Fixed point theory:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
2003
|
| Schriftenreihe: | Springer monographs in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 620 - 667 |
| Beschreibung: | XV, 690 S. Ill., graph. Darst. |
| ISBN: | 9780387001739 |
Internformat
MARC
| LEADER | 00000nam a2200000 c 4500 | ||
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| 001 | BV016972742 | ||
| 003 | DE-604 | ||
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| 008 | 030311s2003 gw ad|| |||| 00||| eng d | ||
| 016 | 7 | |a 966479378 |2 DE-101 | |
| 020 | |a 9780387001739 |9 978-0-387-00173-9 | ||
| 035 | |a (OCoLC)845546552 | ||
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| 084 | |a MAT 549f |2 stub | ||
| 084 | |a 27 |2 sdnb | ||
| 084 | |a MAT 476f |2 stub | ||
| 100 | 1 | |a Granas, Andrzej |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Fixed point theory |c Andrzej Granas ; James Dugundji |
| 264 | 1 | |a New York [u.a.] |b Springer |c 2003 | |
| 300 | |a XV, 690 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer monographs in mathematics | |
| 500 | |a Literaturverz. S. 620 - 667 | ||
| 650 | 4 | |a Fixed point theory | |
| 650 | 0 | 7 | |a Fixpunkttheorie |0 (DE-588)4293945-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Fixpunkttheorie |0 (DE-588)4293945-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Dugundji, James |e Verfasser |4 aut | |
| 856 | 4 | 2 | |m DNB Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010250159&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-010250159 | |
Datensatz im Suchindex
| _version_ | 1826212301150093312 |
|---|---|
| adam_text |
ANDRZEJ GRANAS JAMES DUGUNDJI
FIXED POINT THEORY
WITH 14 ILLUSTRATIONS
%1
SPRINGER
CONTENTS
PREFACE
VII
§0.
INTRODUCTION
1
1. FIXED POINT SPACES
1
2.
FORMING
NEW
FIXED POINT SPACES FROM
OLD 3
3.
TOPOLOGICAL TRANSVERSALITY
4
4.
FACTORIZATION TECHNIQUE
6
I. ELEMENTARY FIXED POINT THEOREMS
§1.
RESULTS BASED
ON
COMPLETENESS
9
1.
BANACH CONTRACTION PRINCIPLE
9
2.
ELEMENTARY DOMAIN INVARIANCE
11
3.
CONTINUATION METHOD
FOR
CONTRACTIVE MAPS
12
4.
NONLINEAR ALTERNATIVE
FOR
CONTRACTIVE MAPS
13
5.
EXTENSIONS
OF
THE
BANACH THEOREM
15
6. MISCELLANEOUS RESULTS
AND
EXAMPLES
17
7. NOTES
AND
COMMENTS
23
§2.
ORDER-THEORETIC RESULTS
25
1.
THE
KNASTER-TARSKI THEOREM
25
2.
ORDER
AND
COMPLETENESS. THEOREM
OF
BISHOP-PHELPS
26
3.
FIXED POINTS
FOR
SET-VALUED CONTRACTIVE MAPS
28
4.
APPLICATIONS
TO
GEOMETRY
OF
BANACH SPACES
29
5.
APPLICATIONS
TO THE
THEORY
OF
CRITICAL POINTS
30
6. MISCELLANEOUS RESULTS
AND
EXAMPLES
31
7. NOTES
AND
COMMENTS
34
X CONTENTS
§3.
RESULTS BASED
ON
CONVEXITY
37
1. KKM-MAPS
AND THE
GEOMETRIC KKM-PRINCIPLE
37
2.
THEOREM
OF VON
NEUMANN
AND
SYSTEMS
OF
INEQUALITIES
40
3.
FIXED POINTS
OF
AFFINE MAPS. MARKOFF-KAKUTANI THEOREM
42
4.
FIXED POINTS
FOR
FAMILIES
OF
MAPS. THEOREM
OF
KAKUTANI
44
5.
MISCELLANEOUS RESULTS
AND
EXAMPLES
46
6. NOTES
AND
COMMENTS
48
§4.
FURTHER RESULTS
AND
APPLICATIONS
51
1. NONEXPANSIVE MAPS
IN
HILBERT SPACE
51
2.
