Stability of a generalized Putnam equation

Consider the rational difference equation xn+1=xn+xn−1+Axn−2xn−3Axnxn−1+xn−2+xn−3,n=0,1,2,…, with initial conditions x−3>0,x−2>0,x−1>0,x0>0x−3>0,x−2>0,x−1>0,x0>0. We prove that c=1c=1 is a globally asymptotically stable equilibrium of this equation if 1≤A<21≤A<2. This result extends a previously known result.