Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

HOME ALL ARTICLES View

Bull. Korean Math. Soc. 2024; 61(3): 745-762

Online first article May 20, 2024      Printed May 31, 2024

https://doi.org/10.4134/BKMS.b230330

Copyright © The Korean Mathematical Society.

Meromorphic solutions of some non-linear difference equations with three exponential terms

Minfeng Chen , Zongsheng Gao, Xiaomin Huang

Guangdong University of Foreign Studies; Beihang University; Guangdong University of Technology

Abstract

In this paper, we study the existence of finite order meromorphic solutions of the following non-linear difference equation \begin{equation*} f^{n}(z)+P_{d}(z,f)=p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}+p_{3}e^{\alpha_{3}z}, \end{equation*} where $n\geq 2$ is an integer, $P_{d}(z,f)$ is a difference polynomial in $f$ of degree $d\leq n-2$ with small functions of $f$ as its coefficients, $p_{j}~(j=1,2,3)$ are small meromorphic functions of $f$ and $\alpha_{j}~(j=1,2,3)$ are three distinct non-zero constants. We give the expressions of finite order meromorphic solutions of the above equation under some restrictions on $\alpha_{j}~(j=1,2,3)$. Some examples are given to illustrate the accuracy of the conditions.

Keywords: Nevanlinna theory, non-linear difference equation, meromorphic solution, finite order

MSC numbers: Primary 39A45, 30D05

Supported by: This work was supported by the National Natural Science Foundation of China (No. 12001117), Guangdong Basic and Applied Basic Research Foundation (No. 2021A1515110654) and by the Basic and Applied Basic Research of Guangzhou Basic Research Program (Nos. 202102020438, 202201010234).