On the dynamics of a nonautonomous rational difference equation

Document Type : Research Paper


Laboratory of Applied Mathematics‎, ‎Badji Mokhtar-Annaba University‎, ‎P.O‎. ‎Box 12‎, ‎Annaba‎, ‎23000‎, ‎Algeria‎


‎In this paper‎, ‎we study the following nonautonomous rational difference equation‎
‎y_{n+1}=\frac{\alpha_n+y_n}{\alpha_n+y_{n-k}},\quad n=0,1,...‎,
‎where $\left\{\alpha_n\right\}_{n\geq0}$ is a bounded sequence of positive numbers‎, ‎$k$ is a positive integer and the initial values $y_{-k},...,y_0$ are positive real numbers‎. ‎We give sufficient conditions under which the unique equilibrium $\bar{y}=1$ is globally asymptotically stable‎. ‎Furthermore‎, ‎we establish an oscillation result for positive solutions about the equilibrium point‎. ‎Our work generalizes and improves earlier results in the literature‎.