Dynamics of a Rational Difference Equation

The main goal of the paper is to investigate boundedness, invariant intervals, semicycles, and global attractivity of all nonnegative solutions of the equation xn+1=(α+βxn+γxn-k)/(1+xn-k), n∈ℕ0, where the parameters α,β,γ∈[0,∞), k≥2 is an integer, and the initial conditions x-k,…,x0∈[0,∞). It is shown that the unique positive equilibrium of the equation is globally asymptotically stable under the condition β≤1. The result partially solves the open problem proposed by Kulenović and Ladas in work (2002)