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 Uspekhi Mat. Nauk, 1972, Volume 27, Issue 1(163), Pages 81–146 (Mi umn5007)

Limit-compact and condensing operators

Abstract: The paper contains a survey of investigations concerned with three new concepts: limit-compact operators, measures of non-compactness, and condensing operators. A measure of non-compactness is a function of a set that is invariant under the transition to the closed convex hull of the set. If a certain measure of non-compactness is defined in a space, a condensing operator is defined, roughly speaking, as an operator that decreases the measure of non-compactness of any set whose closure is not compact. The more general concept of a limit-compact operator is defined by means of a property common to all condensing operators; it can be formulated in terms not related to measures of non-compactness. The theory of limit-compact operators can be regarded as a simultaneous generalization of the theory of completely continuous and contracting operators. For non-linear operators the main result is the construction of the theory of the rotation of limit-compact vector fields and, in particular, the proof of a number of new fixed-point principles (Chapter 3 of the present paper). In the theory of linear operators a number of results are obtained that are related to the concept of a Fredholm operator and the Fredholm spectrum of an operator (Chapter 2). The theory of measures of non-compactness and condensing operators has found different applications in general topology, in the theory of ordinary differential equations, functional-differential equations, partial differential equations, the theory of extrema of functionals, etc. The paper contains several examples concerning differential equations in a Banach space and functional-differential equations of neutral type. These examples do not have a special significance but are chosen merely to illustrate the methods. They are therefore investigated with neither maximal generality nor completeness.

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English version:
Russian Mathematical Surveys, 1972, 27:1, 85–155

Bibliographic databases:

UDC: 517.43
MSC: 46A50, 54D45, 46A03, 47A53

Citation: B. N. Sadovskii, “Limit-compact and condensing operators”, Uspekhi Mat. Nauk, 27:1(163) (1972), 81–146; Russian Math. Surveys, 27:1 (1972), 85–155

Citation in format AMSBIB
\Bibitem{Sad72} \by B.~N.~Sadovskii \paper Limit-compact and condensing operators \jour Uspekhi Mat. Nauk \yr 1972 \vol 27 \issue 1(163) \pages 81--146 \mathnet{http://mi.mathnet.ru/umn5007} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=428132} \zmath{https://zbmath.org/?q=an:0232.47067|0243.47033} \transl \jour Russian Math. Surveys \yr 1972 \vol 27 \issue 1 \pages 85--155 \crossref{https://doi.org/10.1070/RM1972v027n01ABEH001364} 

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