Meromorphic solutions of one type of systems of algebraic differential equations (Q2721805)

Consider a system of differential equations NEWLINE\[NEWLINE\begin{cases} \Omega_{11} /\Omega_{12}= R_1(z,w_1,w_2),\\ \Omega_{21}/ \Omega_{22}= R_2(z,w_1,w_2), \end{cases} \tag{1}NEWLINE\]NEWLINE where \(\Omega_{kl} (k,l=1,2)\) are differential polynomials of \(w_1\) and \(w_2\) and \(R_1\) and \(R_2\) are rational functions of \(w_1\) and \(w_2\). Let meromorphic functions \(a_j(z)\), \(b_m(z)\) and \(c_n(z)\) be coefficients of \(\Omega_{kl}\), \(R_1\) and \(R_2\) and let NEWLINE\[NEWLINES(r)= \sum_j T(r,a_j)+ \sum_m T(r,b_m) +\sum_n T(r,c_n),NEWLINE\]NEWLINE where \(T(\cdot)\) represents the Nevanlinna characteristic function. For a system of solutions \((w_1,w_2),\) \(w_k(k=1,2)\) verifying \(\varlimsup S(r)/T(r,w_k)=0\) \((r\to +\infty)\) is called admissible. By the value distribution theory the author finds some necessary conditions for the existence of a system of admissible solutions \((w_1,w_2)\) of (1) and proves that \(\exists M_1\) and \(M_2>0\) such that for \(r(>0)\) sufficiently large, \(M_1T(r,w_1)\leq T(r,w_2) \leq M_2T(r,w_1)\) if \((w_1,w_2)\) is a system of admissible solutions.