Global Attractivity in a Second-Order Nonlinear Difference Equation

Consider the difference equation xn+1 = xnƒ(xn−1), n = 0, 1, 2, ..., (1) where the function ƒ satisfies the following conditions: • ƒ ∈ [[0, ∞), (0, ∞)] and ƒ() is nonincreasing in ; • The equation ƒ() = 1 has a unique positive solution; • If denotes the unique positive solution of ƒ() = 1, then [xƒ(x) − x](x − x) > 0 for x ≠ x Then x is a global attractor of all positive solutions of Eq. (1).