Global asymptotic stability of the higher order equation xn+1=axn+bxn-kA+Bxn-k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{n+1} = \frac{ ax_{n}+bx_{n-k}}{A+Bx_{n-k}}$$\end{document}

In this paper, we investigate the local and global stability and the period two solutions of all nonnegative solutions of the difference equation, xn+1=axn+bxn-kA+Bxn-k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} x_{n+1} = \frac{ ax_{n}+bx_{n-k}}{A+Bx_{n-k}} \end{aligned}$$\end{document}where a, b, A, B are all positive real numbers, k≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 1$$\end{document} is a positive integer, and the initial conditions x-k,x-k+1,...,x0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{-k},x_{-k+1},...,x_{0}$$\end{document} are nonnegative real numbers. It is shown that the zero equilibrium point is globally asymptotically stable under the condition a+b≤A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a+b \le A$$\end{document}, and the unique positive solution is also globally asymptotically stable under the condition a-b≤A≤a+b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a-b \le A \le a+b$$\end{document}. By the end, we study the global stability of such an equation through numerically solved examples.