On convergence of the solutions of the difference equation xn+1 = 1+ xn−1 xn ✩

In this note we study the difference equation xn+1 = 1+ xn−1 xn , n= 0, 1, . . . , where initial values x−1, x0 ∈ (0,+∞), and obtain the set of all initial values x−1, x0 ∈ (0,+∞) such that the positive solutions {xn}∞n=−1 of that equation converges to the unique equilibrium x = 2. This answers the open problem 4.8.9 proposed by M.R.S. Kulenovic and G. Ladas in [M.R.S. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures, Chapman and Hall/CRC, 2002]. © 2006 Elsevier Inc. All rights reserved.