一个二阶方程的振动性质及其应用

本文研究了一个二阶微分方程的解及其振动性质,利用它获得了一般二阶自共轭方程的振动性判别准则,特别对其中一类二阶微分方程的振动性给出了定量的判别方法.推广了E.Hille的工作,使E.Hille的研究结论成为本文结果的极特殊的情形,同时对一类具有“积分小”系数或可化为具有“积分小”系数的二阶微分方程的振动性与非振动性给出了简便、精确的判别方法.     

可化为一个“积分小”系数的二阶非线性微分方程解的渐近性和振动性

本文讨论可化为一个“积分小”系数的二阶非线性微分方程的渐近性和振动性，得到若干充分性定理，推广和改进了文［１－４］的有关结果。     

具有正负系数的非线性中立型方程的振动定理

本文研究了一类具有正负系数的二阶非线性中立型时滞微分方程的振动问题.利用变限积分方法,获得了此类方程若干新的振动定理,结果推广和改进了最近Manojlovic等人在文献[4]中得到的结果.     

具有变号系数中立型时滞微分方程的振动性

本文讨论一类一阶具有变号系数中立型时滞微分方程的振动性，建立了此类方程一切解振动的几族充分条件。将其结论用于具有正系数中立型时滞微分方程及具有正负系数中立型时滞微分方程，改进了[1-8]的相应定理。     

一类非线性中立型微分方程解的振动性

本文研究一类具有分布时滞的三阶非线性泛函微分方程解的振动行为,利用推广的黎卡提变换,通过积分平均方法,获得了泛函微分方程一些新的振动性判据,推广和改进了最近文献中的一些结果.     

一阶变系数滞后型微分方程解的振动性积分条件与渐近性

本文给出了一阶变系数滞后型微分方程振动解存在的积分条件,同时利用Banach不动点定理讨论了该类方程解的渐近性。     

非线性中立双曲型时滞偏微分方程解的振动性质

讨论一类多滞量非线性中立双曲型时滞偏微分方程解的振动性质，应用积分不等式和泛函微分方程的某些结果，获得了其一切解振动的一系列充分条件。结论充分表明了时滞量的决定性作用，指出了其与普通双曲型偏微分方程质的差异。     

二阶非线性时滞微分方程的振动性

研究了一类二阶非线性时滞微分方程的振动性.利用广义Riccati变换,积分平均技巧和不等式,建立此微分方程若干新的振动准则,所得结果推广且改进了最近一些文献中的某些振动结果.     

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．     

非线性中立抛物型偏微分方程系统的振动性定理

研究一类非线性中立抛物型时滞偏微分方程系统解的振动性质,利用积分不等式和泛函微分方程的某些结果,获得了该类系统在第一类边值条件下所有解振动的若干充分条件.结论充分表明振动是由时滞量引起的.     

Existence and uniqueness of solutions of system governed by second order quasilinear integro-partial differential equation of parabolic type

Recently, Teo and Ahmed [17] have established the existence and uniqueness of solutions for a class of systems governed by second order quasilinear parabolic integro-partial differential equations. In their system equation, all but the second order coefficients are assumed to be bounded and measurable while more restrictive assumptions are imposed on the second order coefficients. In this paper, their results are generalized so that the second order coefficients can also be assumed to be bounded and measurable. However, the parabolic integro-partial differential equation is in divergence form and the solution of the system under consideration is in the sense of Aronson [1]. Our main result is presented in Theorem 3.6 which is proved using several fundamental results reported by Aronson [1].     

Nonoscillation theorems for certain second order nonlinear differential equations

Some necessary and sufficient conditions for nonoscillation are established for the second order nonlinear differential equation where p > 0 is a constant. These results are extensions of the earlier results of Hille, Wintner, Opial, Yan for second order linear differential equations and include the recent results of Li and Yeh, Kusano and Yoshida, Yang and Lo for half-linear differential equations. Authors’ address: School of Mathematical Sciences, South China Normal University, Guangzhou 510631, P.R. China     

Oscillation and nonoscillation of solutions of second order linear dynamic equations with integrable coefficients on time scales

We obtain Willett-Wong-type oscillation and nonoscillation theorems for second order linear dynamic equations with integrable coefficients on a time scale. The results obtained extend and are motivated by oscillation and nonoscillation results due to Willett [20] and Wong [21] for the second order linear differential equation. As applications of the new results obtained, we give the complete classification of oscillation and nonoscillation for the difference equations

     

Gaussian-type upper bound for the evolution kernels on nilpotent meta-abelian groups

We give a Gaussian-type upper bound for the transition kernels of the time-inhomogeneous diffusion processes on a nilpotent meta-abelian Lie group N generated by the family of time dependent second order left-invariant differential operators. These evolution kernels are related to the heat kernel for the left-invariant second order differential operators on higher rank NA groups.     

New Kamenev-type oscillation criteria for half-linear partial differential equations

We establish new Kamenev-type oscillation criteria for the half-linear partial differential equation with damping under quite general conditions. These results are extensions of the recent results developed by Sun [Y.G. Sun, New Kamenev-type oscillation criteria of second order nonlinear differential equations with damping, J. Math. Anal. Appl. 291 (2004) 341-351] for second order ordinary differential equations in a natural way, and improve some existing results in the literature. As applications, we illustrate our main results using two different types of half-linear partial differential equations.     

一类二阶微分方程的渐近行为

In the paper the asympotic behavior of the solution of the following second order differential equationy" ＋ q(t)f(y) ＝ 0is investigated. Some results for asymptotic behavior of the nonoscillation solution are obtained.     

Philos-type oscillation theorems for second order damped elliptic equations

Using general means, we establish several new Philos-type oscillation theorems for the second order damped elliptic differential equation
Σi,j=1 N Di[aij（x）Djy]＋Σi=1 N bi（x）Diy＋c（x）f（y）=0
under quite general assumptions. The obtained results are extensions of the well-known oscillation results due to Kamenev, Philos, Yan for second order linear ordinary differential equations and improve recent results of Xu, Jia and Ma.     

Oscillations of certain second order nonlinear differential equations

A second order nonlinear differential equation

Estimates for The Hyper-Order of Solutions of Certain Linear Differential Equations

In this article, the authors study the growth of certain second order linear differential equation f″＋A（z）f′＋B（z）f=0 and give precise estimates for the hyperorder of solutions of infinite order. Under similar conditions, higher order differential equations will be considered.     

Periodic solutions of second order nonlinear difference equations with discrete -Laplacian

We use Brouwer degree to prove existence and multiplicity results for the periodic solutions of some nonlinear second order difference equations involving discrete -Laplacian. We obtain in particular upper and lower solutions theorems, Ambrosetti–Prodi type multiplicity results, sharp existence conditions for nonlinearities which are bounded from below or from above and necessary and sufficient conditions for the existence of positive periodic solutions when the nonlinearity is singular at 0.