Darbo-type theorem for quasimeasure of noncompactness

TL;DR

This paper develops a new quasimeasure of noncompactness inspired by classical theorems, extending fixed point results to integral equations with applications in Banach spaces.

Contribution

It introduces a novel quasimeasure of noncompactness and extends Darbo's fixed point theorem using measure of nonconvexity for the analysis of integral equations.

Findings

Established a quasimeasure for $C^b(X,E)$ spaces.

Proved a Darbo-type fixed point theorem using the new quasimeasure.

Demonstrated existence of fixed points for Hammerstein operators with specific kernels.

Abstract

The paper introduces the concept of quasimeasure of noncompactness. Motivated by the Arzel\`a-Ascoli theorem for $C^b(X,E)$, where $X$ is an Euclidean space and $E$ an arbitrary Banach space, we construct a quasimeasure for this space and study its properties. An analogon for Darbo fixed point theorem is obtained with the additional aid of measure of nonconvexity. The paper ends with possible application in integral equations. We prove that a Hammerstein operator with Carath\'{e}odory kernel and nonlinearity of a certain type has a fixed point.