Boundedness of Solutions of \(x_{n+1} = \frac{a_n' + b_n' y_n}{C_n'x_n}\) and \(y_{n+1} = \frac{a_n+b_nx_n + c_n y_n}{A_n+B_nx_n + C_ny_n}\) with Non-constant Coefficients

We will say that a sequence of real numbers \(\{s_n\}_{n=0}^{\infty }\) is positively bounded if there exists positive constants \(m\) and \(M\) , such that \(m < s_n < M\) for all \(n=0,1,\ldots \) . We will say that a sequence \(\{s_n\}_{n=0}^{\infty }\) is identically zero if \(s_n = 0\) for all \(n=0,1,\ldots \) . We establish the boundedness character of families contained within: \(\begin{aligned} \left\{ \begin{array}{ll} x_{n+1} = \dfrac{a_n' + b_n' y_n}{C_n'x_n}\\ y_{n+1} = \dfrac{a_n+b_nx_n + c_n y_n}{A_n+B_nx_n + C_ny_n} \end{array} \right. \text { for } n=0,1,\ldots \end{aligned}\) with positive initial conditions \(x_0\) , \(y_0\) and where \(\{b_i'\}\) and \(\{C_i'\}\) are positively bounded sequences and the other coefficient sequences are either identically zero or positively bounded in such a way that the denominator of each equation is nonzero.