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

TL;DR

This paper investigates the convergence properties of certain mappings in metric spaces, demonstrating conditions under which their uniform limits are light mappings and establishing equicontinuity of inverse mappings for specific classes.

Contribution

It introduces new conditions involving modulus inequalities and finite mean oscillation that ensure the lightness of uniform limits and inverse equicontinuity in metric space mappings.

Findings

Uniform limits of mappings satisfying modulus inequalities are light.

Finite mean oscillation ensures the lightness of the limit mappings.

Theorems on inverse equicontinuity for specific classes of homeomorphisms.

Abstract

For mappings in metric spaces satisfying one inequality with respect to modulus of families of curves, there is proved a lightness of the uniform limit of these mappings. It is proved that, the uniform limit of these mappings is light mapping, whenever a function which corresponds to distortion of families of curves, is of finite mean oscillation at every point. Besides that, for one class of homeomorphisms of metric spaces, there are obtained theorems about equicontinuity of inverse mappings.