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

AbstractIn this note we study the difference equationxn+1=1+xn−1xn,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]