We study and solve one class of discrete matrix equations with cubic nonlinearity. The existence of a two-parameter family of monotone and bounded solutions is proved. Under certain additional conditions, we determine the asymptotic behavior of the constructed solutions. The obtained results are extended to the corresponding inhomogeneous discrete matrix equations and to some more general cases of nonlinearity.

中文翻译：

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关于一类具有三次非线性的离散矩阵方程的可解性

我们研究并求解一类具有三次非线性的离散矩阵方程。证明了单调有界解的双参数族的存在性。在某些附加条件下，我们确定构造解的渐近行为。获得的结果扩展到相应的非齐次离散矩阵方程和一些更一般的非线性情况。