On the iterates of Chatterjea contraction

The question of the relation between contractive conditions may be extended to the problem of their iterates. It was shown by B. Fisher that for a Banach contraction there exists an iterate that fulfills Chatterjea contractive condition, but the reverse relation holds under some restriction imposed on the metric space which assures that there exists an iterate of a Chatterjea contraction that is a Banach contraction. However, the proposed restriction holds only for the identity mapping which is not a Chatterjea contraction except for the singleton domain. We offer a possible adjustment of this approach with several examples answering some open questions on this topic.