A quantitative version of the Arzelà-Ascoli theorem based on the degree of nondensifiability and applications

Abstract

We present a novel result that, in a certain sense, generalizes the Arzelà-Ascoli theorem. Our main tool will be the so called degree of nondensifiability, which is not a measure of noncompactness but canbe used as an alternative tool in certain fixed problems where such measures do not work out. To justify our results, we analyze the existence of continuous solutions of certain Volterra integral equations defined by vector valued functions.

Keywords

• Arzelà-Ascoli theorem
• degree of nondensifiability
• α-dense curves
• measures of noncompactness
• Volterra integral equations