关于偶数阶中立型微分方程的线性化振动的一个注记

就偶数阶中立型时滞微分方程的所有有界解的振动性而言,研究了偶数阶非 线性中立型时滞微分方程［ｘ（ｔ）－ｐ（ｔ）ｘ（ｔ－τ）］－ｑ（ｔ）ｈ（ｘ（ｔ－σ））＝０和与它相应的 线性方程［ｘ（ｔ）－ｐ_０ｘ（ｔ－τ）］－ｑ-０ｘ（ｔ－τ）＝０在振动性上的等价性．所得结果改进 了 Ｌａｄａｓ Ｇ,和 Ｑｉａｎ Ｃ．的一个结果．     

奇数阶中立型时滞微分方程的线性化振动性

考虑奇数阶中立型非线性微分方程ｄ＾ｎ／ｄｔ＾ｎ（ｘ（ｔ）－Ｐ（ｔ）ｇ（ｘ（ｔ－τ）））＋Ｑ（ｔ）ｈ（ｘ（ｔ－σ））＝０。本文在允许Ｐ（ｔ）－１振动的条件下给出了该方程的几个线性化振动结果。     

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。     

临界状态下具渐近周期系数的一阶时滞微分方程的振动性

首先证明了在临界情形ｌｉｍｉｎｆ「ｐ（ｔ）－ｒ（ｔ）」＝０且∫ｔ－ｒｒ（ｓ）ｄｓ＝１／ｅ下一阶时滞微分方程ｘ’（ｔ）＋ｐ（ｔ）ｘ（ｔ－τ）＝０（＊）所有解振动等价于Ｒｉｃｃａｔｉ不等式ｗ（ｔ）＋ｒ（ｔ）ｗ＾２（ｔ）＋２ｅ＾２（ｐ（ｔ）＝ｒ（ｔ））≤０无最终正解,然后据此给出了方程（＊）在临界状态下两个振动及非振动准则。     

二阶非线性摄动微分方程的振动性定理

讨论了二阶非线性摄动微分方程（ａ（ｔ）ｘ’（ｔ）’＋ｐ（ｔ）ｘ’（ｔ）＋Ｑ（ｔ,ｘ（ｔ））＝Ｒ（ｔ,ｘ（ｔ）,ｘ’（ｔ））（１）的解的振动性质。应用分析方法,建立了方程（１）的三个新的振动性定理。     

一类中立型微分方程的比较定理及应用

考虑中立型微分方程［ｘ（ｔ）－ｘ（ｔ－τ）］（ｎ）＋Ｑ（ｔ）ｘ（ｔ－σ）＝０，ｔｔ０，其中ｎ１为奇数，τ∈（０，∞），σ∈Ｒ＋＝［０，∞），Ｑ∈Ｃ（［ｔ０，∞），Ｒ＋）．本文获得了此方程存在最终正解以及所有解振动的新的比较定理．并据此获得了所有解振动的“ｓｈａｒｐ″条件以及存在最终正解的一般性结果，改进了文［４］的主要结果．     

脉冲中立型时滞微分方程解的振动性

本文讨论一阶脉冲中立型时滞微分方程［ｙ（ｔ）＋Ｐｙ（ｔ－σ）］′＋Ｑ（ｔ）ｙ（ｔ－σ）＝０，ｔ０，ｔ≠ｔｋ，ｋ＝１，２，…，ｙ（ｔ＋ｋ）－ｙ（ｔ－ｋ）＝ｂｋｙ（ｔｋ），ｋ＝１，２，…，｛（Ｅ）这里τ，σ，Ｐ均为常数，τ＞０，σ＞０，Ｑ（ｔ）∈Ｃ（［０，∞），Ｒ＋），ｂｋ＞－１，ｋ＝１，２，…．分三种情况，Ｐ－１；－１＜Ｐ＜０；Ｐ＞０给出了方程（Ｅ）所有解振动的充分条件．     

变系数高阶中立型泛函数分方程的振动性与渐近性

考虑变系数高阶中立型泛函微分方程ｄ＾ｎ／ｄｔ＾ｎ（ｘ（ｔ）＋ｐ（ｔ）ｘ（ｔ－τ（ｔ））＋ｍ∑ｉ＝１Ｐｉ（ｔ）ｘ（ｔ－τｉ（ｔ）＝０，在－１＜ｐ（ｔ）≤０情形下解的振动性与渐近性，取消了Ｐｉ（ｔ）≥ｑｉ＞０的限制，改进以往的相应结果，本文结果时高阶泛函方程Ｘ＾（ｎ）（ｔ）＋ｍ∑ｉ＝１ｐｉ（ｔ）ｘ（ｔ－τｉ（ｔ））＝０也是适用的。     

强迫二阶非线性泛函微分方程解的振动性与渐近性

本文讨论强迫二阶非线性泛函微分方程（ａ（ｔ）ｘ＇（ｔ））＇＋ｐ（ｔ）ｆ（ｘ（ｔ），ｘ（ｑ（ｔ）））＝ｒ（ｔ）解的振动性与渐近性。所得结论改进和推广了已知的一些结果。     

具偏差变元微分方程解的振动性

该文建立了具偏差变元微分方程 ｘ’（ｔ）＋ ｐ（ｔ）ｘ（τ（ｔ））＝ ０的新的振动准则,所得定理改进了文献［１－６］中相应的结果．     

On some method for solving a nonlinear heat equation

Some exact solutions to a nonlinear heat equation are constructed. An initial-boundary value problem is examined for a nonlinear heat equation. To construct solutions, the problem for a partial differential equation of the second order is reduced to a similar problem for a first order partial differential equation.     

