Cascade search principle and its applications to the coincidence problems of n one-valued or multi-valued mappings

The problem of the construction of a multi-cascade with a given limit subset A is considered in a metric space X. A multi-cascade is a discrete multi-valued dynamic system with the translation semigroup (Z⩾0,+). The cascade search principle using so-called search functionals is suggested. It gives a solution of the problem. Also, an estimation is obtained for the distance between any initial point x and every correspondent limit point. Several applications of one-valued and multi-valued versions of the mentioned cascade search principle are given for the cases when the limit subset A is (1) the full (or expanded) preimage of a closed subspace under a mapping from X to another metric space; (2) the coincidence set (or expanded coincidence set) of n mappings from X to another metric space (n>1); (3) the common preimage (or the expanded one) of a closed subspace under n mappings; and (4) the common fixed point set of n mappings of the space X into itself (n⩾1). Generalizations of the previous authors results are obtained. And, in particular cases, generalizations of some recent results by A.V. Arutyunov on coincidences of two mappings and a generalization of Banach fixed point principle are obtained.