Global Behavior of the Difference Equation xn+1 = xn-1g(xn)

We consider the following difference equation x(n+1) = x(n-1)g(x(n)), n = 0, 1,..., where initial values x(-1),x(0) epsilon [0,+infinity) and g: [0, +infinity). (0, 1] is a strictly decreasing continuous surjective function. We show the following. (1) Every positive solution of this equation converges to a, 0,a, 0,..., or 0,a, 0,a,... for some... [0,+8). (2) Assume... (0,+8). Then the set of initial conditions (x(-1),x(0)) epsilon (0, +infinity) x (0, +infinity) such that the positive solutions of this equation converge to a, 0,a, 0,..., or 0,a, 0,a,... is a unique strictly increasing continuous function or an empty set.