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On equicontinuity of mappings with variable domains

N. Ilkevych, D. Romash, E. Sevost'yanov

Abstract

The paper is devoted to the study of the global behavior of mappings. We consider families mappings with different definition domains which satisfy moduli inequalities of Poletskii type. Under some additional assumptions we have proved that such families are uniformly equicontinuous.

On equicontinuity of mappings with variable domains

Abstract

The paper is devoted to the study of the global behavior of mappings. We consider families mappings with different definition domains which satisfy moduli inequalities of Poletskii type. Under some additional assumptions we have proved that such families are uniformly equicontinuous.
Paper Structure (5 sections, 13 theorems, 54 equations, 2 figures)

This paper contains 5 sections, 13 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On image of the continuum under homeomorphisms
  4. Proof of the main results
  5. Main Lemma for mappings with a branching

Key Result

theorem 1.1

Let $p\in (n-1, n]$ and let $f_m\in \frak{F}_{Q, A, p, \delta}(D_0, \frak{D}),$$m=1,2,\ldots ,$ be a sequence such that: 1) the sequence of domains $D_m$ is regular with respect to $D_0;$ 2) for every $m\in {\Bbb N},$ a domain $D_m$ is locally connected on its boundary; 3) the family $f_m(D_m)$ is e for every $x_0\in \overline{D_0}$ and some $\beta(x_0)>0.$ Then the family $f_m,$$m=1,2,\ldots,$ is

Figures (2)

  • Figure 1: To prove Lemma \ref{['lem1']}
  • Figure 2: To prove Lemma \ref{['lem3']}

Theorems & Definitions (17)

  • theorem 1.1
  • theorem 1.2
  • lemma 2.1
  • lemma 2.2
  • proof
  • proposition 2.1
  • theorem 2.3
  • proposition 3.2
  • proposition 3.3
  • proof
  • ...and 7 more