APPLICATIONS
OF THE
BANACH PRINCIPLE
TO
INTEGRAL
AND
DIFFERENTIAL
EQUATIONS
55
3.
APPLICATIONS
OF THE
ELEMENTARY DOMAIN INVARIANCE
57
4.
ELEMENTARY KKM-PRINCIPLE
AND ITS
APPLICATIONS
64
5.
THEOREMS
OF
MAZUR-ORLICZ
AND
HAHN-BANACH
70
6. MISCELLANEOUS RESULTS
AND
EXAMPLES
74
7. NOTES
AND
COMMENTS
81
II.
THEOREM
OF
BORSUK
AND
TOPOLOGICAL TRANSVERSALITY
§5.
THEOREMS
OF
BROUWER
AND
BORSUK
85
1. PRELIMINARY REMARKS
85
2.
BASIC TRIANGULATION
OF
S
N
86
3.
A
COMBINATORIAL LEMMA
88
4.
THE
LUSTERNIK-SCHNIRELMANN-BORSUK THEOREM
90
5.
EQUIVALENT FORMULATIONS.
THE
BORSUK-ULAM THEOREM
92
6.
SOME SIMPLE CONSEQUENCES
94
7. BROUWER'S THEOREM
95
8. TOPOLOGICAL KKM-PRINCIPLE
96
9. MISCELLANEOUS RESULTS
AND
EXAMPLES
98
10.
NOTES
AND
COMMENTS
104
§6.
FIXED POINTS FOR COMPACT MAPS IN NORMED LINEAR SPACES
112
1.
COMPACT
AND
COMPLETELY CONTINUOUS OPERATORS
112
2.
SCHAUDER PROJECTION
AND
APPROXIMATION THEOREM
116
3.
EXTENSION
OF THE
BROUWER
AND
BORSUK THEOREMS
119
4.
TOPOLOGICAL TRANSVERSALITY. EXISTENCE
OF
ESSENTIAL MAPS
120
5.
EQUATION
X
=
F{X).
THE
LERAY-SCHAUDER PRINCIPLE
123
6.
EQUATION
X =
XF(X).
BIRKHOFF-KELLOGG THEOREM
125
CONTENTS
XI
7. COMPACT FIELDS
126
8. EQUATION
Y
= X
-
F(X).
INVARIANCE
OF
DOMAIN
128
9.
MISCELLANEOUS RESULTS
AND
EXAMPLES
131
10.
NOTES
AND
COMMENTS
137
§7.
FURTHER RESULTS
AND
APPLICATIONS
142
1. APPLICATIONS
OF
THE TOPOLOGICAL KKM-PRINCIPLE
142
2.
SOME APPLICATIONS
OF
THE ANTIPODAL THEOREM
151
3.
THE
SCHAUDER THEOREM
AND
DIFFERENTIAL EQUATIONS
154
4.
TOPOLOGICAL TRANSVERSALITY
AND
DIFFERENTIAL EQUATIONS
156
5.
APPLICATION
TO THE
GALERKIN APPROXIMATION THEORY
158
6. THE INVARIANT SUBSPACE PROBLEM
160
7. ABSOLUTE RETRACTS
AND
GENERALIZED SCHAUDER THEOREM
162
8.
FIXED POINTS
FOR
SET-VALUED KAKUTANI MAPS
166
9. THEOREM
OF
RYLL-NARDZEWSKI
171
10.
MISCELLANEOUS RESULTS
AND
EXAMPLES
175
11.
NOTES
AND
COMMENTS
190
III.
HOMOLOGY AND FIXED POINTS
§8.
SIMPLICIAL HOMOLOGY
197
1. SIMPLICIAL COMPLEXES
AND
POLYHEDRA
197
2.
SUBDIVISIONS
200
3.
SIMPLICIAL MAPS
AND
SIMPLICIAL APPROXIMATIONS
201
4.
VERTEX SCHEMES, REALIZATIONS,
AND
NERVES
OF
COVERINGS
203
5.
SIMPLICIAL HOMOLOGY
205
6. CHAIN TRANSFORMATIONS
AND
CHAIN HOMOTOPIES
208
7. INDUCED HOMOMORPHISM
212
8. TRIANGULATED SPACES
AND
POLYTOPES
214
9. RELATIVE HOMOLOGY
215
10.
