高阶齐次线性微分方程解的零点

研究了一类高阶齐次线性微分方程解的零点收敛指数,并得到当方程的系数A_0为整函数,其泰勒展式为缺项级数,并且A_0起控制作用时,方程f~((k))+A_(k-2)f~((k-2))+…+A_1f′+A_0f=0的任意两个线性无关解f_1,f_2满足max{λ(f_1),λ(f_2)}=∞,其中λ(f)表示亚纯函数.f的零点收敛指数.     

二阶微分方程解的零点分布

应用角域Nevanlinna理论和Ahlfors覆盖曲面理论, 研究了二阶微分方程f’’+A(z)f=0的解的零点分布. 证明了在复平面上至少存在一条半直线, 使得二阶微分方程解在该直线上的零点的径向收敛指数为无穷. 用新的方法证明了伍胜健在文献[5]中的一个定理.     

二阶微分方程解关于超级的零点分布

设f1和f2是二阶微分方程f″+A(z)f=0.的两个线性无关的解,其中A(z)是无穷级整函数且超级δ2(A)<∞.令E=f1,f2.文章研究了E的超级为无穷的Borel方向和零点聚值线之间的关系.所得结果推广并改进了文献[7]中的一个结果.     

On the Frequency of Zeros of Solutions of Second Order Linear Differential Equations

We consider the equation \(\rm f^{\prime\prime}+{A}(z){f}=0\) with linearly independent solutions f₁,₂, where A(z) is a transcendental entire function of finite order. Conditions are given on A(z) which ensure that max{λ(f₁),λ(f₂)} = ∞, where λ(g) denotes the exponent of convergence of the zeros of g. We show as a special case of a further result that if P(z) is a non-constant, real, even polynomial with positive leading coefficient then every non-trivial solution of \(\rm f^{\prime\prime}+{e}^P{f}=0\) satisfies λ(f) = ∞. Finally we consider the particular equation \(\rm f^{\prime\prime}+({e}^Z-K){f}=0\) where K is a constant, which is of interest in that, depending on K, either every solution has λ(f) = ∞ or there exist two independent solutions f₁, f₂ each with λ(f_i) ≤ 1.     

Bounds for Zeros of Solutions of Second Order Differential Equations with Polynomial Coefficients

We consider the equation ${u''=P(z)u\;\;(z\in\mathbb{C})}$ where P(z) is a polynomial. Let z _k (u), k = 1, 2,... be the zeros of a solution u(z) to that equation. Bounds for the sums $$\sum_{k=1}^{j} \frac {1} {|z_k(u)|}\;(j=1, 2, \ldots)$$ are established. Some applications of these bounds are also considered.     

高阶亚纯系数非齐次线性微分方程解的零点

讨论一类高阶亚纯系数非齐次线性微分方程解的零点问题,当方程的系数A0是亚纯函数且满足δ(∞,A0)=δ(0)和lim(r→∞)log T(r,Ao)/log r=∞时,如果f1和f2是方程f((k))+A(κ—1)f((k—1))+…+Aof=F的两个线性无关解,得到max{λ(f1),λ(f2)}=∞.还考虑了σ(F)=∞或Ad(1dκ—1)满足lim(r→∞)log m(r,Ad)/log r=∞的情况.     

一阶线性中立型微分方程解的稳定性

本文讨论方程其中c、p_i∈c([t_0,∞),R),0≤c(t)≤1.τ≥0,i=1,2,…,n.通过对方程(1)的非振动解及振动解的渐近性研究,我们得到了方程(1)的平凡解渐近稳定的新的充分条件.     

一阶迭代泛函微分方程的解析解

利用Schrder变换,给出了一类新的时滞变量依赖于状态变量的迭代泛函微分方程的解析解.不但讨论了Schrder变换中常数α在单位圆内和单位根的情况,也给出了α在单位圆上接近于单位根时的结果.     

二阶亚纯系数微分方程解的零点分布

应用角域Nevanlinna理论,研究了二阶亚纯系数微分方程f′′+A(z)f=0的解的零点聚值线和Borel方向之间的关系.推广了文献[5]中的一个定理.     

