On Malmquist type theorem of systems of complex difference equations

The main purpose of this paper is to give the Malmquist type result of the meromorphic solutions of a system of complex difference equations of the following form: $$\left \{ \textstyle\begin{array}{l} \sum_{\lambda_{1} \in I_{1}, \mu_{1}\in J_{1}}\alpha_{\lambda_{1}, \mu_{1}}(z) (\prod_{\nu=1}^{n}f(z+c_{\nu})^{l_{\lambda_{1}, \nu}}\prod_{\nu=1}^{n}g(z+c_{\nu})^{m_{\mu_{1}, \nu}} ) = \frac{\sum_{i=0}^{p}a_{i}(z)g(z)^{i} }{\sum_{j=0}^{q}b_{j}(z)g(z)^{j}}, \\ \sum_{\lambda_{2} \in I_{2}, \mu_{2}\in J_{2}}\beta_{\lambda_{2}, \mu_{2}}(z) (\prod_{\nu =1}^{n}f(z+c_{\nu})^{l_{\lambda_{2}, \nu}}\prod_{\nu=1}^{n}g(z+c_{\nu})^{m_{\mu_{2}, \nu}} ) = \frac{\sum_{k=0}^{s}d_{k}(z)f(z)^{k} }{\sum_{l=0}^{t}e_{l}(z)f(z)^{l}}, \end{array} \right . $$ where \(c_{1}, c_{2}, \ldots, c_{n}\) are distinct, nonzero complex numbers, the coefficients \(\alpha_{\lambda_{1}, \mu_{1}}(z)\) (\(\lambda _{1} \in I_{1}\), \(\mu_{1}\in J_{1}\)), \(\beta_{\lambda_{2}, \mu_{2}}(z)\) (\(\lambda_{2} \in I_{2}\), \(\mu_{2}\in J_{2}\)), \(a_{i}(z)\) (\(i=0,1,\ldots, p\)), \(b_{j}(z)\) (\(j=0,1,\ldots, q\)), \(d_{k}(z)\) (\(k=0,1,\ldots, s\)), and \(e_{l}(z)\) (\(l=0,1,\ldots, t\)) are small functions relative to \(f(z)\) and \(g(z)\), \(I_{i} = \{\lambda _{i}=(l_{\lambda_{i}, 1}, l_{\lambda_{i}, 2}, \ldots, l_{\lambda_{i},n})| l_{\lambda_{i}, \nu}\in{N} \cup\{0\},\nu= 1,2,\ldots,n\}\) (\(i=1,2\)) and \(J_{j} = \{\mu_{j}=(m_{\mu_{j}, 1}, m_{\mu_{j}, 2}, \ldots, m_{\mu_{j},n})| m_{\mu_{j}, \nu}\in{N} \cup\{0\},\nu=1,2, \ldots,n\}\) (\(j=1, 2\)) are finite index sets. The growth of meromorphic solutions of a related system of complex functional equations is also investigated.