On the Behaviour of the Solutions of a Second-Order Difference Equation

We study the difference equation x n + 1 = α − x n / x n − 1 , n ∈ ℕ 0 , where α ∈ ℝ and where x − 1 and x 0 are so chosen that the corresponding solution ( x n ) 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 .

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