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arXiv · 1605.05229

Darbo-type theorem for quasimeasure of noncompactness

Mateusz Krukowski
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Abstract

The paper introduces the concept of quasimeasure of noncompactness. Motivated by the Arzelà-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 Carathéodory kernel and nonlinearity of a certain type has a fixed point.

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Mateusz Krukowski. 2016-05-17. Darbo-type theorem for quasimeasure of noncompactness. https://arxiv.org/abs/1605.05229

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