The main objective of this paper is to study the boundedness character, the periodic character, the convergence and the global stability of positive solutions of the difference equation $$x_{n+1}=\left(A+\sum _{i=0}^k{\alpha }_i{x}_{n-i}\right)/\sum _{i=0}^k{\beta }_i{x}_{n-i},n=0,1,2,\cdots$$ where the coefficients $A$, ${\alpha }_i$, ${\beta }_i$ and the initial conditions $x_{-k},x_{-k+1},\cdots ,x_{-1},x_0$ are positive real numbers, while $k$ is a positive integer number.

The main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability of positive solutions of the difference equation $$x_{n+1}=\frac{{\alpha }_0{x}_n+{\alpha }_1{x}_{n-l}+{\alpha }_2{x}_{n-k}}{{\beta }_0{x}_n+{\beta }_1{x}_{n-l}+{\beta }_2{x}_{n-k}},n=0,1,2,\cdots$$ where the coefficients ${\alpha }_i,{\beta }_i\in (0,\infty )$ for $i=0,1,2,$ and $l$, $k$ are positive integers. The initial conditions $x_{-k},\cdots ,x_{-l},\cdots ,x_{-1},x_0$ are arbitrary positive real numbers such that $l<k$. Some numerical experiments are presented.

Some new oscillation and nonoscillation criteria for the second order neutral delay difference equation $$\Delta \left(c_n\Delta (y_n+p_n{y}_{n-k})\right)+q_n{y}_{n+1-m}^{\beta }=0,n\ge n_0$$ where $k$, $m$ are positive integers and $\beta$ is a ratio of odd positive integers are established, under the condition ${\sum }_{n=n_0}^{\infty }\frac{1}{c_n}<\infty .$

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.