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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 Banaś
exaly   +2 more sources

Application of Measure of Noncompactness to Infinite Systems of Differential Equations

Canadian Mathematical Bulletin, 2013
In this paper we determine theHausdorff measure of noncompactness on the sequence space n(ϕ) ofW. L. C. Sargent. Further we apply the technique of measures of noncompactness to the theory of infinite systems of differential equations in the Banach ...
Mohammad Mursaleen
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Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness

, 2017
This book offers a comprehensive treatment of the theory of measures of noncompactness. It discusses various applications of the theory of measures of noncompactness, in particular, by addressing the results and methods of fixed-point theory. The concept
Jleli, Mohamed, Vetro, Calogero, Mursaleen, Mohammad, Samet, Bessem, Banaś, Józef*, Banaś, Józef   +5 more
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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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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 ...
Mallet-Paret, John, Nussbaum, Roger D.
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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 and Their Applications

2017
In this chapter, we present a survey of theory and applications of measures of noncompactness. The standard measures of noncompactness are discussed and their properties are compared. Some results concerning standard measures of noncompactness in different spaces including \(C([a,b];\mathbb {R})\), \(L^p([a,b];\mathbb {R})\), Banach spaces with ...
Mohammad Mursaleen, Syed M. H. Rizvi, Bessem Samet   +2 more
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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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