Coincidence points in the cases of metric spaces and metric maps

In the first half of the paper, we are concerned with the problems of existence (and searching) of coincidence points and the common preimage of a closed subset (in particular, a common root) in the case of a finite system of mappings of one metric space to another one. The second half of the paper is devoted to fiberwise variants of Arutyunov's theorem on coincidence points.Obtaining the main results of the paper is based on the use of the class of almost exactly (α,β)(α,β)-search functionals that is wider than Fomenko's class of (α,β)(α,β)-search functionals.