Stability of a generalized Putnam equation

Consider the rational difference equation x n + 1 = x n + x n − 1 + A x n − 2 x n − 3 A x n x n − 1 + x n − 2 + x n − 3 , n = 0 , 1 , 2 , … , with initial conditions x − 3 > 0 , x − 2 > 0 , x − 1 > 0 , x 0 > 0 . We prove that c = 1 is a globally asymptotically stable equilibrium of this equation if 1 ≤ A < 2 . This result extends a previously known result.