MISCELLANEOUS RESULTS
AND
EXAMPLES
219
11.
NOTES
AND
COMMENTS
221
§9.
THE
LEFSCHETZ-HOPF THEOREM
AND
BROUWER DEGREE
223
1.
ALGEBRAIC PRELIMINARIES
223
2.
THE LEFSCHETZ-HOPF FIXED POINT THEOREM
226
3.
THE
EULER NUMBER
OF A
MAP. PERIODIC POINTS
229
4.
APPLICATIONS
231
5.
THE BROUWER DEGREE
OF
MAPS
S
N
-
S
N
234
XLI CONTENTS
6. THEOREM
OF
BORSUK-HIRSCH
236
7. MAPS
OF
EVEN-
AND OF
ODD-DIMENSIONAL SPHERES
237
8.
DEGREE
AND
HOMOTOPY. THEOREM
OF
HOPF
239
9. VECTOR FIELDS
ON
SPHERES
241
10.
MISCELLANEOUS RESULTS
AND
EXAMPLES
243
11.
NOTES
AND
COMMENTS
245
IV. LERAY-SCHAUDER DEGREE
AND
FIXED POINT INDEX
§10.
TOPOLOGICAL DEGREE IN
R
N
249
1.
PL
MAPS
OF
POLYHEDRA
250
2.
POLYHEDRAL DOMAINS
IN
R
N
.
DEGREE
FOR
GENERIC MAPS
251
3.
LOCAL CONSTANCY
AND
HOMOTOPY INVARIANCE
254
4.
DEGREE
FOR
CONTINUOUS MAPS
258
5.
SOME PROPERTIES
OF
DEGREE
260
6. EXTENSION
TO
ARBITRARY OPEN SETS
262
7. AXIOMATICS
263
8.
THE
MAIN THEOREM
ON THE
BROUWER DEGREE
IN
R
N
266
9. EXTENSION
OF THE
ANTIPODAL THEOREM
268
10.
MISCELLANEOUS RESULTS
AND
EXAMPLES
270
11.
NOTES
AND
COMMENTS
274
§11.
ABSOLUTE NEIGHBORHOOD RETRACTS
279
1. GENERAL PROPERTIES
279
2.
ARS AND
ANRS
280
3.
LOCAL PROPERTIES
281
4.
PASTING ANRS TOGETHER
283
5.
THEOREM
OF
HANNER
285
6. HOMOTOPY PROPERTIES
287
7. GENERALIZED LERAY-SCHAUDER PRINCIPLE
IN
ANRS
289
8.
MISCELLANEOUS RESULTS
AND
EXAMPLES
292
9. NOTES
AND
COMMENTS
300
§12.
FIXED POINT INDEX
IN
ANRS
305
1. FIXED POINT INDEX
IN
R
N
305
2.
AXIOMS
FOR THE
INDEX
308
3.
THE
LERAY-SCHAUDER INDEX
IN
NORMED LINEAR SPACES
309
4.
COMMUTATIVITY
OF THE
INDEX
312
5.
FIXED POINT INDEX
FOR
COMPACT MAPS
IN
ANRS
315
CONTENTS XLLL
6. THE LERAY-SCHAUDER CONTINUATION PRINCIPLE
IN
ANRS
317
7.
SIMPLE CONSEQUENCES AND INDEX CALCULATIONS
321
8.
LOCAL INDEX
OF
AN ISOLATED FIXED POINT
326
9. MISCELLANEOUS RESULTS AND EXAMPLES
329
10.
NOTES AND COMMENTS
333
§13.
FURTHER RESULTS AND APPLICATIONS
338
1.
BIFURCATION RESULTS
IN
ANRS
338
2.
APPLICATION OF THE INDEX
TO
NONLINEAR PDES
344
3.
THE LERAY-SCHAUDER DEGREE
348
4.
EXTENSIONS OF THE BORSUK AND BORSUK-ULAM THEOREMS
352
5.
THE LERAY-SCHAUDER INDEX
IN
LOCALLY CONVEX SPACES
354
6.
MISCELLANEOUS RESULTS AND APPLICATIONS
357
7.
