  This paper investigates the local asymptotic stabilization of a very general
class of instable autonomous nonlinear difference equations which are subject
to perturbed dynamics which can have a different order that that of the nominal
difference equation. In the general case, the controller consists of two
combined parts, namely, the feedback nominal controller which stabilizes the
nominal (i.e. perturbation - free) difference equation plus an incremental
controller which completes the stabilization in the presence of dynamics in the
uncontrolled difference equation. A stabilization variant consists of using a
single controller to stabilize the nominal difference equation and also the
perturbed one under a smallness-type characterization of the perturbed
dynamics. The study is based on Banach fixed point principle and it is also
valid with slight modification for the stabilization of unstable oscillatory
solutions.
