Results on solutions of several systems of the product type complex partial differential difference equations

Abstract

This article is devoted to exploring the solutions of several systems of the first-order partial differential difference equations (PDDEs) with product type u(z+c)[α1u(z)+β1uz1+γ1uz2+α2v(z)+β2vz1+γ2vz2]=1,v(z+c)[α1v(z)+β1vz1+γ1vz2+α2u(z)+β2uz1+γ2uz2]=1,\left\{\begin{array}{l}u\left(z+c){[}{\alpha }_{1}u\left(z)+{\beta }_{1}{u}_{{z}_{1}}+{\gamma }_{1}{u}_{{z}_{2}}+{\alpha }_{2}v\left(z)+{\beta }_{2}{v}_{{z}_{1}}+{\gamma }_{2}{v}_{{z}_{2}}]=1,\\ v\left(z+c){[}{\alpha }_{1}v\left(z)+{\beta }_{1}{v}_{{z}_{1}}+{\gamma }_{1}{v}_{{z}_{2}}+{\alpha }_{2}u\left(z)+{\beta }_{2}{u}_{{z}_{1}}+{\gamma }_{2}{u}_{{z}_{2}}]=1,\end{array}\right. where c=(c1,c2)∈C2c=\left({c}_{1},{c}_{2})\in {{\mathbb{C}}}^{2}, αj,βj,γj∈C,j=1,2{\alpha }_{j},{\beta }_{j},{\gamma }_{j}\in {\mathbb{C}},\hspace{0.33em}j=1,2. Our theorems about the forms of the transcendental solutions for these systems of PDDEs are some improvements and generalization of the previous results given by Xu, Cao and Liu. Moreover, we give some examples to explain that the forms of solutions of our theorems are precise to some extent.

Keywords

• partial differential difference equation
• second order
• nevanlinna theory
• 30d35
• 35m30
• 39a45