On the global behavior of the rational difference equation \(y_{n+1}=\frac{\alpha_n+y_{n-r}}{\alpha_n+y_{n-k}}\)

In this article, we study the global behavior of the following higher-order nonautonomous rational difference equation \[ y_{n+1}=\frac{\alpha_n+y_{n-r}}{\alpha_n+y_{n-k}},\quad n=0,1,..., \] where \(\left\{\alpha_n\right\}_{n\geq0}\) is a bounded sequence of positive numbers, \(k,r\) are nonnegative integers such that \(r