On the equicontinuity of mappings by prime ends finitely connected on some set

This paper is devoted to the equicontinuity of families of mappings in terms of prime ends. Here, we develop Caratheodory’s ideas on boundary extension using the terminology of moduli of families of paths. We consider mappings with inverse Poletskii-type modulus inequalities in which the characteristic of quasiconformality is either integrable over spheres or satisfies a Lehto-Dini-type integral divergence condition. Under some very specific restrictions on the cluster set and the image under the mapping, we obtain a result on their global behavior in the closure of a domain. This global behavior is described for the case where the Euclidean metric is defined in the pre-image under the mapping, and the prime ends metric is defined in the corresponding image domain. The conditions under which the main result of the paper is obtained are very specific, since we refuse the closeness (preservation of the boundary of the domain) of the mappings.