An Arzel\`a-Ascoli theorem for the Hausdorff measure of noncompactness

TL;DR

This paper extends the classical Arzelà-Ascoli theorem to relate the Hausdorff measure of noncompactness with non-uniform equicontinuity in spaces of continuous functions, using Jung's Chebyshev radius.

Contribution

It provides a quantitative generalization of the Arzelà-Ascoli theorem connecting noncompactness measures and equicontinuity in function spaces.

Findings

Establishes a link between Hausdorff measure of noncompactness and equicontinuity.

Utilizes Jung's Chebyshev radius in the proof.

Offers a new quantitative perspective on classical compactness criteria.

Abstract

We generalize the Arzel\`a-Ascoli theorem in the space of continuous maps on a compact interval with values in Euclidean N-space by providing a quantitative link between the Hausdorff measure of noncompactness in this space and a natural measure of non-uniform equicontinuity. The proof hinges upon a classical result of Jung's on the Chebyshev radius.