Non-Existence of Entire Solution of a Type of System of Equations

Abstract:

      In this paper, we will prove that the system of differential-difference equations \begin{eqnarray}\begin{cases}{(f(z)f'(z))}^n+p_1^2(z)g^m(z+\eta)=Q_1(z), \\\nonumber {(g(z)g'(z))}^n+p_2^2(z)f^m(z+\eta)=Q_2(z),\end{cases}\end{eqnarray} has no transcendental entire solution $(f(z),g(z))$ with $\rho(f,g)<\infty$ such that $\lambda(f)<\rho(f)$ and $\lambda(g)<\rho(g),$ where $P_1(z), Q_1(z), P_2(z)$ and $ Q_2(z)$ are non-vanishing polynomials.