关于复微分方程f″+Af′+Bf=0解的增长性(2)

考虑微分方程f″+Af′+Bf=0,其中A(z),B(z)都是亚纯函数.如果A(z)有一个有穷亏值,当赋予B(z)某些条件时,上述方程的每一个非零解具有无穷级.这些结果将伍鹏程和朱军前期的工作扩展到B(z)的级等于1/2的情形.     

关于一类二阶线性复微分方程解的增长性

利用亚纯函数的Nevanlinna值分布理论,研究了一类二阶复微分方程f″+A(z)f′+B(z)f=0解的增长性,其中A(z)是方程ω″+P(z)ω=0的非平凡解,P(z)是n次多项式.证明了B(z)在适当条件的假设下,方程的每一个非平凡解为无穷级的结果,推广了以前一些文献的结论.     

一类亚纯系数微分方程解的增长性

研究了一类亚纯系数微分方程的复振荡问题.通过应用亚纯函数的分解和模的增长估计,得到了该类方程解的级及其超级的精确估计.     

一类高阶复微分方程解的增长性

本文研究高阶线性微分方程f~((k))+A_(k-1)f~((k-1))+···+A1f′+A0f=0解的增长性,其中Aj(j=0,···,k-1)为整函数.当存在某个系数A_s是方程ω′′+P(z)ω=0的一个非零解时,我们得到上述方程具有无穷级解的判定条件,并对解的超级进行了估计.这里的P(z)为非零多项式,当P(z)为特定形式的多项式时,A_s可取为Airy函数,Weber-Hermite函数或指数函数.     

一类高阶非齐次线性微分方程亚纯解的零点

利用亚纯函数的值分布理论研究了下列高阶线性微分方程解的增长性及解的零点增长性,f^((k))+A_(k-1)f((k))+A_(k-1)f^((k-1))+…+A_1f′+A_0f=F(z)其中A_0,A_1,…,A_(k-1),F≠0是亚纯函数.证明了如果A_0以∞为亏值或Borel例外值,那么方程的所有非零解的零点收敛指数均为无穷,至多除去一个例外解,获得的结果推广了以前一些文献的结论.     

一类高阶微分方程亚纯解的增长性

研究了几种类型的高阶线性亚纯系数微分方程的亚纯解的增长性,对方程的亚纯解的增长率得到了精确估计.     

一类非齐次复微分方程解的增长性

研究了一类线性非齐次微分方程f(k)+ak-1f(k-1)+…+a1f-′(eQ(z)-a0)f=eQ(z)+F(z)解的增长性,其中aj(j=0,1,…,k-1)为常数,Q(z)为非常数多项式,F(z)为级小于deg Q的整函数.     

关于一类有理系数常微分方程的亚纯解

In this paper, we give an estimation of the number of meromorphic solu-tions and linearly independent meromorphic solutions of some class ordinary differential equations with rational coefficients (dw)/(dz)=a₀(z)+a₁(z)w+…+a_n(z)wⁿ (n≥3).     

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。     

一类齐次线性微分方程复振荡的特例和结果

发现一类k(≥2)阶齐次线性微分方程存在唯一的情形具有k个无零点的线性无关解,同时得到这类方程复振荡的一个普遍结果,由此,完整解决了在超越整函数时这类方程的复振荡理论,而在多项式时回答了Bank的问题。     

On the growth of solutions to the complex differential equation f'+Af'+Bf=0

In this paper, we consider the differential equation f'+ Af'+ Bf = 0, where A(z) and B(z) ≡ 0are entire functions. Assume that A(z) has a finite deficient value, then we will give some conditions on B(z)which can guarantee that every solution f ≡ 0 of the equation has infinite order.     

Polynomially bounded solutions to the Loewner differential equation in several complex variables

We determine the form of polynomially bounded solutions to the Loewner differential equation that is satisfied by univalent subordination chains of the form f(z,t)=etAz+?, where A∈L(Cn,Cn) has the property m(A)>0. Here m(A)=min{R〈A(z),z〉:‖z‖=1}. We also give sufficient conditions for g(z,t)=L(f(z,t)) to be polynomially bounded, where f(z,t) is an A-normalized polynomially bounded Loewner chain solution to the Loewner differential equation.     

On convergence of the solutions of a functional differential equation

A sufficient condition is given for the solutions of a functionally perturbed linear system of ordinary differential equations to have limits at ± ∞.     

On some solutions of the extended confluent hypergeometric differential equation

The extended confluent hypergeometric equation is defined (Section 1) as a linear second-order differential equation with (Section 2) a regular singularity at the origin and an (Section 3) irregular singularity of arbitrary degree M+1 at infinity; the original confluent hypergeometric equation is the particular case M=0, whereas the case M=1 is reducible (Section 3.2) to the former. Six types of solutions of the extended confluent hypergeometric equation of degree M, are obtained viz.: (i) functions of the first kind, i.e., regular ascending power series expansions, with infinite radius of convergence about the origin, for all values of the coefficients and degree M (Section 2.1); (ii) functions of the second kind, i.e. power series expansions with a logarithmic singularity, at the origin, for some values of the coefficients and all M (Section 2.2): (iii) only one asymptotic power series expansion exists, (Section 2.3) for M=0; (iv) concerning normal integrals (Section 3.2), valid as asymptotic expansions in the neighbourhood of the point-at-infinity, one exists for degree zero M=0 and two for degree unity M=1; (v) for degree greater than one M>1, two Laurent series expansions (Section 3.3) valid in the neighbourhood of infinity are obtained; (vi) an integral representation (Section 4) using the complex Laplace transform (Section 4.1) is obtained for (Sections 4.2–4.3) degree unity or zero M⩽1, using paths in a complex cut-plane (Fig. 1).     

On solutions set of a multivalued stochastic differential equation

We analyse multivalued stochastic differential equations driven by semimartingales. Such equations are understood as the corresponding multivalued stochastic integral equations. Under suitable conditions, it is shown that the considered multivalued stochastic differential equation admits at least one solution. Then we prove that the set of all solutions is closed and bounded.     

Controlling solutions to a linear differential equation

Both global attainability and global reducibility problems for a controllable system equivalent to a linear differential equation are positively solved. Furthermore, the existence of a linear equation having a given Cauchy matrix on a given segment and coinciding with given equations from the left and right of that segment is proved. The results obtained allow one to construct a linear equation with the fundamental solutions system possessing preassigned properties.     

Meromorphic solutions of an auxiliary ordinary differential equation using complex method

In this paper, we employ the complex method to obtain first all meromorphic solutions of an auxiliary ordinary differential equation and then find all meromorphic exact solutions of the classical Korteweg–de Vries equation, Boussinesq equation, ( 3 + 1)‐dimensional Jimbo–Miwa equation, and Benjamin–Bona–Mahony equation. Our results show that the method is more simple than other methods. Copyright © 2013 John Wiley & Sons, Ltd.