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 ∈ { 0 , 1 , … } and the initial conditions are real numbers. We use the linearization of this equation in the form x n + l = ∑ i = 1 − l k g i x n − i , n = 0 , 1 , … where l ∈ { 1 , 2 , … } and the functions g i : R k + l → R to investigate the asymptotic behavior of the solutions of the considered equation. We illustrate our results with various examples.