Consider the difference equation
with positive real numbers, and nonnegative initial conditions .
By a change of variables, it is reduced to
where , .
Since this equation is semiconjugate to a Möbius transformation [cf. A. Andruch-Sobiło, M. Małgorzata, Opusc. Math. 26, No. 3 387–394 (2006; Zbl 1131.39003)], a formula for the solutions is available in terms of the parameter and the initial data. The authors use this formula to prove that every positive solution of (1) converges to zero if , and converges to a periodic solution of period two if . [For recent results concerning the same equation, see H. Sedaghat, J. Difference Equ. Appl. 15, No. 3, 215–224 (2009; Zbl 1169.39006)].