On the behaviour of the solutions of a second-order difference equation

Abstract

We study the difference equation xn+1 = α - xn/xn-1, n ∈ ℕ0, where α ∈ ℛ and where x-1 and x0 are so chosen that the corresponding solution (xn) of the equation is defined for every n ∈ ℕ. We prove that when α = 3 the equilibrium x = 2 of the equation is not stable, which corrects a result due to X. X. Yan, W. T. Li, and Z. Zhao. For the case α = 1, we show that there is a strictly monotone solution of the equation, and we also find its asymptotics. An explicit formula for the solutions of the equation are given for the case α = 0.