一阶微分方程的多重周期正解

本文是关于一阶时滞微分方程多重周期正解的存在性问题的研究,利用Leggett—Williams不动点定理和Guo—Krasnosel’skii锥拉伸锥压缩不动点定理,得到了一些新的结论。     

一阶微分方程周期边值问题的解的存在性

该文用拓扑度理论证明了一阶微分方程周期边值问题关于下解与上解之间解的存在性的两个定理，这里下解α(t)与上解β(t)可以不满足关系α(0)≤α(2π)，β(0)≥β(2π)     

Positive Periodic Solutions of Systems of First Order Ordinary Differential Equations

Consider the n-dimensional nonautonomous system ?(t) = A(t)G(x(t)) ? B(t)F(x(t ? τ(t))) Let u = (u ₁,…,u _n ), $f^{i}_{0}={\rm lim}_{\|{\rm u}\|\rightarrow 0}{f^{i}(\rm u)\over \|u\|}$ , $f^{i}_{\infty}={\rm lim}_{\|{\rm u}\|\rightarrow \infty}{f^{i}(\rm u)\over \|u\|}$ , i = l,…,n, F = (f ¹…,f ⁿ ), ${\rm F_{0}}={\rm max}_{i=1,\ldots,n}{f^{i}_{0}}$ and ${\rm F_{\infty}}={\rm max}_{i=1,\ldots,n}{f^{i}_{\infty}}$ . Under some quite general conditions, we prove that either F₀ = 0 and F_∞ = ∞, or F₀ = ∞ and F_∞ = 0, guarantee the existence of positive periodic solutions for the system for all λ > 0. Furthermore, we show that F₀ = F_∞ = 0, or F_∞ = F_∞ = ∞ guarantee the multiplicity of positive periodic solutions for the system for sufficiently large, or small λ, respectively. We also establish the nonexistence of the system when either F₀ and F_∞ > 0, or F₀ and F_∞, < for sufficiently large, or small λ, respectively. We shall use fixed point theorems in a cone.     

The Separation of Zeros of Solutions of Higher Order Linear Differential Equations With Entire Coefficients

Suppose that a homogeneous linear differential equation has entire coefficients of finite order and a fundamental set of solutions each having zeros with finite exponent of convergence. Upper bounds are given for the number of zeros of these solutions in small discs in a neighbourhood of infinity.     

Asymptotic and Oscillatory Behavior of Solutions of First Order Neutral Differential Equations

This paper has made researches on first order neutral differential equations with variable coefficients and several deviations. The asymptotic behavior of nonoscillatory solutions of the equations are discussed. Necessary and sufficient conditions and several sufficient conditions for the oscillations of the equations are obtained. The relevent results in [1—3] are improved and generalized .     

依赖于一阶导数的二阶脉冲微分方程边值问题的正解(英)

本文研究一类二阶脉冲微分方程：■的正解存在性．其中,0＜η＜1,0＜α＜1,f：[0,1]×[0,∞)×R→[0,∞),I_i：[0,∞)×R→R,J_i：[0,∞)×R→R,(i=1,2,…,k)均为连续函数．本文所用方法是文献[5]推广的Krasnoselskii不动点定理,此定理为解决依赖于一阶导数的边值问题提供了理论依据．基于此定理,获得了问题正解存在性定理．特别地,我们获得此类问题的Green函数,使问题的解决更直观和简单．     

The Distribution of Zeroes of Solutions of First Order Neutral Differential Equations

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Infinitesimal Classification of Systems of Two First Order Partial Differential Equations in Two Variables

Systems of two first order PDE with respect to two functions of two variables are considered. A classification of symbols of such systems with respect to the natural action of the group of local changes of dependent and independent variables is given. Bibliography: 6 titles.     

Some Exact Solutions of Two Fifth Order KdV-type Nonlinear Partial Differential Equations

We consider the generalized integrable fifth order nonlinear Korteweg-de Vries (fKdV) equation. The extended Tanh method has been used rigorously, by computational program MAPLE, for solving this fifth order nonlinear partial differential equation. The general solutions of the fKdV equation are formed considering an ansatz of the solution in terms of tanh. Then, in particular, some exact solutions are found for the two fifth order KdV-type equations given by the Caudrey-Dodd-Gibbon equation and the another fifth order equation. The obtained solutions include solitary wave solution for both the two equations.     

一阶非线性中立型方程组的非振动解的存在性

利用Krasnoselskii不动点定理,得到了一阶非线性中立型方程组[x_i(t) sum from j=1 to n c_(ij)x_j(t-σ)]′ f_i(t,x_l(g_(il)(t)),…,x_n(g_(in)(t)))=0,i=1,2,…,n存在趋于具均为正(负)分量的常向量的非振动解的充分必要条件.     

二阶齐次与非齐次线性微分方程亚纯解的零点与增长性质估计(英文)

本文研究了方程f′′+A(z)f′+B(z)f=0与f′′+A(z)f′+B(z)f=F亚纯解的零点与增长性,其中A(z),B(z)(■0),F(z)(■0)为亚纯函数,得到了方程亚纯解的增长级、下级、超级、二级不同零点收敛指数等的精确估计,改进了KwonKi-Ho、陈宗煊与杨重骏、Benharrat Beladi等的结果.