶半线性微分方程解的振动性

研究了一类含有连续分布时滞和阻尼项的二阶半线性微分方程运用Riccati变换和H函数方法,获得了该方程解的振动性的若干充分条件.     

具变号系数的四阶非线性微分方程的振动性

研究了四阶非线性微分方程x(4)(t)+p(t)f(x(t))=0的振动性,对振动因子p(t)变号的情况,给出了两个重要的引理,并得到方程振动的一个充分性定理.所得结论推广了四阶非线性微分方程当系数不变号时原有的振动性结论.     

Oscillation of partial population model with diffusion and delay

By using the upper- and lower-solution method of partial functional differential equations and the oscillation theory of functional differential equation, the oscillation of a population equation with diffusion and delay is studied and a sufficient condition for all positive solutions of the equation to oscillate about the positive equilibrium is obtained. Finally, a model arising from ecology is given to illustrate the obtained results.