On the difference equation x n +1 = Af(x n ) + f(x n −1)

Abstract

The difference equation (Formula presented.) where f : R \ {0} → R is a piecewise nonincreasing continuous function, is investigated for various values of the parameter A. If A > 0, then sufficient conditions are given to ensure that all solutions converge to the (unique) equilibrium of the equation. If A ≤ 0, it is shown that period two solutions exist and their (local) exponential stability and Lyapunov instability are discussed. Moreover in some specific cases it is shown that these periodic solutions are (globally) asymptotically stable. © 2004, Taylor & Francis Group, LLC.