Convergence properties in the nonhyperbolic case xn+1 = xn−1 1+f (xn)

Consider the difference equation xn+1 = xn−1 1+f (xn) where f is in a certain class of increasing continuous functions. In particular, the class includes all functions of the form f (x) = αxβ with α >0 and β >0. The set of initial conditions (x0, x1) in the first quadrant that converge to any given boundary point of the first quadrant forms a unique increasing continuous function. Furthermore, all of the positive solutions xn are stable under small perturbations of the initial point (x0, x1). © 2006 Elsevier Inc. All rights reserved