复微分方程、差分方程解的性质与函数微分、差分的唯一性

In this project, the properties of meromorphic solutions of complex differential equations and difference equations are study. We mainly study the existence of admissible meromorphic solutions of nonlinear differential equations and nonlinear difference equations. We also study the growth and the value distribution of solutions of linear difference equations and some type of nonlinear difference equations. The theory of complex difference equations has become an active research direction in recent years. It has extensive applications in physics. Many physical models are complex difference equations. The theory of complex difference equations is an inevitable development of real difference equations, and has an important significance in the theoretical and practical research. The Bruck conjecture is a problem not fully resolved in the uniqueness theory of meromorphic functions until now. In this project, we will apply the properties of solutions of differential equations to study this conjecture. In addition, we will study the uniqueness problem of meromorphic functions and their difference operators, which become an active topic in recent years. In our early research works, we have applied the properties of solutions of difference equations to study the difference analogue of Bruck conjecture and have obtained some results. In this project, we continue to deepen and extend the research on this subject.

本项目研究复微分方程、差分方程整体亚纯解的性质。我们将主要研究非线性微分方程、非线性差分方程允许亚纯解的存在性；研究线性差分方程和某些非线性差分方程解的增长性和值分布问题。 复差分方程理论是近几年来兴起的热门研究方向，它在物理学中有广泛的应用，许多物理模型就是差分方程，它是实域差分方程向复域的必然发展，在理论和实际研究上都有重要的意义。Bruck猜想是亚纯函数唯一性理论中至今仍未完全解决的一个问题，本项目将应用微分方程解的性质对其进行研究。 此外，我们还将研究亚纯函数与其差分算子的唯一性问题，这是近几年国际上兴起的一个热点。在前期工作中，我们借助差分方程解的性质研究了Bruck猜想的差分模拟，已得到一些结果。本项目将继续深化和发展这方面的研究。

复微分、差分方程理论是复分析中的一个重要分支，它在物理学中有广泛的应用，许多物理模型就是微分、差分方程。本项目对几类非线性微分方程、差分方程整体亚纯解的存在性进行了研究，给出了判定方程整体亚纯解存在性的较弱条件。对系数具有相同增长级的线性差分方程和Malmquist型非线性差分微分方程，研究了它们的超越亚纯解的增长性和值分布，给出了经济人口模型对应的差分方程有理解的存在性和特征。对某个系数为特殊函数的线性微分方程，得到了其无穷级解存在性的判定条件，同时精确估计了系数为指数型多项式的非齐次线性微分方程解的超级。关于亚纯函数唯一性理论，我们主要研究了亚纯函数与其差分算子分担小函数，亚纯函数与其线性差分多项式、q-平移差分算子分担公共值的唯一性，取得了较好的进展。在本项目的研究中还培养了5名硕士研究生从事本方向的研究。