Entire Solutions for Complex Systems of the Second-Order Partial Differential Difference Equations of Fermat Type

Abstract

This article is mainly concerned with the existence and the forms of entire solutions for several systems of the second-order partial differential difference equations of Fermat type α∂2f1z1,z2/∂z12+β∂2f1z1,z2/∂z22n1+f2z1+c1,z2+c2m1=1α∂2f2z1,z2/∂z12+β∂2f2z1,z2/∂z22n2+f1z1+c1,z2+c2m2=1 and ∂2f1z1,z2/∂z122+f2z1+c1,z2+c22=1∂2f2z1,z2/∂z122+f1z1+c1,z2+c22=1. Our results about the existence and the forms of solutions for these systems generalize the previous theorems given by Xu and Cao, Gao, Liu, and Yang. In addition, we give some examples to explain the existence of solutions of this system in each case.