Cascade Search for Preimages and Coincidences: Global and Local Versions

In previous papers of the author, the cascade search principle was proposed, which makes it possible to construct a set-valued self-map of a metric space $X$ from a set-valued functional or a collection of set-valued maps of $X$ so that the new map generates a multicascade, i.e., a set-valued discrete dynamical system whose limit set coincides with the zero set of the given functional, with the coincidence set of the given collection, or with the common preimage of a closed subspace under the maps from this collection. Stability issues of cascade search were studied. This paper is devoted to a generalization and local modifications of the cascade search principle and their applications to problems concerning local search and approximation of common preimages of subspaces and coincidence sets for finite collections of set-valued maps of metric spaces.