Entire solutions of system of Fermat-type difference and partial differential-difference equations in $ \mathbb{C}^2 $

Abstract

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 $$\begin{cases} f_1(z)^2+(\Delta_cf_2(z))^2=1\cr f_2(z)^2+(\Delta_cf_1(z))^2=1,\end{cases}$$ $$\begin{cases} a_1^2f_1(z)^2+(a_2f_2(z+c)+a_3f_2(z))^2=1\cr a_1^2f_2(z)^2+(a_2f_1(z+c)+a_3f_1(z))^2=1,\end{cases}$$ $$\begin{cases} (a_1f_1(z+c)+a_2f_1(z))^2+(a_3f_2(z+c)+a_4f_2(z))^2=1\cr (a_1f_2(z+c)+a_2f_2(z))^2+(a_3f_1(z+c)+a_4f_1(z))^2=1,\end{cases}$$ 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.