On semi-homogeneous maps of degree $k$

TL;DR

This paper investigates properties of semi-homogeneous operators of degree k in Banach spaces, providing criteria for their compactness and continuity, with applications in pure and applied mathematics.

Contribution

It introduces necessary and sufficient conditions for semi-homogeneous operators to have vanishing measures of noncompactness and for their derivatives to be completely continuous.

Findings

Criteria for superposition operators to be improving

Conditions for the complete continuity of Fréchet derivatives

Characterizations of semi-homogeneous operators' properties

Abstract

We study properties of continuous semi-homogeneous operators of degree $k$ via various functions (e.g. measures of noncompactness) on all bounded subsets of a Banach space. We prove necessary and sufficient conditions for these functions to vanish on the image of the unit ball under these operators. In particular, we give criteriafor superposition operators to be improving and criteria for the complete continuity of the Fr\'echet derivatives. The results obtained can be applied in various areas of both pure and applied mathematics.