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−j xn−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.  2004 Elsevier Inc. All rights reserved.