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.     

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.     

Subnormal Solutions of Second Order Linear Differential Equations With Periodic Coefficients

We find the form of all subnormal solutions of equation (1.4). Our results generalize and improve a well-known result of Wittich about equation (1.1). Several examples are given. Higher order equations are discussed.     

亚纯系数的高阶线性微分方程的解的增长性

本文研究亚纯系数的高阶线性微分方程,当方程系数满足一定条件时,得到方程的每一非零亚纯解具有无穷级且超级为n.此外,还研究了非齐次线性微分方程的亚纯解.     

Some Oscillation Theorems for Higher Order Linear Differential Equations with Entire Coefficients of Small Growth

We show that if A and B are entire of order less than 1/6, and are not both polynomials, then the linear differential equation $y(3)+Ay^\prime+By=0$ can never have a fundamental set of solutions each having zeros with finite exponent of convergence. We go on to consider higher order equations where one coefficient is dominant in the sense that either it has larger order than any other coefficient, or it is the only transcendental coefficient.     

一类二阶整函数非齐次线性微分方程的复振荡

研究具有整函数函数系数的二阶非齐次线性微分方程:f″+A(z)e~(az)f′+B(z)e~(P(z))f=F(z)解的复振荡,其中P(z)为非常数多项式且deg(P)=n,A(z),B(x),F(z)均为整函数且max{ρ(A),ρ(B)}n.我们将看到方程的任一非零解具有无穷增长级.     

关于高阶整函数系数微分方程解的增长性的进一步结果

本文研究一类高阶整函数系数微分方程的增长性问题，当存在某个系数对方程的解的性质起主要支配作用时，得到了齐次与非齐次方程解的超级的精确估计及方程的解与小函数的关系。     

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.     

一类高阶线性微分方程解的增长性

本文研究了在Aj(z),aj(j=0,1,…,k-1)满足一些条件下方程f(k)+Ak-1(z)eak-1f(k-1)+…+A0(z)ea0zf=0解的超级和在Aj(z),Pj(j)(j=0,1,…,k-1)满足一些条件下方程f(k)+Ak-1(z)ePk-1(z)f(k-1)+…+Aj(z)eajzf(j)+…+A0(z)eP0(z)f=0解的级。     

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.     

一类高阶整函数系数微分方程解的增长级、下级与超级

该文研究了一类高阶整函数系数微分方程解的增长性,对方程f~(k)+A_(k-1)(z)e~(ak-1z).f~(k-1)+…+A_0(z)e~(a0z)f=0与方程f~(k)+(A_(k-1)(z)e~(ak-1z)+D_(k-1)(z))f~(k-1)+…+(A_0(z)e~(a0z)+D_0(z))f=0中a_j(0≤j≤k-1)幅角主值不全相等的情形,得到了解的增长级、下级与超级的精确估计.     

系数的级相同的高阶微分方程解的增长性

In this paper,we consider the growth of solutions of some homogeneous and nonhomogeneous higher order differential equations.It is proved that under some conditions for entire functions F,A_(ji) and polynomials P_j(z),Q_j(z)(j=0,1,…,k-1;i=1,2)with degree n≥1,the equation f~(k)+(A_(k-1,1)(z)e~(p_(k-1)(z))+A_(k-1,2)(z)e~(Q_(k-1(z)))/~f~(k-1)+…+(A_(0,1)(z)e~(P_o(z))+A_(0,2)(z)e~(Q_0(z)))f=F,where k≥2,satisfies the properties:When F ≡0,all the non-zero solutions are of infinite order;when F=0,there exists at most one exceptional solution fo with finite order,and all other solutions satisfy λ(f)=λ(f)=σ(f)=∞.     

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.     

一类系数为间隙级数的高阶线性微分方程解的增长性

In this paper, we investigate the growth of solutions of a class of higher order linear differential equations with coefficients being gap series. In this case, we remove the condition that the order of coefficients in equations is less than 1/2, and obtain some results which improve the previous results.     

关于高阶整函数系数微分方程解的增长性

本文研究了微分方程f(k) (Ak-1(z)eak-1z Dk-1(z))f(k-1) … (A0(z)ea0z D0(z))f=0解的增长性问题,针对方程中aj(0≤j≤k-1)的幅角主值不全相等的情形,得到了方程解的增长级和超级的精确估计.     

一类亚纯系数高阶线性微分方程解的增长性

In this paper, we investigate the growth of solutions of higher order linear differ-ential equations with meromorphic coefficients. Under certain conditions, we obtain precise estimation of growth order and hyper-order of solutions of the equation.     

一类高阶微分方程解的增长性

该文研究了一类高阶微分方程解的增长性, 推广并完善了G. Gundersen^[7], J.K. Langley^[8], 和 陈宗煊^[10]的一些结果.     

高阶线性微分方程的解的定性状态

本文的目的是考查高阶线性微分方程解的定性状态，建立方程分类的某些条件．我们还给出了方程解的振动判据．     

Linear Second Order Ordinary Differential Equations with Nonsmooth Coefficients

This paper deals with a Dirichlet boundary value problem for a linear second order ordinary differential operator, whose coefficients belong to certainL^p-spaces. Its solution is to be understood in the sense of Sobolev, so that the Fredholm alternative holds. The main purpose of this paper is, in case of unique solvability, to introduce a Green's function by means of which the solution can be given explicitly by integrals. We give the precise definition of the Green's function via Riesz' Representation Theorem and establish some of its basic properties. As a preliminary tool the Cauchy initial value problem is considered.     

一类高阶周期系数线性微分方程解的性质

本文主要研究了一类高阶周期系数线性微分方程解的超级,e-型级,相关性等问题,并得到了e-型级与超级之间的一些关系,以及这两种级与系数的精确关系．本文是首次使用e-型级来估计方程解的增长性,这种估计比级,超级更为精确．