Global behavior of the difference equation \(x_{n + 1} = x_{n - 1} g(x_n)\) (Q1724778)

Summary: We consider the following difference equation \(x_{n + 1} = x_{n - 1} g(x_n)\), \(n = 0,1, \ldots\), where initial values \(x_{- 1}, x_0 \in [0, + \infty)\) and \(g : [0, + \infty) \rightarrow(0,1]\) is a strictly decreasing continuous surjective function. We show the following. (1) Every positive solution of this equation converges to \(a, 0, a, 0, \ldots\), or \(0, a, 0, a, \ldots\) for some \(a \in [0, + \infty)\). (2) Assume \(a \in(0, + \infty)\). Then the set of initial conditions \((x_{- 1}, x_0) \in(0, + \infty) \times(0, + \infty)\) such that the positive solutions of this equation converge to \(a, 0, a, 0, \ldots\), or \(0, a, 0, a, \ldots\) is a unique strictly increasing continuous function or an empty set.