Oscillations for even-order neutral difference equations

Consider the even-order neutral difference equation

Lyapunov-type inequality for a class of even-order differential equations

In this paper, we generalize the well-known Lyapunov-type inequality for second-order linear differential equations to a class of even-order linear differential equations, the results of this paper are new and generalize some known inequalities in the literatures.     

二阶中立型微分方程的振动准则

研究了一类具有连续偏差变元的二阶非线性中立型方程,得到了该类方程的振动准则     

Oscillatory of unstable type second-order neutral difference equations

We consider the problem of oscillation and nonoscillation solutions for unstable type second-order neutral difference equation: Δ²(x(n) − p(n)x(n −τ)) =q(n)x(g(n)). (1) In this paper, we obtain some conditions for the bounded solutions of Eq(1) to be oscillatory and for the existence of the nonoscillatory solutions.     

一类二阶非线性中立型方程的振动准则

获得了一类具连续分布滞量的非线性二阶中立型泛函微分方程所有解振动的若干充分条件.     

Linearized oscillation theorems for neutral difference equations

Consider the nonlinear neutral difference equation {fx149-1} We establish a linearized oscillation theorem which is a discrete result of the open problem by Gyori and Ladas. This work is partially supported by NNSF (NO: 19671027) of China.     

一阶中立型时滞微分方程的振动性

分别获得了中立型时滞微分方程「ｘ（ｔ）＋ｐｘ（ｔ－γ」’＋ｑ（ｔ）ｘ（ｔ－σ）＝０的振动性和非振动性的充分条件,其中ｐ〉０,ｑ（ｔ）是一个正的周期函数。     

Oscillatory behavior of the second-order nonlinear neutral difference equations

In this paper, we consider the oscillation of the second-order neutral difference equation $$\Delta ^2 \left( {x_n - px_{n - \tau } } \right) + q_n f\left( {x_{n - \sigma _n } } \right) = 0$$ as well as the oscillatory behavior of the corresponding ordinary difference equation $$\Delta ^2 z_n + q_n f\left( {R\left( {n,\lambda } \right)z_n } \right) = 0$$ .     

Existence theorems for a class of nonlocal differential equations

A class of nonlocal second-order ordinary differential equations of the form

y^″(x)=f(x,y(x),(y○λ)(x),y^′(x))     

Oscillation theorems for second-order superlinear neutral differential equations

The purpose of this paper is to investigate the oscillation of the second-order neutral differential equations of the form (E) $$ (r(t)|z'(t)|^{\alpha - 1} z'(t))' + q(t)|x(\sigma (t))|^{\alpha - 1} x(\sigma (t)) = 0, $$ where z(t) = x(t) + p(t)x(τ(t)). The obtained comparison principles essentially simplify the examination of the studied equations. Further, our results extend and improve the results in the literature.     

Some oscillation theorems for a class of quasilinear elliptic equations

Oscillation criteria are obtained for quasilinear elliptic equations of the form (E)below. We are mainly interested in the case where the coefficient function oscillates near infinity. Generalized Riccati inequalities are employed to establish our results.