Convergence properties in the nonhyperbolic case

Consider the difference equation 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).