Notes on Solutions for Some Systems of Complex Functional Equations in ℂ2

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 ∂f1z1,z2/∂z1+∂f1z1,z2/∂z2n1+P1zf2z1+c1,z2+c2m1=Q1z,∂f1z1,z2/∂z1+∂f1z1,z2/∂z2n2+P2zf1z1+c1,z2+c2m2=Q2z, where P1z,P2z,Q1z,andQ2z are polynomials in ℂ2; n1,n2,m1,and m2 are positive integers, and c=c1,c2∈ℂ2. 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.