Operator Theory: Advances and Applications

# Measures of Noncompactness and Condensing Operators

Authors: Akhmerov, Kamenskii, Potapov, Rodkina, Sadovskii

Free Preview

### Buy this book

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
• Free shipping for individuals worldwide
• Usually dispatched within 3 to 5 business days.
About this book

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.

## Table of contents (4 chapters)

Table of contents (4 chapters)
• 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

### Buy this book

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
• Free shipping for individuals worldwide
• Usually dispatched within 3 to 5 business days.

## Bibliographic Information

Bibliographic Information
Book Title
Measures of Noncompactness and Condensing Operators
Authors
Series Title
Operator Theory: Advances and Applications
Series Volume
55
Copyright
1992
Publisher
Birkhäuser Basel
Copyright Holder
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