  We present different extensions of the Banach contraction principle in the
$G$-metric space setting. More precisely, we consider mappings for which the
contractive condition is satisfied by a power of the mapping and for which the
power depends on the specified point in the space. We first state the result in
the continuous case and later, show that the continuity is indeed not
necessary. Imitating some techniques obtained in the metric case, we prove that
under certain conditions, it is enough for the contractive condition to be
verified on a proper subset of the space under consideration. These results
generalize well known comparable results.
