Notes on Solutions for Some Systems of Complex Functional Equations in ℂ2. (2nd July 2021) - Li, Hong; Xu, Hong Yan
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 ∂ f 1 z 1, z 2 / ∂ z 1 + ∂ f 1 z 1, z 2 / ∂ z 2 n 1 + P 1 z f 2 z 1 + c 1, z 2 + c 2 m 1 = Q 1 z, ∂ f 1 z 1, z 2 / ∂ z 1 + ∂ f 1 z 1, z 2 / ∂ z 2 n 2 + P 2 z f 1 z 1 + c 1, z 2 + c 2 m 2 = Q 2 z, where P 1 z, P 2 z, Q 1 z, and Q 2 z are polynomials in ℂ 2 ; n 1, n 2, m 1, and   m 2 are positive integers, and c = c 1, c 2 ∈ 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.