复微分方程、差分方程及差分值分布

Complex differential equations and complex difference equations are very interesting topics in the studying of complex analysis, there have been a lot of achievements about the modular distribution of solutions of complex equations, value distribution theory and the zero distribution of difference operators. However, few researches are made for the angular distribution of solutions of the complex differential equations and difference equations as well as the difference operators. This project aims to study the angular distribution of solutions of complex differential eqautions and difference equations, and singular directions of difference operators. We shall focus on the following three aspects:. (1) We shall study singular directions of difference operators by analysing the logarithmic derivative lemma and Wiman-Valiron theory for difference operators; .(2) We shall study the growth、oscillation and angular distribution of solutions of complex differential equations and difference equations by using some deep results of value distribution theory of meromorphic functions, such as deficient value、asymptotic value、angular distribution and so on;. (3) We shall study the relation between the coefficients with solutions for complex differential equations in the unit disc, and discuss the relation between the solutions of the equations with the function spaces theory.

复微分方程和差分方程是目前复分析研究的热点课题，人们在复方程解的模分布性质、复差分值分布及差分的零点分布方面，取得了丰富的成果。但复方程解的辐角分布性质和差分的奇异方向的研究还处于初始阶段。本项目将重点研究复方程解的辐角分布性质和差分的奇异方向，拟从以下三个方面开展研究：.（1）通过研究差分对数导数引理和Wiman-Valiron理论，讨论差分的公共奇异方向；.（2）利用亚纯函数值分布理论中的深刻结果（比如亏值、渐近值、辐角分布等）去探索复域微分方程、差分方程解的增长性、复振荡和辐角分布性质；.（3）探索单位圆上复微分方程系数和复微分方程解之间的关系，讨论方程解与函数空间的关系。

本项目涉及亚纯函数的值分布理论及其应用，利用亚纯函数的值分布理论及解析函数理论，研究了复微分方程、非线性复微分-差分方程解的性质、亚纯函数的q-差分算子的辐角分布，具体涉及以下几个方面。.（1）复平面上复微分方程解的性质：找到了一些使得复微分方程解具有快速增长的系数条件，同时也刻画了使得方程解为无穷级的角度大小；利用系数函数下级条件刻画方程解的零点收敛指数；利用权函数的性质描述方程解的Fock空间性质。.（2）单位圆上复微分方程解的性质：利用新的插值公式，完全解决方程解的零点的Heittokangas问题；刻画了方程解的增长性和方程解的函数空间性质，比如Hardy型空间、Bloch型空间、Morrey空间等，得到比已有结果更加广泛的结果。.（3）圆环域上复微分方程解的增长性：利用圆环域上亚纯函数的值分布理论，讨论系数带奇点的复微分方程解的[p,q]级。.（4）非线性复微分-差分方程解的性质：考虑几类非线性复微分-差分方程，获得了方程解的增长级估计和方程解的表达式。.（5）整函数的q-差分算子的辐角分布理论：结合差分值分布理论和亚纯函数辐角分布理论，证明了整函数与其q-差分算子的公共Julia方向、Borel方向的存在性。