In this paper we investigate the global convergence result, boundedness and periodicity of solutions of the recursive sequence $$x_{n+1}=\frac{a_0{x}_n+a_1{x}_{n-1}+\cdots +a_k{x}_{n-k}}{b_0{x}_n+b_1{x}_{n-1}+\cdots +b_k{x}_{n-k}},n=0,1,\cdots$$ where the parameters $a_i$ and $b_i$ for $i=0,1,\cdots ,k$ are positive real numbers and the initial conditions $x_{-k},x_{-k+1},\cdots ,x_0$ are arbitrary positive numbers.

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.

In this paper, sharp a priori estimate of the periodic solutions is obtained for the discrete analogue of the continuous time ratio-dependent predator-prey system, which is governed by nonautonomous difference equations, modelling the dynamics of the $n-1$ competing preys and one predator having nonoverlapping generations. Based on more precise a priori estimate and the continuation theorem of the coincidence degree, an easily verifiable sufficient criterion of the existence of positive periodic solutions...

In this paper we recall discrete order preserving property related to the discrete Riccati matrix equation. We present results obtained by applying this property to the solutions of the Riccati matrix differential equation.