Attractivity and global stability for linearizable difference equations

Consider the difference equation x(n+1) = f(x(n), ..., x(n-k)), n = 0, 1, ... where k is an element of {0, 1, ...} and the initial conditions are real numbers. We use the linearization of this equation in the form x(n+l) = Sigma(k)(i=1-l)g(i)x(n-i), n = 0, 1, ... where l is an element of {1,2, ...} and the functions g(i) : Rk+l -> R to investigate the asymptotic behavior of the solutions of the considered equation. We illustrate our results with various examples. (C) 2008 Elsevier Ltd. All rights reserved.