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.

We investigate diverse separation properties of two convex polyhedral sets for the case when there are parameters in one row of the constraint matrix. In particular, we deal with the existence, description and stability properties of the separating hyperplanes of such convex polyhedral sets. We present several examples carried out on PC. We are also interested in supporting separation (separating hyperplanes support both the convex polyhedral sets at given faces) and permanent separation (a hyperplane...