Stability of cascade search

We find sufficient conditions on a searching multi-cascade for a modification of the set of limit points of the cascade that satisfy an assessing inequality for the distance from each of these points to the initial point to be small, provided that the modifications of the initial point and the initial set-valued functionals or maps used to construct the multi-cascade are small. Using this result, we prove the stability (in the above sense) of the cascade search for the set of common pre-images of a closed subspace under the action of $n$ set-valued maps, $n\geqslant1$ (in particular, for the set of common roots of these maps and for the set of their coincidences). For $n=2$ we obtain generalizations of some results of A. V. Arutyunov; the very statement of the problem comes from a recent paper of his devoted to the study of the stability of the subset of coincidences of a Lipschitz map and a covering map.