# Measures of Noncompactness and Condensing Operators

Authors: Akhmerov, Kamenskii, Potapov, Rodkina, Sadovskii

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A condensing (or densifying) operator is a mapping under which the image of any set is in a certain sense more compact than the set itself. The degree of noncompactness of a set is measured by means of functions called measures of noncompactness. The contractive maps and the compact maps [i.e., in this Introduction, the maps that send any bounded set into a relatively compact one; in the main text the term "compact" will be reserved for the operators that, in addition to having this property, are continuous, i.e., in the authors' terminology, for the completely continuous operators] are condensing. For contractive maps one can take as measure of noncompactness the diameter of a set, while for compact maps can take the indicator function of a family of non-relatively com­ pact sets. The operators of the form F( x) = G( x, x), where G is contractive in the first argument and compact in the second, are also condensing with respect to some natural measures of noncompactness. The linear condensing operators are characterized by the fact that almost all of their spectrum is included in a disc of radius smaller than one. The examples given above show that condensing operators are a sufficiently typical phenomenon in various applications of functional analysis, for example, in the theory of differential and integral equations. As is turns out, the condensing operators have properties similar to the compact ones.

• Measures of Noncompactness

Akhmerov, R. R. (et al.)

Pages 1-52

• The Linear Theory

Akhmerov, R. R. (et al.)

Pages 53-97

• The Fixed-Point Index of Condensing Operators

Akhmerov, R. R. (et al.)

Pages 99-150

• Applications

Akhmerov, R. R. (et al.)

Pages 151-228

eBook $59.99 price for USA in USD (gross) • ISBN 978-3-0348-5727-7 • Digitally watermarked, DRM-free • Included format: PDF • ebooks can be used on all reading devices • Immediate eBook download after purchase Softcover$79.99
price for USA in USD
• ISBN 978-3-0348-5729-1
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## Bibliographic Information

Bibliographic Information
Book Title
Measures of Noncompactness and Condensing Operators
Authors
Series Title
Series Volume
55
1992
Publisher
Birkhäuser Basel
Springer Basel AG
eBook ISBN
978-3-0348-5727-7
DOI
10.1007/978-3-0348-5727-7
Softcover ISBN
978-3-0348-5729-1
Series ISSN
0255-0156
Edition Number
1
Number of Pages
VIII, 252
Topics