On the global behavior of the rational difference equation yn+1 = αn+yn−r αn+yn−k

Abstract

In this article, we study the global behavior of the following higher-order non-autonomous rational difference equation yn+1 =αn + yn−r αn + yn−k , n = 0, 1, ..., where {αn}n≥0 is a bounded sequence of positive numbers, k,r are nonnegative integers such that r < k, and the initial values y−k, ..., y0 are nonnegative real numbers. We show that all positive solutions are oscillatory about the equilibrium point ¯y = 1. Furthermore, we establish sufficient conditions under which the unique equilibrium attracts all positive solutions