We study k th order systems of two rational difference equations $$x_n=\frac{\alpha +{\sum }_{i=1}^k{\beta }_i{x}_{n-1}+{\sum }_{i=1}^k{\gamma }_i{y}_{n-1}}{A+{\sum }_{j=1}^k{B}_j{x}_{n-j}+{\sum }_{j=1}^k{C}_j{y}_{n-j}},y_n=\frac{p+{\sum }_{i=1}^k{\delta }_i{x}_{n-i}+{\sum }_{i=1}^k{\epsilon }_i{y}_{n-i}}{q+{\sum }_{j=1}^k{D}_j{x}_{n-j}+{\sum }_{j=1}^k{E}_j{y}_{n-j}}n\in \mathbb{N}$$ . In particular, we assume non-negative parameters and non-negative initial conditions, such that the denominators are nonzero. We develop several approaches which allow us to extend well known boundedness results on the k th order rational difference equation to the setting of systems in certain cases.

In the paper we study the subject of stability of systems with $h$-differences of Caputo-, Riemann-Liouville- and Grünwald-Letnikov-type with $n$ fractional orders. The equivalent descriptions of fractional $h$-difference systems are presented. The sufficient conditions for asymptotic stability are given. Moreover, the Lyapunov direct method is used to analyze the stability of the considered systems with $n$-orders.

We study the asymptotic behavior of the solutions of a scalar convolution sum-difference equation. The rate of convergence of the solution is found by determining the asymptotic behavior of the solution of the transient renewal equation.

We study k th order systems of two rational difference equations $$x_n=\frac{\alpha +{\sum }_{i=1}^k{\beta }_i{x}_{n-i}+{\sum }_{i=1}^k{\gamma }_i{y}_{n-i}}{A+{\sum }_{j=1}^k{B}_j{x}_{n-j}+{\sum }_{j=1}^k{C}_j{y}_{n-j}},n\in \mathbb{N},$$ In particular we assume non-negative parameters and non-negative initial conditions. We develop several approaches which allow us to prove that unbounded solutions exist for certain initial conditions in a range of the parameters.

The asymptotic and oscillatory behavior of solutions of Volterra summation equation and second order linear difference equation are studied.