On the positive nonoscillatory solutions of the difference equation xn+1 = α + \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left( {\tfrac{{x_{n - k} }} {{x_{n - m} }}} \right) $$\end{document}p

The aim of this paper is to show that the following difference equation: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ x_{n + 1} = \alpha + \left( {\tfrac{{x_{n - k} }} {{x_{n - m} }}} \right)^p ,n = 0,1,2,..., $$\end{document} where α > −1, p > 0, k,m ∈ N are fixed, 0 ≤ m < k, x−k, x−k+1, ..., x−m, ..., x−1, x0 are positive, has positive nonoscillatory solutions which converge to the positive equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \bar x $$\end{document} = α + 1. It is interesting that the method described in the paper, in some cases can also be applied when the parameter α is variable.