亚纯函数差分算子值分布及复微分差分方程的研究

It is a great and important theoretical achievement that the difference analogue of Nevanlinna theory obtained in recent years, which greatly promotes the research advances on complex difference equations, Painleve equations, complex differential-difference equations and uniqueness theory and so on. Therefore, it is a international frontier research hotspot as which have enormous theoretical significance and practical value..During the development of the research on difference operators of meromorphic functions, there are some results about the uniqueness of meromorphic functions and their first order difference operators with three CM shared values. However, for the uniqueness problems of meromorphic functions and their first or high order difference operators under some more weakened shared-value conditions comparing to "3CM", it have not been deep studied so far..This project will be devoted to investigate the following two parts. The first part is about the uniqueness problems of meromorphic functions and their first or high order difference operators with some shared values, basing on the related research in my doctoral dissertation. The other part is about the delay differential equations, including how to transform those equations to complex differential-difference equations and what properties of the meromorphic solutions of the complex differential-difference equations. The work of this part will be accomplished together with a project number whose research background is biomathematics..Our aims of this project is to obtain some uniqueness theorems concerning to the first or high order difference operators of meromorphic functions, and obtain some new results on the meromorphic solutions of some certain kinds of complex differential-difference equations with slow-growing variable coefficients.

亚纯函数值分布复域差分模拟理论是近年来取得的一个重大理论成果，极大推进了复差分方程、Painleve方程、复微分差分方程、唯一性理论等诸多领域的研究进展，具有重要的理论意义和实用价值，是一个国际前沿研究热点。.在亚纯函数差分算子唯一性方面，目前虽已在亚纯函数与其一阶差分算子具3个CM分担值时的唯一性研究取得若干成果，但在亚纯函数与其一阶差分算子具有更弱分担条件时的唯一性问题以及涉及高阶差分算子的唯一性问题等方面尚未有系统的研究。.本项目计划一方面在项目申请人博士论文的研究基础上进一步深入研究亚纯函数与其一阶差分算子或其高阶差分算子具有分担值时的唯一性问题，另一方面开展对复微分差分方程的研究，并联合具有生物数学背景的项目成员，深入探讨时滞微分方程的复化问题及其亚纯解的性质。预期获得若干涉及一阶或高阶差分算子的唯一性定理、某些带有慢增长变系数的复微分差分方程亚纯解的新结果。

一是深入研究了亚纯函数与其一阶差分算子的唯一性问题，证明了超级严格小于1的非常数亚纯函数f与其一阶差分算子△f分担0、无穷CM，分担1IM时，则f与△f恒等。该结果较大程度地改进了针对陈宗煊和仪洪勋提出的一个猜想的系列已有结果，把分担条件从“3CM”弱化为“2CM+1IM”。该结果的证明方法具有较好原创性，利用李平的一个经典引理和亚纯函数值分布差分模拟理论，将亚纯函数f(z)、平移算子f(z+c)和差分算子△f三者有机结合起来综合考虑。二是研究了亚纯函数与其高阶差分算子的唯一性问题，证明超级严格小于1的超越整函数与其n阶差分算子分担0CM、1IM时两者也恒等。该结果推广了Zhang-Liao的一个结果[Sci. China A, 2014]和Gao等人的一个结果[Analy. Math. 2019]。三是基于复微分差分方程的研究背景研究了亚纯函数的导数与平移算子的唯一性问题，证明了亚纯函数f的k阶导数f^(k)与平移算子f(z+c)分担分担0、∞CM，分担1IM时两者恒等。该结果改进了Qi-Yang的相关结果[Comput. Methods Funct. Theory, 2013]。..上述研究成果进一步丰富了亚纯函数值分布理论，并以论文形式在SCI期刊发表学术论文3篇、在CSCD核心期刊发表学术论文1篇。