Journal ArticleDOI
Limit-compact and condensing operators
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In this paper, the authors present a survey of work concerning limit-compact operators, measures of noncompactness, and condensing operators, which is a generalization of the theory of completely continuous and contracting 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.read more
Citations
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Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces
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On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions
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On the axiomatic theory of spectrum II
Vladimír Kordula,Vladimír Müller +1 more
TL;DR: In this paper, the authors give an axiomatic theory for spectral regularities, which is a generalization of the notion of subspectrum, which was introduced by W.Żelazko.
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Topological methods in the fixed-point theory of multi-valued maps
TL;DR: In this paper, the authors present a generalization of the Vietoris-Begle-Sklyarenko theorem to the fixed-point theory of multi-valued vector fields.
References
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Topologie et équations fonctionnelles
Jean Leray,Jules Schauder +1 more
TL;DR: In this paper, Gauthier-Villars implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions).