Notes on Solutions for Some Systems of Complex Functional Equations in ℂ 2

Abstract

The purpose of this article is to give the details of finding the transcendental entire solutions with finite order for the systems of nonlinear partial differential-difference equations $\left\{\begin{array}{c}{\left(\left(\left(\partial f_1\left(z_1,z_2\right)\right)/\partial z_1\right)+\left(\left(\partial f_1\left(z_1,z_2\right)\right)/\partial z_2\right)\right)}^{n_1}+P_1\left(z\right)f_2{\left(z_1+c_1,z_2+c_2\right)}^{m_1}=Q_1\left(z\right),\hfill \\ {\left(\left(\left(\partial f_1\left(z_1,z_2\right)\right)/\partial z_1\right)+\left(\left(\partial f_1\left(z_1,z_2\right)\right)/\partial z_2\right)\right)}^{n_2}+P_2\left(z\right)f_1{\left(z_1+c_1,z_2+c_2\right)}^{m_2}=Q_2\left(z\right),\hfill \end{array}\right.$ where $P_1\left(z\right),P_2\left(z\right),Q_1\left(z\right),\text{and}{Q}_2\left(z\right)$ are polynomials in ${ℂ}^2$ ; $n_1,n_2,m_1,\text{and}{m}_2$ are positive integers, and $c=\left(c_1,c_2\right)\in {ℂ}^2$ . We obtain that there exist some pairs of the transcendental entire solutions of finite order for the above system, which is a very powerful supplement to the previous theorems given by Xu and Cao and Xu and Yang.