Attractivity and global stability for linearizable difference equations

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