NOTES AND COMMENTS
364
V. THE LEFSCHETZ-HOPF THEORY
§14.
SINGULAR HOMOLOGY
369
1.
SINGULAR CHAIN COMPLEX AND HOMOLOGY FUNCTORS
369
2.
INVARIANCE OF HOMOLOGY UNDER BARYCENTRIC SUBDIVISION
378
3.
EXCISION
384
4.
AXIOMATIZATION
386
5.
COMPARISON OF HOMOLOGIES. KIINNETH THEOREM
391
6.
HOMOLOGY AND TOPOLOGICAL DEGREE
397
7.
MISCELLANEOUS RESULTS AND EXAMPLES
402
8. NOTES AND COMMENTS
409
§15.
LEFSCHETZ THEORY
FOR
MAPS
OF
ANRS
413
1.
THE LERAY TRACE
413
2.
GENERALIZED LEFSCHETZ NUMBER
418
3.
LEFSCHETZ MAPS AND LEFSCHETZ SPACES
420
4.
LEFSCHETZ THEOREM
FOR
COMPACT MAPS
OF
ANRS
423
5.
ASYMPTOTIC FIXED POINT THEOREMS FOR ANRS
425
6. BASIC CLASSES OF LOCALLY COMPACT MAPS
426
7.
ASYMPTOTIC LEFSCHETZ-TYPE RESULTS IN ANRS
429
8.
PERIODICITY INDEX OF
A
MAP. PERIODIC POINTS
431
9. MISCELLANEOUS RESULTS AND EXAMPLES
434
10.
NOTES AND COMMENTS
437
XIV CONTENTS
§16.
THE
HOPF INDEX THEOREM
441
1.
NORMAL FIXED POINTS
IN
POLYHEDRAL DOMAINS
441
2.
HOMOLOGY
OF
POLYHEDRA WITH ATTACHED CONES
444
3.
THE
HOPF INDEX THEOREM
IN
POLYHEDRAL DOMAINS
447
4.
THE
HOPF INDEX THEOREM
IN
ARBITRARY ANRS
448
5.
THE
LEFSCHETZ-HOPF FIXED POINT INDEX
FOR
ANRS
450
6.
SOME CONSEQUENCES
OF
THE
INDEX
451
7.
MISCELLANEOUS RESULTS
AND
EXAMPLES
456
8. NOTES
AND
COMMENTS
458
§17.
FURTHER RESULTS
AND
APPLICATIONS
463
1.
LOCAL INDEX THEORY
FOR
ANRS
463
2.
FIXED POINTS
FOR
SELF-MAPS
OF
ARBITRARY COMPACTA
465
3.
FORMING NEW LEFSCHETZ SPACES FROM
OLD BY
DOMINATION
467
4.
FIXED POINTS
IN
LINEAR TOPOLOGICAL SPACES
469
5.
FIXED POINTS
IN
NES(COMPACT) SPACES
471
6. GENERAL ASYMPTOTIC FIXED POINT RESULTS
474
7*.
DOMINATION
OF
ANRS
BY
POLYTOPES
475
8. MISCELLANEOUS RESULTS
AND
EXAMPLES
483
9. NOTES
AND
COMMENTS
488
VI.
SELECTED TOPICS
§18.
FINITE-CODIMENSIONAL CECH COHOMOLOGY
491
1.
PRELIMINARIES
492
2.
CONTINUOUS FUNCTORS
500
3.
THE
CECH COHOMOLOGY GROUPS IF
N
(X)
506
4.
THE
FUNCTOR
H
00
'
71
:
(,
~) - AB
511
5.
COHOMOLOGY THEORY
ON 513
6. MISCELLANEOUS RESULTS
AND
EXAMPLES
521
7.
NOTES
AND
COMMENTS
523
§19.
VIETORIS FRACTIONS
AND
COINCIDENCE THEORY
531
1.
PRELIMINARY REMARKS
531
2.
CATEGORY
OF
FRACTIONS
532
3.
VIETORIS MAPS
AND
FRACTIONS
534
4.
INDUCED HOMOMORPHISMS
AND THE
LEFSCHETZ NUMBER
536
5.
