肖丽鹏.高阶线性微分方程解的零点分布及Zygmund-型空间[J].数学研究及应用,2022,42(6):599~610
高阶线性微分方程解的零点分布及Zygmund-型空间
Zero Distribution of Solutions of Higher-Order Linear Differential Equations and Zygmund Type Space
投稿时间:2021-11-30  修订日期:2022-06-25
DOI:10.3770/j.issn:2095-2651.2022.06.005
中文关键词:  线性微分方程  一致分离序列  Zygmund-型空间
英文关键词:linear differential equation  uniformly separated sequence  Zygmund type space
基金项目:国家自然科学基金(Grant No.11661043).
作者单位
肖丽鹏 江西师范大学数学与统计学院, 江西 南昌 330022 
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中文摘要:
      本文主要考虑以下两个问题: (1) 建立非齐次线性微分方程$$f'''+A_2(z)f''+A_1(z)f'+A_0(z)f=A_3(z),$$ 系数增长性与解的零点的几何分布的相互关系, 其中 $A_0(z),\ldots, A_3(z)$为单位圆内的解析函数; (2) 找到一些使方程$$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_1(z)f'+A_0(z)f=0,$$ 所有解属于Zygmund-型空间的充分条件. 我们得到的结果推广了Heittokangas, Gr\"{o}hn, Korhoneon 和 R\"{a}tty\"{a}的部分结果.
英文摘要:
      The aim of this paper is to consider the following two problems: (1)~~Establish interrelationships between the growth of coefficients and the geometric distribution of zeros of solutions of non-homogeneous linear differential equation $$f'''+A_2(z)f''+A_1(z)f'+A_0(z)f=A_3(z),$$ where $A_0(z),\ldots, A_3(z)$ are analytic functions in the unit disc $\mathbb{D}$; (2)~~Find some sufficient conditions on the analytic coefficients of the differential equation $$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_1(z)f'+A_0(z)f=0,$$ for all solutions to belong to the Zygmund type space. The results we obtain are a generalization of some earlier results by Heittokangas, Gr\"{o}hn, Korhoneon and R\"{a}tty\"{a}.
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