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Nonlinear Spectral Theory : De Gruyter Series in Nonlinear Analysis & Applications - Jurgen Appell

Nonlinear Spectral Theory

De Gruyter Series in Nonlinear Analysis & Applications


Published: 26th April 2004
For Ages: 22+ years old
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In view of the eminent importance of spectral theory of linear operators in many fields of mathematics and physics, it is not surprising that various attempts have been made to define and study spectra also for nonlinear operators. This book provides a comprehensive and self-contained treatment of the theory, methods, and applications of nonlinear spectral theory.

The first chapter briefly recalls the definition and properties of the spectrum and several subspectra for bounded linear operators. Then some numerical characteristics for nonlinear operators are introduced which are useful for describing those classes of operators for which there exists a spectral theory. Since spectral values are closely related to solvability results for operator equations, various conditions for the local or global invertibility of a nonlinear operator are collected in the third chapter. The following two chapters are concerned with spectra for certain classes of continuous, Lipschitz continuous, or differentiable operators. These spectra, however, simply adapt the corresponding definitions from the linear theory which somehow restricts their applicability. Other spectra which are defined in a completely different way, but seem to have useful applications, are defined and studied in the following four chapters. The remaining three chapters are more application-oriented and deal with nonlinear eigenvalue problems, numerical ranges, and selected applications to nonlinear problems.

The only prerequisite for understanding this book is a modest background in functional analysis and operator theory. It is addressed to non-specialists who want to get an idea of the development of spectral theory for nonlinear operators in the last 30 years, as well as a glimpse of the diversity of the directions in which current research is moving.

"This text will certainly serve as an excellent reference book for many years."EMS Newsletter 3/2005 "The book can be considered as a systematic introduction to this area, emphasizing the diversity of directions in which current research in nonlinear spectral ist developing."S. Cobzas in: Studia Universitatis Babes-Boyai 1/2005 "This carefully written book is a comprehensive treatment of the subject including new results several of them archieved bythe author. [...] This research monograph is recommended to scientists interested in the theory of nonlinar operators and their applications."R. Schumann in ZAA 4/2004

Prefacep. vii
Introductionp. 1
Spectra of Bounded Linear Operatorsp. 11
The spectrum of a bounded linear operatorp. 11
Compact and [alpha]-contractive linear operatorsp. 16
Subdivision of the spectrump. 23
Essential spectra of bounded linear operatorsp. 30
Notes, remarks and referencesp. 33
Some Characteristics of Nonlinear Operatorsp. 40
Some metric characteristics of nonlinear operatorsp. 40
A list of examplesp. 43
Compact and [alpha]-contractive nonlinear operatorsp. 51
Special subsets of scalarsp. 61
Notes, remarks and referencesp. 64
Invertibility of Nonlinear Operatorsp. 67
Proper and ray-proper operatorsp. 67
Coercive and ray-coercive operatorsp. 73
Further properties of nonlinear operatorsp. 76
The mapping spectrump. 78
Excursion: topological degree theoryp. 85
Notes, remarks and referencesp. 92
The Rhodius and Neuberger Spectrap. 94
The Rhodius spectrump. 94
Frechet differentiable operatorsp. 96
The Neuberger spectrump. 100
Special classes of operatorsp. 103
Notes, remarks and referencesp. 107
The Kachurovskij and Dorfner Spectrap. 109
Lipschitz continuous operatorsp. 109
The Kachurovskij spectrump. 111
The Dorfner spectrump. 118
Restricting the Kachurovskij spectrump. 120
Semicontinuity properties of spectrap. 121
Continuity properties of resolvent operatorsp. 123
Notes, remarks and referencesp. 125
The Furi-Martelli-Vignoli Spectrump. 130
Stably solvable operatorsp. 130
FMV-regular operatorsp. 135
The FMV-spectrump. 139
Subdivision of the FMV-spectrump. 141
Special classes of operatorsp. 144
The AGV-spectrump. 148
Notes, remarks and referencesp. 155
The Feng Spectrump. 159
Epi and k-epi operatorsp. 159
Feng-regular operatorsp. 166
The Feng spectrump. 170
Special classes of operatorsp. 173
A comparison of different spectrap. 178
Notes, remarks and referencesp. 181
The Vath Phantomp. 184
Strictly epi operatorsp. 184
The phantom and the large phantomp. 186
The point phantomp. 192
Special classes of operatorsp. 199
A comparison of spectra and phantomsp. 206
Notes, remarks and referencesp. 212
Other Spectrap. 214
The semilinear Feng spectrump. 214
The semilinear FMV-spectrump. 223
The pseudo-adjoint spectrump. 228
The Singhof-Weyer spectrump. 232
The Weber spectrump. 240
Spectra for homogeneous operatorsp. 243
The Infante-Webb spectrump. 253
Notes, remarks and referencesp. 263
Nonlinear Eigenvalue Problemsp. 268
Classical eigenvaluesp. 268
Eigenvalue problems in conesp. 275
A nonlinear Krejn-Rutman theoremp. 280
Other notions of eigenvaluep. 285
Connected eigenvaluesp. 294
Notes, remarks and referencesp. 296
Numerical Ranges of Nonlinear Operatorsp. 303
Linear operators in Hilbert spacesp. 303
Linear operators in Banach spacesp. 307
Numerical ranges of nonlinear operatorsp. 313
Numerical ranges and Jordan domainsp. 327
Notes, remarks and referencesp. 329
Some Applicationsp. 336
Solvability of nonlinear equationsp. 336
Solvability of semilinear equationsp. 341
Applications to boundary value problemsp. 350
Bifurcation and asymptotic bifurcation pointsp. 360
The p-Laplace operatorp. 366
Notes, remarks and referencesp. 369
Referencesp. 375
List of Symbolsp. 395
Indexp. 401
Table of Contents provided by Rittenhouse. All Rights Reserved.

ISBN: 9783110181432
ISBN-10: 3110181436
Series: De Gruyter Series in Nonlinear Analysis & Applications
Audience: Professional
For Ages: 22+ years old
Format: Hardcover
Language: English
Number Of Pages: 419
Published: 26th April 2004
Publisher: De Gruyter
Country of Publication: DE
Dimensions (cm): 24.0 x 17.0  x 2.39
Weight (kg): 0.83