On the Equicontinuity of Unclosed Orlicz-Sobolev Classes by Prime Ends

We study the equicontinuity of Orlicz-Sobolev classes by prime ends. We investigate the case when the mappings are open, discrete, but not boundary preserving (closed). It is established that, under certain conditions on the cluster set, the family of above mappings is equicontinuous in the terms of prime ends whenever the corresponding Orlicz function satisfies the Calderon condition, and the inner dilatation of mappings has a majorant satisfying the integral divergence Lehto-Dini condition in the closure of a domain. A similar result was also obtained in the case where the corresponding majorant has a finite mean oscillation (FMO) at each point of the closure of a domain.