论文标题
$ \ mathbb {c}^2 $中Fermat-Type差异和部分差分方程的整个解决方案
Entire solutions of system of Fermat-type difference and partial differential-difference equations in $ \mathbb{C}^2 $
论文作者
论文摘要
在本文中，我们主要研究了以下有限顺序的整个解决方案的存在和形式。 $$ \ begin {cases} a_1^2f_1（z）^2+（a_2f_2（z+c）+a_3f_2（z））^2 = 1 = 1 \ cr a_1^2f_2（z）^2+（a_2f_1（a_2f_1（z+c）+c）+a_3f_1（z）+a_3f_1（z）） （a_1f_1（z+c）+a_2f_1（z））^2+（a_3f_2（z+c）+a_4f_2（z））^2 = 1 = 1 \ cr （a_1f_2（z+c）+a_2f_2（z））^2+（a_3f_1（z+c）+a_4f_1（z））^2 = 1，\ end \ end {cases} $$ and $ $ \ begin {cases}} （\ partial^{i} f_1（z）+\ partial^{j} f_1（z））^{n_1}+f_2（z+c）^{m_1} = 1 \ cr （\ partial^{i} f_2（z）+\ partial^{j} f_2（z））^{n_2}+f_1（z+c）^{m_2} = 1 \ 1 \ end end {cases} $$在几个复杂变量中。我们的某些结果是Zheng-Xu \ cite {Zheng-Xu＆Analysis Math＆2021}的改进和扩展，Xu-Cao \ cite \ cite {Xu＆Cao＆2018}，XU \ textit {et。 al。} \ cite {xu-liu-li-jmaa-2020}和li \ textit {et。 al。} \ cite {li-zhang-xu＆2021＆aims}。此外，我们提供了一些与本文内容相关的示例。
In this paper we mainly study the existence and the form of entire solutions with finite order for the following system of Fermat-type difference and partial differential-difference equations $$\begin{cases} f_1(z)^2+(Δ_cf_2(z))^2=1\cr f_2(z)^2+(Δ_cf_1(z))^2=1,\end{cases}$$ $$\begin{cases} a_1^2f_1(z)^2+(a_2f_2(z+c)+a_3f_2(z))^2=1\cr a_1^2f_2(z)^2+(a_2f_1(z+c)+a_3f_1(z))^2=1,\end{cases}$$ $$\begin{cases} (a_1f_1(z+c)+a_2f_1(z))^2+(a_3f_2(z+c)+a_4f_2(z))^2=1\cr (a_1f_2(z+c)+a_2f_2(z))^2+(a_3f_1(z+c)+a_4f_1(z))^2=1,\end{cases}$$ and $$\begin{cases} (\partial^{I}f_1(z)+\partial^{J}f_1(z))^{n_1}+f_2(z+c)^{m_1}=1\cr (\partial^{I}f_2(z)+\partial^{J}f_2(z))^{n_2}+f_1(z+c)^{m_2}=1\end{cases}$$ in several complex variables. Some of our results are improvements and extensions of the previous theorems given by Zheng-Xu \cite{Zheng-Xu & Analysis math & 2021}, Xu-Cao \cite{Xu & Cao & 2018}, Xu \textit{et. al.} \cite{Xu-Liu-Li-JMAA-2020} and Li \textit{et. al.} \cite{Li-Zhang-Xu & 2021 & AIMS}. Moreover, we give some examples which are relevant to the content of the paper.
