Existence of a coincidence point in a critical case when the covering constant and the Lipschitz constant are equal

We consider two mappings acting between metric spaces and such that one of them is covering and the other satisfies the enhanced Lipschitz property. It is assumed here that the covering constant and the Lipschitz constant of these mappings are equal. We prove the result of the existence of a coincidence point of single-valued mappings in the case when the series of iterations of the function that provides execution of the enhanced Lipschitz property converges. We prove the similar result for set-valued mappings. We provide examples of functions for which the series of their iterations converges or diverges.