Global asymptotic stability of a family of difference equations

We consider the family of difference equations of the form x(n+1) = Sigma i is an element of Z(k+2) - {j,l}(xn-i) + (xn-jxn-1) + A/Sigma i is an element of Z(k+2) - {s,t}(xn-i + xn-sxn-t) + A, n = 0,1..., where k is an element of {0, 1,...,}, j, l, s, t is an element of Z(k+2) ={0, 1,..., k + 2} with j not equal l and s not equal t, A is nonnegative and the initial values x(-k-2), x(-k-1),..., x(0) are positive real numbers. For these difference equations, we prove that the unique equilibrium (x) over bar = 1 is globally asymptotically stable. (C) 2004 Elsevier Inc. All rights reserved.