On convergence of mappings in metric spaces with direct and inverse modulus conditions

Abstract

For mappings in metric spaces satisfying one inequality with respect to the modulus of families of curves, we establish the property of lightness of the limit mapping. It is shown that the uniform limit of these mappings is a light mapping, whenever the function responsible for the distortion of the families of curves, is of finite mean oscillation at every point. In addition, for one class of homeomorphisms of metric spaces, we prove theorems on the equicontinuity of the families of inverse mappings.