The Study of Solutions of Several Systems of Nonlinear Partial Differential Difference Equations

Abstract

Our main aim is to describe the entire solutions of several systems of α1f1z2+α2f2z+c2=1,β1f2z2+β2f1z+c2=1,α1∂f1/∂z1n1+α2f2z+cm1=1,β1∂f2/∂z1n2+β2f1z+cm2=1, and α1∂f1/∂z12+α2f2z+c2=1,β1∂f2/∂z12+β2f1z+c2=1,α1∂f1/∂z12+α2f2z+c+α3f1z2=1,β1∂f2/∂z12+β2f1z+c+β3f2z2=1, where αj,βjj=1,2,3 are nonzero constants in ℂ and mj,njj=1,2 are positive integers. We obtain several theorems on the existence and the forms of solutions for these systems, which are some improvements and supplements of the previous theorems given by Xu and Cao, Gao, and Liu and Yang. Moreover, we give some examples to explain the existence of solutions for such systems.