COINCIDENCE SPACES
537
6. SOME GENERAL COINCIDENCE THEOREMS
539
CONTENTS XV
7. FIXED POINTS FOR COMPACT AND ACYCLIC SET-VALUED MAPS 542
8.
MISCELLANEOUS RESULTS AND EXAMPLES 544
9. NOTES AND COMMENTS 547
§20.
FURTHER RESULTS AND SUPPLEMENTS
551
1. DEGREE FOR EQUIVARIANT MAPS IN
R
N
551
2.
THE INFINITE-DIMENSIONAL E^-COHOMOLOGY 558
3.
LEFSCHETZ THEOREM FOR A^-MAPS OF COMPACTA 565
4.
MISCELLANEOUS RESULTS AND EXAMPLES 570
5.
NOTES AND COMMENTS 573
APPENDIX: PRELIMINARIES
588
A. GENERALITIES 588
B.
TOPOLOGICAL SPACES 590
C. LINEAR TOPOLOGICAL SPACES 599
D.
ALGEBRAIC PRELIMINARIES 608
E. CATEGORIES AND FUNCTORS 616
BIBLIOGRAPHY
620
I. GENERAL REFERENCE TEXTS 620
II.
MONOGRAPHS, LECTURE NOTES, AND SURVEYS 621
III.
ARTICLES 625
IV. ADDITIONAL REFERENCES 650
LIST OF STANDARD SYMBOLS 668
INDEX OF NAMES 672
INDEX OF TERMS 678 |
| any_adam_object | 1 |
| author | Granas, Andrzej Dugundji, James |
| author_facet | Granas, Andrzej Dugundji, James |
| author_role | aut aut |
| author_sort | Granas, Andrzej |
| author_variant | a g ag j d jd |
| building | Verbundindex |
| bvnumber | BV016972742 |
| callnumber-first | Q - Science |
| callnumber-label | QA329 |
| callnumber-raw | QA329.9.D833 2003 |
| callnumber-search | QA329.9.D833 2003 |
| callnumber-sort | QA 3329.9 D833 42003 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 320 SK 280 SK 620 |
| classification_tum | MAT 584f MAT 549f MAT 476f |
| ctrlnum | (OCoLC)845546552 (DE-599)BVBBV016972742 |
| dewey-full | 515/.724821 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.7248 21 |
| dewey-search | 515/.7248 21 |
| dewey-sort | 3515 47248 221 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV016972742 |
| illustrated | Illustrated |
| indexdate | 2025-03-10T13:02:46Z |
| institution | BVB |
| isbn | 9780387001739 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010250159 |
| oclc_num | 845546552 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-824 DE-19 DE-BY-UBM DE-91G DE-BY-TUM DE-521 DE-634 DE-11 DE-188 |
| owner_facet | DE-355 DE-BY-UBR DE-824 DE-19 DE-BY-UBM DE-91G DE-BY-TUM DE-521 DE-634 DE-11 DE-188 |
| physical | XV, 690 S. Ill., graph. Darst. |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer monographs in mathematics |
| spelling | Granas, Andrzej Verfasser aut Fixed point theory Andrzej Granas ; James Dugundji New York [u.a.] Springer 2003 XV, 690 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Literaturverz. S. 620 - 667 Fixed point theory Fixpunkttheorie (DE-588)4293945-8 gnd rswk-swf Fixpunkttheorie (DE-588)4293945-8 s DE-604 Dugundji, James Verfasser aut DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010250159&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Granas, Andrzej Dugundji, James Fixed point theory Fixed point theory Fixpunkttheorie (DE-588)4293945-8 gnd |
| subject_GND | (DE-588)4293945-8 |
| title | Fixed point theory |
| title_auth | Fixed point theory |
| title_exact_search | Fixed point theory |
| title_full | Fixed point theory Andrzej Granas ; James Dugundji |
| title_fullStr | Fixed point theory Andrzej Granas ; James Dugundji |
| title_full_unstemmed | Fixed point theory Andrzej Granas ; James Dugundji |
| title_short | Fixed point theory |
| title_sort | fixed point theory |
| topic | Fixed point theory Fixpunkttheorie (DE-588)4293945-8 gnd |
| topic_facet | Fixed point theory Fixpunkttheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010250159&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT granasandrzej fixedpointtheory AT dugundjijames fixedpointtheory |