Continuity of modulus of noncompact convexity for minimalizable measures of noncompactness

TL;DR

This paper investigates the properties of the modulus of noncompact convexity linked to minimalizable measures of noncompactness, proving its subhomogeneity and continuity in Banach spaces with the Radon-Nikodym property.

Contribution

It establishes the subhomogeneity and continuity of the modulus of noncompact convexity for minimalizable measures in Banach spaces with the Radon-Nikodym property.

Findings

The modulus of noncompact convexity is subhomogeneous.

The modulus is continuous on the specified interval.

Results apply to Banach spaces with the Radon-Nikodym property.

Abstract

We consider the modulus of noncompact convexity $\Delta_{X,\phi}(\varepsilon)$ associated with the minimalizable measure of noncompactness $\phi$. We present some properties of this modulus, while the main result of this paper is showing that $\Delta_{X,\phi }(\varepsilon)$ is a subhomogenous and continuous function on $[0,\phi (\bar{B}_X))$ for an arbitrary minimalizable measure of compactness $\phi$ in the case of a Banach space $X$ with the Radon-Nikodym property.