On the dynamics of a nonautonomous rational difference equation

‎In this paper‎, ‎we study the following nonautonomous rational difference equation‎‎\[‎‎y_{n+۱}=\frac{\alpha_n+y_n}{\alpha_n+y_{n-k}},\quad n=۰,۱,...‎,‎\]‎‎where \left\{\alpha_n\right\}_{n\geq۰} is a bounded sequence of positive numbers‎, ‎k is a positive integer and the initial values y_{-k},...,y_۰ are positive real numbers‎. ‎We give sufficient conditions under which the unique equilibrium \bar{y}=۱ 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.

Keywords:

‎nonautonomous difference equation‎ , ‎global asymptotic stability‎ , ‎oscillation‎