Regular measures of noncompactness and Ascoli–Arzelà type compactness criteria in spaces of vector-valued functions

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 equicontinuity-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–Arzelà type compactness criteria.