Existence of monotone solutions of some difference equations with unstable equilibrium

We consider the difference equation xn+1 = F(xn,xn−1), n = 0, 1, . . . , where the function F(u,v) is continuous on I 2 for some interval I 0, F(0, 0) = 0, and F(x, x) = x for x ∈ I \{0}. Assuming that F(u,c) is decreasing in u on I and that F(c,v) is increasing in v on I, for any c ∈ I , we establish a necessary and sufficient condition for existence of monotone solutions converging to the equilibrium x = 0.  2002 Elsevier Science (USA). All rights reserved