JULIA LIMITING DIRECTIONS OF ENTIRE SOLUTIONS OF COMPLEX DIFFERENTIAL EQUATIONS

For entire or meromorphic function f,a value θ∈[0,2π)is called a Julia limiting direction if there is an unbounded sequence{z_(n)}in the Julia set satisfying limn→∞ arg z_(n)=θ.Our main result is on the entire solution f of P(z,f)+F(z)f^(s)=0,where P(z,f)is a differential polynomial of f with entire coefficients of growth smaller than that of the entire transcendental F,with the integer s being no more than the minimum degree of all differential monomials in P(z,f). We observe that Julia limiting directions of f partly come from the directions in which F grows quickly.

ON ALGEBRAIC SOLUTIONS OF THE SECOND-ORDER COMPLEX DIFFERENTIAL EQUATIONS WITH ENTIRE ALGEBRAIC ELEMENT COEFFICIENTS

The main purpose of this paper is to study the problems on the existence of algebraic solutions for some second-order complex differential equations with entire algebraic function element coeifficients. Several theorems on the existence of solutions are obtained, which perfect the solution theory of linear complex differential equations.

On Meromorphic Solutions of Nonlinear Complex Differential Equations

Applying the Nevanlinna theory of meromorphic function, we investigate the non-admissible meromorphic solutions of nonlinear complex algebraic differential equation and gain a general result. Meanwhile, we prove that the meromorphic solutions of some types of the systems of nonlinear complex differential equations are non-admissible. Moreover, the form of the systems of equations with admissible solutions is discussed.

MEROMORPHIC SOLUTIONS OF A TYPE OF SYSTEM OF COMPLEX DIFFERENTIAL-DIFFERENCE EQUATIONS

Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form {j=1∑nαj（z）f1（λj1）（z＋cj）=R2（z,f2（z））,j=1∑nβj（z）f2（λj2）（z＋cj）=R1（z,f1（z））. where λij （j = 1, 2,…, n; i = 1, 2） are finite non-negative integers, and cj （j = 1, 2,… , n） are distinct, nonzero complex numbers, αj（z）, βj（z） （j = 1,2,… ,n） are small functions relative to fi（z） （i =1, 2） respectively, Ri（z, f（z）） （i = 1, 2） are rational in fi（z） （i =1, 2） with coefficients which are small functions of fi（z） （i = 1, 2） respectively.

A RESULT OF SYSTEMS OF NONLINEAR COMPLEX ALGEBRAIC DIFFERENTIAL EQUATIONS

In this article, we mainly investigate the behavior of systems of complex differential equations when we add some condition to the quality of the solutions, and obtain an interesting result, which extends Gaekstatter and Laine＇s result concerning complex differential equations to the systems of algebraic differential equations.

Weierstrass Integrability of Complex Differential Equations

We characterize the complex differential equations of the form dy/dx=a_(n)(x)y^)n_+a_(n-1)(x)y^(n-1)+…+a_(1)(x)y+a_(0)(x) where a_(j)(x) are meromorphic functions in the variable x for j = 0,..., n that admit either a Weierstrass first integral or a Weierstrass inverse integrating factor.

On the Order of the Solutions of Systems of Complex Algebraic Differential Equations

This paper is concerned with the order of the solutions of systems of high-order complex algebraic differential equations.By means of Zalcman Lemma,the systems of equations of[1]is extended to more general form.

On the Conjecture of Gackstatter and Laine Concerning Algebraic Differential Equations

In this paper we investigate the form of a class of complex differential equations withadmissible solution and obtain a main result.

Supplement to “on the Conjecture of Gackstatter and Laine Concerning Algebraic Differential Equations”

This paper investigates the form of complex a lgebraic differential equation with admissible meromorphic solutions and obtains two results which are more precise thatn that of the paper [2].

PERTURBATIONS OF ZEROS OF SOLUTIONS TO SECOND ORDER DIFFERENTIAL EQUATIONS WITH POLYNOMIAL COEFFICIENTS

Let P（z） and P（z） be polynomials of the same degree. We consider the equations u＂ = P（z）u and u＂ = P（z）u （z ∈ C） whose solutions are u（z） and u（z）, respectively. Let Zk（U） and zk（u）, k = 1, 2,..., be the zeros of u（z） and u（z）, respectively. We derive bounds for the quantity sup j inf k｜1/zk（u）-1/zj（u）｜

A Contour Integral Method for Linear Differential Equations in Complex Plane

This paper presents the contour integral method for solving the linear constant coefficient ordinary differential equations in complex plane,and obtains the uniform expressions of the general solutions.Firstly,by using Residue Theorem,the general form of the contour integral representation for the homogeneous complex differential equation is obtained,which can be degenerated to classical results in real line.As for inhomogeneous complex differential equations with constant coefficients,we construct the integral expression of the particular solution for any continuous forcing term,and give rigorous proof via Residue Theorem.Thus the general solutions of inhomogeneous complex differential equations are also given.The main purpose of this paper is to give a foundation for a complete theory of linear complex differential equations with constant coefficients by a contour integral method.The results can not only solve the inhomogeneous complex differential equation well,but also explain the forms that are difficult to be understood in the classical solutions.

THEORY OF SINGULAR POINTS OF ORDINARY DIFFERENTIAL EQUATIONS IN COMPLEX DOMAIN

In this paper, the topological of integral surfaces near certain of Lyapunov type singularpoints and certain type of nodes of ordinary differential equations in complex domain are studied.We introduce Briot-Bouquet transformation, in order to study the topological structure of integralsurfaces near higher order singular points. At last we give an estimate of the makimum numberof isolated limit integral surfaces passing through certain type of higher order singular points.

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.

Oscillation of Solutions of Linear Differential Equations

This paper is devoted to studying the growth problem, the zeros and fixed points distribution of the solutions of linear differential equations f″＋e^-zf′＋Q（z）f=F（z）,whereQ（z）≡h（z）e^cz and c∈R.

ANALYTIC BOUNDARY VALUE PROBLEMS ON CLASSICAL DOMAINS

In this paper analytic boundary value problems for some classical domains in Cn are developed by using the harmonic analysis due to L.K. Hua. First it is discussed for the version of one variable in order to induce the relation between the analytic boundary value problem and the decomposition of function space L2 on the boundary manifold. Then an easy example of several variables, the version of torus in C2, is stated. For the noncommutative classical group L1, the characteristic boundary of a kind of bounded symmetric domain in C4, the boundary behaviors of the Cauchy integral are obtained by using both the harmonic expansion and polar coordinate transformation. At last we obtain the conditions of solvability of Schwarz problem on L1, if so, the solution is given explicitly.