Approximation of Coincidence Points and Common Fixed Points of a Collection of Mappings of Metric Spaces

On a complete metric space $X$, we solve the problem of constructing an algorithm (in general, nonunique) of successive approximations from any point in space to a given closed subset $A$. We give an estimate of the distance from an arbitrary initial point to the corresponding limit points. We consider three versions of the subset $A$: (1) $A$ is the complete preimage of a closed subspace $H$ under a mapping from $X$ into the metric space $Y$; (2) $A$ is the set of coincidence points of $n$ ($n>1$) mappings from $X$ into $Y$; (3) $A$ is the set of common fixed points of $n$ mappings of $X$ into itself ($n=1,2,\dots$). The problems under consideration are stated conveniently in terms of a multicascade, i.e., of a generalized discrete dynamical system with phase space $X$, translation semigroup equal to the additive semigroup of nonnegative integers, and the limit set $A$. In particular, in case (2) for $n=2$, we obtain a generalization of Arutyunov's theorem on the coincidences of two mappings. In case (3) for $n=1$, we obtain a generalization of the contraction mapping principle.