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Entire solutions for several systems of nonlinear difference and partial differential-difference equations of Fermat-type

2019; Elsevier BV; Volume: 483; Issue: 2 Linguagem: Inglês

10.1016/j.jmaa.2019.123641

1096-0813

Hong Yan Xu, San Yang Liu, Qiao Ping Li,

Mathematics and Applications

The purpose of this paper is mainly to study the existence and the forms of entire solutions for the following systems of nonlinear difference and partial differential-difference equations of Fermat-type{f1(z1,z2)2+f2(z1+c1,z2+c2)2=1,f2(z1,z2)2+f1(z1+c1,z2+c2)2=1,{(∂f1(z1,z2)∂z1)n1+f2(z1+c1,z2+c2)m1=1,(∂f2(z1,z2)∂z1)n2+f1(z1+c1,z2+c2)m2=1, and{(∂f1(z1,z2)∂z1)2+[f2(z1+c1,z2+c2)−f1(z1,z2)]2=1,(∂f2(z1,z2)∂z1)2+[f1(z1+c1,z2+c2)−f2(z1,z2)]2=1. Our results about the existence and the forms of solutions for these systems are some improvements and supplements of the previous theorems given by Xu and Cao [31], Gao [4], Liu, Yang [17], [18], [20]. Moreover, we give some examples to utilize the correctness of our conclusions about the existence and forms of solutions for such systems.