Topological Fixed Point Theory of Multivalued Mappings

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Springer Science & Business Media, Nov 11, 2013 - Mathematics - 403 pages
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This book is an attempt to give a systematic presentation of results and meth ods which concern the fixed point theory of multivalued mappings and some of its applications. In selecting the material we have restricted ourselves to study ing topological methods in the fixed point theory of multivalued mappings and applications, mainly to differential inclusions. Thus in Chapter III the approximation (on the graph) method in fixed point theory of multi valued mappings is presented. Chapter IV is devoted to the homo logical methods and contains more general results, e. g. , the Lefschetz Fixed Point Theorem, the fixed point index and the topological degree theory. In Chapter V applications to some special problems in fixed point theory are formulated. Then in the last chapter a direct application's to differential inclusions are presented. Note that Chapter I and Chapter II have an auxiliary character, and only results con nected with the Banach Contraction Principle (see Chapter II) are strictly related to topological methods in the fixed point theory. In the last section of our book (see Section 75) we give a bibliographical guide and also signal some further results which are not contained in our monograph. The author thanks several colleagues and my wife Maria who read and com mented on the manuscript. These include J. Andres, A. Buraczewski, G. Gabor, A. Gorka, M. Gorniewicz, S. Park and A. Wieczorek. The author wish to express his gratitude to P. Konstanty for preparing the electronic version of this monograph.
 

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Contents

BACKGROUND IN TOPOLOGY 1 Extension and embedding properties
1
Homotopical properties of spaces
10
Approximative and proximative retracts
19
Hyperspaces of metric spaces
22
The Cech homology cohomology functor
28
Maps of spaces of finite type
35
The Cech homology functor with compact carriers
36
Vietoris maps
38
Admissible mappings
199
The Lefschetz fixed point theorem for admissible mappings
203
admissible mappings
207
nAdmissible mappings
211
Category of morphisms
224
The Lefschetz fixed point theorem for morphisms
231
Homotopical classification theorems for morphisms
232
The fixed point index for morphisms
235

Homology of open subsets of Euclidean spaces
42
The ordinary Lefschetz number
46
The generalized Lefschetz number
49
The coincidence problem
53
MULTIVALUED MAPPINGS
61
Upper semicontinuous mappings
67
Lower semicontinuous mappings
71
Michaels selection theorem
74
GSelectionable mappings
77
Directionally continuous selections
81
Measurable selections
85
Borsuk and Hausdorff continuity of multivalued mappings
93
Banach contraction principle for multivalued maps
96
APROXIMATION METHODS IN FIXED POINT THEORY OF MULTIVALUED MAPPINGS
105
Existence of approximations
110
Homotopy
117
The fixed point index in AX
120
Topological degree in R
122
Topological degree for mappings with noncompact values in R
130
Topological degree in normed spaces
143
Topological degree of vector fields with noncompact values in Banach spaces
147
Topological essentiality
152
Random fixed points
155
HOMOLOGICAL METHODS IN FIXED POINT THEORY OF MULTIVALUED MAPPINGS
159
Strongly acyclic maps
163
The fixed point index for acyclic maps of Euclidean Neighbourhood Retracts
166
The Nielsen number
173
nAcyclic mappings
182
Theorem on antipodes for nacyclic mappings
187
Theorem on invariance of domain
193
nAcyclic compact vector fields in normed spaces
196
Noncompact morphisms
242
nMorphisms
246
Multivalued maps with nonconnected values
247
A fixed point index of decompositions for finite polyhedra
262
Fixed point index of decompositions for compact ANRS
267
Fixed point index of decompositions for arbitrary ANRS
274
Spheric mappings
276
CONSEQUENCES AND APPLICATIONS
281
Fixed point property and families of multivalued map pings
285
The Lefschetz fixed point theorem for pairs of spaces
289
Repulsive and ejective fixed points
291
Condensing and kset contraction mappings
296
Compacting mappings
302
Fixed points of differentiable multivalued maps
304
The generalized topological degree for acyclic mappings
312
The bifurcation index
317
Multivalued dynamical systems
322
Minimax theorems for ANRs
331
KKMmappings
338
Topological dimension of the set of fixed points
343
On the basis problem in normed spaces
345
FIXED POINT THEORY APPROACH TO DIFFERENTIAL INCLUSIONS
347
Solution sets for differential inclusions
352
The 1 s c case
359
Periodic solutions for differential inclusions in R
363
Differential inclusions on proximate retracts
369
Implicit differential inclusions
373
Concluding remarks and comments
378
RE FERENCES
381
INDEX
397
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