Regular measures of noncompactness and Ascoli-Arzel\` a type compactness criteria in spaces of vector-valued function

TL;DR

This paper develops new measures of noncompactness for spaces of vector-valued functions using a novel equicontinuity concept, leading to refined compactness criteria akin to Ascoli-Arzelà theorems.

Contribution

It introduces regular measures of noncompactness in vector-valued function spaces based on a new equicontinuity concept, extending classical results.

Findings

Derived inequalities for measures of noncompactness

Established new compactness criteria similar to Ascoli-Arzelà

Provided precise formulas for specific classes of subsets

Abstract

In this paper we estimate the Kuratowski and the Hausdorff measures of noncompactness of bounded subsets of spaces of vector-valued bounded functions and of vector-valued bounded differentiable functions. To this end, we use a quantitative characteristic modeled on a new equiconti\-nuity-type concept and classical quantitative characteristics related to pointwise relative compactness. We obtain new regular measures of noncompactness in the spaces taken into consideration. The established inequalities reduce to precise formulas in some classes of subsets. We derive Ascoli-Arzel\` a type compactness criteria.