A study of solutions for several classes of systems of complex nonlinear partial differential difference equations in ℂ2

Abstract

This article is devoted to describing the entire solutions of several systems of the first-order nonlinear partial differential difference equations (PDDEs). Using the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we establish some interesting results to reveal the existence and the forms of the finite-order transcendental entire solutions of several systems of the first-order nonlinear PDDEs: f(z1+c1,z2+c2)(a1gz1+a2gz2)=m1,g(z1+c1,z2+c2)(a3fz1+a4fz2)=m2,f(z1+c1,z2+c2)(a1fz1+a2gz1)=m1,g(z1+c1,z2+c2)(a3fz2+a4gz2)=m2,\begin{array}{l}\left\{\begin{array}{l}f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})({a}_{1}{g}_{{z}_{1}}+{a}_{2}{g}_{{z}_{2}})={m}_{1},\\ g\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})({a}_{3}{f}_{{z}_{1}}+{a}_{4}{f}_{{z}_{2}})={m}_{2},\end{array}\right.\\ \left\{\begin{array}{l}f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})({a}_{1}{f}_{{z}_{1}}+{a}_{2}{g}_{{z}_{1}})={m}_{1},\\ g\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})({a}_{3}{f}_{{z}_{2}}+{a}_{4}{g}_{{z}_{2}})={m}_{2},\end{array}\right.\end{array} and f(z1+c1,z2+c2)(a1fz2+a2gz1)=m1,g(z1+c1,z2+c2)(a3fz1+a4gz2)=m2,\left\{\begin{array}{l}f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})({a}_{1}{f}_{{z}_{2}}+{a}_{2}{g}_{{z}_{1}})={m}_{1},\\ g\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})({a}_{3}{f}_{{z}_{1}}+{a}_{4}{g}_{{z}_{2}})={m}_{2},\end{array}\right. where a1,a2,a3,a4,c1,c2∈C{a}_{1},{a}_{2},{a}_{3},{a}_{4},{c}_{1},{c}_{2}\in {\mathbb{C}}, m1,m2∈C−{0}{m}_{1},{m}_{2}\in {\mathbb{C}}-\left\{0\right\}. Moreover, some examples are given to explain that there are significant differences in the forms of solutions from some previous systems of functional equations.

Keywords

• nevanlinna theory
• entire function
• system of functional equations
• several complex variables
• 30d35
• 35m30