Global asymptotic stability of a family of difference equations

We consider the family of difference equations of the form xn+1=∑i∈Zk+2−{j,l}xn−i+xn−jxn−l+A∑i∈Zk+2−{s,t}xn−i+xn−sxn−t+A,n=0,1,…, where k∈{0,1,…}, j,l,s,t∈Zk+2≡{0,1,…,k+2} with j≠l and s≠t, A is nonnegative and the initial values x−k−2,x−k−1,…,x0 are positive real numbers. For these difference equations, we prove that the unique equilibrium x¯=1 is globally asymptotically stable.