一类二阶变系数线性微分方程的解法

介绍如何通过变换把二阶变系数线性微分方程转化为一阶非线性微分方程,进而利用待定系数法对其求解,并对二阶变系数线性微分方程与一阶常系数非线性微分方程的内在的关系进行讨论.     

具非线性边界条件的Volterra型泛函微分方程边值问题奇摄动

首先利用微分不等式理论和一些分析技巧,探讨了一类具非线性边界条件的二阶Volterra型泛函微分方程边值问题解的存在性问题.然后通过对右端边界层函数和外部解的构造,进一步研究了一类具小参数的二阶Votterra型非线性边值问题.利用微分中值定理和上、下解方法得到了边值问题解的存在性,并给出了解的关于小参数的一致有效渐近展开式.     

Existence theorem of periodic solutions for a delay nonlinear differential equation with piecewise constant arguments

Existence criteria are proved for the periodic solutions of a first order nonlinear differential equation with piecewise constant arguments.     

An instability theorem for a certain sixth order nonlinear delay differential equation

The aim of this paper is to establish an instability theorem for a certain sixth order nonlinear delay differential equation. The proof of the theorem is based on the use of Lyapunov–Krasovskii functional approach. By this work, we improve an instability result obtained in the literature for a certain sixth order nonlinear differential equation without delay to the instability of the zero solution of a certain sixth order nonlinear delay differential equation.     

Unicity of meromorphic solutions of partial differential equations

In this survey, results on the existence, growth, uniqueness, and value distribution of meromorphic (or entire) solutions of linear partial differential equations of the second order with polynomial coefficients that are similar or different from that of meromorphic solutions of linear ordinary differential equations have been obtained. We have characterized those entire solutions of a special partial differential equation that relate to Jacobian polynomials. We prove a uniqueness theorem of meromorphic functions of several complex variables sharing three values taking into account multiplicity such that one of the meromorphic functions satisfies a nonlinear partial differential equations of the first order with meromorphic coefficients, which extends the Brosch??s uniqueness theorem related to meromorphic solutions of nonlinear ordinary differential equations of the first order.     

非正则奇异性偏微分方程组的形式解

本文研究了在复空间Ct×Cx上一类具有非正则奇异性的偏微分方程组的问题.利用形式Borel变换的方法,在一定的条件下,获得了在复空间Ct×Cx的原点附近的某个邻域内存在唯一的形式解,且该形式解属于某Gevery类,并给出了该Gevery类的指标,推广了非正则奇异性偏微分方程的研究结果.     

OSCILLATION OF FIRST ORDER NONLINEAR IMPULSIVE DIFFERENTIAL EQUATIONS WITH MIXED ARGUMENT

This paper devotes to study the oscillatory behavior of solutions of a first order nonlinear impulsive differential equation with mixed argument. First, without assuming the deviating argument to be retarded or advanced, a sufficient condition is established for all solutions of the differential equation to be oscillatory. Next, a sufficient condition for the differential equation to have nonoscillatoty solution is given. Finally, a sufficient and necessary condition for all solutions of the differential equation to be oscillatory is obtained.     

Asymptotic behavior and decay rate estimates for a class of semilinear evolution equations of mixed order

In this article we present a unified approach to study the asymptotic behavior and the decay rate to a steady state of bounded weak solutions of nonlinear, gradient-like evolution equations of mixed first and second order. The proof of convergence is based on the Lojasiewicz-Simon inequality, the construction of an appropriate Lyapunov functional, and some differential inequalities. Applications are given to nonautonomous semilinear wave and heat equations with dissipative, dynamical boundary conditions, a nonlinear hyperbolic-parabolic partial differential equation, a damped wave equation and some coupled system.     

OSCILLATORY AND ASYMPTOTIC BEHAVIOR OF A CLASS OF FIRST ORDER NEUTRAL DIFFERENTIAL EQUATION WITH PIECEWISE CONSTANT DELAY

The oscillatory and asymptotic behavior of a class of first order nonlinear neutral differential equation with piecewise constant delay and with diverse deviating arguments are considered. We prove that all solutions of the equation are nonoscillatory and give sufficient criteria for asymptotic behavior of nonoscillatory solutions of equation.