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

We consider the following difference equation xn+1 = xn−1g(xn), n = 0,1, …, where initial values x−1, x0 ∈ [0, +∞) and g : [0, +∞)→(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 a ∈ [0, +∞). (2) Assume a ∈ (0, +∞). Then the set of initial conditions (x−1, x0)∈(0, +∞)×(0, +∞) 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.