Abstract:
The present paper is a continuation of the previous papers of the author dealing with this subject. We study the cascade search of a given subset $A$, i.e., the construction on the metric space $X$ of a multicascade with given limit subset $A$ in $X$. A multicascade is a multivalued dynamical system with translation semigroup equal to the additive semigroup of nonnegative integers. We propose a finer (than in the author's previous papers) version of cascade search for cases in which (1) $A$ is the complete preimage of a closed subspace under the multivalued mapping of metric spaces; (2) $A$ is the set of coincidence points $n$, $n>1$, of the multivalued mappings. An estimate of the distance from the initial to any corresponding limit point is given. In particular, in case (2), a new generalization of a recent theorem due to Arutyunov is obtained for $n=2$.
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