Attractivity and global stability for linearizable difference equations

Consider the difference equation xnC1 D f .xn; : : : ; xn􀀀k/; n D 0; 1; : : : where k 2 f0; 1; : : :g and the initial conditions are real numbers. We use the linearization of this equation in the form xnCl D Xk iD1􀀀l gixn􀀀i; n D 0; 1; : : : where l 2 f1; 2; : : :g and the functions gi V RkCl ! R to investigate the asymptotic behavior of the solutions of the considered equation. We illustrate our results with various examples.