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Measures of Noncompactness [PDF]

, 1992
In this chapter we consider the basic notions connected with measures of noncompactness (MNCs for brevity) and condensing (or densifying) operators. We define and study in detail the three main and most frequently used MNCs: the Hausdorff MNC χ the Kuratowski MNC α, and the MNC β.
R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina, B. N. Sadovskii   +4 more
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On measures of weak noncompactness

Annali Di Matematica Pura Ed Applicata, 1988
The authors give an axiomatic definition of measures of weak noncompactness which is in some sense parallel to \textit{B. N. Sadovskij}'s definition of measures of (strong) noncompactness [see e.g. Usp. Mat. Nauk 27, No.1, 81-146 (1972; Zbl 0243.47033)]. The first explicit measure of weak noncompactness is due to \textit{F. S. de Blasi} [Bull.
Józef Banas
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Inequivalent measures of noncompactness

Annali Di Matematica Pura Ed Applicata, 2010
Let \(X\) be a Banach space and \({\mathcal B}(X)\) denote the set of all bounded subsets of \(X\). We say that a map \(\beta:{\mathcal B}(X)\to [0,\infty)\) is a homogeneous measure of noncompactness on \(X\) if for all \(S,T\in{\mathcal B}(X)\): (1) \(\beta(S)= 0\) iff \(\overline S\) is compact, (2) \(\beta(S)\leq\beta(T)\) for all \(S\subset T ...
John Mallet-Paret, Roger D Nussbaum
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An application of a measure of noncompactness in the study of asymptotic stability [PDF]

Applied Mathematics Letters, 2003
Using the technique of fixed-point theorem of Darbo type associated with measures of noncompactness, we obtain an existence result for some functional-integral equation. Moreover, the choice of suitable measure of noncompactness allows us to characterize
J Banas
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Applications of the Hausdorff measure of noncompactness in some sequence spaces of weighted means [PDF]

Computers and Mathematics With Applications, 2010
In the present paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain operators on some sequence spaces of weighted means.
M Mursaleen, Abdullah K Noman
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Measures of Noncompactness

1997
As we have seen in Chapter I, compactness plays an essential role in the proof of the Schauder fixed point theorem. However, there are some important problems where the operators are not compact.
J. M. Ayerbe Toledano, T. Domínguez Benavides, G. López Acedo   +2 more
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Measure of Noncompactness and Spectral Theory

Mathematische Nachrichten, 1984
Using the theory of measure of noncompactness the author has extended the results of J. Leray on the spectral theory to the case of noncompact operators in Fréchet spaces. Applying these results the author has estimated the radius of the essential spectrum of operators and has obtained some results in operator theory.
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Compactness in measure and measure of noncompactness

Siberian Mathematical Journal, 1997
In the class of Banach function spaces with order continuous norm, the author reduces the notion of compactness in measure for a subset of a function space to some equality between two numerical characteristics of the subset.
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Measures of noncompactness of interpolated polynomials

Forum Mathematicum, 2022
Abstract We study interpolation of the measure of noncompactness of homogeneous polynomials on Banach spaces. We prove that, for a large class of interpolation functors, preserving interpolation of measures of noncompactness of interpolated linear operators between Banach couples can be lifted to polynomials.
Mastyło, Mieczysław, da Silva, Eduardo Brandani   +1 more
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On controllability and measures of noncompactness

AIP Conference Proceedings, 2014
This article deals with an infinitely dimensional nonlinear dynamical systems given in a state space form. Among such systems we distinguish a wide class of semilinear systems, for which we present a set of controllability conditions. These conditions for controllability are based on a fixed point theorems and measures of noncompactness.
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