Topological Fixed Point Theory of Multivalued Mappings

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Springer Science & Business Media, Jan 1, 1999 - Mathematics - 399 pages
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This volume presents a broad introduction to the topological fixed point theory of multivalued (set-valued) mappings, treating both classical concepts as well as modern techniques. A variety of up-to-date results is described within a unified framework. Topics covered include the basic theory of set-valued mappings with both convex and nonconvex values, approximation and homological methods in the fixed point theory together with a thorough discussion of various index theories for mappings with a topologically complex structure of values, applications to many fields of mathematics, mathematical economics and related subjects, and the fixed point approach to the theory of ordinary differential inclusions. The work emphasises the topological aspect of the theory, and gives special attention to the Lefschetz and Nielsen fixed point theory for acyclic valued mappings with diverse compactness assumptions via graph approximation and the homological approach. Audience: This work will be of interest to researchers and graduate students working in the area of fixed point theory, topology, nonlinear functional analysis, differential inclusions, and applications such as game theory and mathematical economics.
  

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Contents

BACKGROUND IN TOPOLOGY
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
trSelectionable 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 Rn
122
Topological degree for mappings with noncompact values in Rn
130
Topological degree in normed spaces
143
Topological degree of vector fields with noncompact
147
values in Banach spaces
149
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 l s c case
359
Periodic solutions for differential inclusions in Rn
363
Differential inclusions on proximate retracts
369
Implicit differential inclusions
373
Concluding remarks and comments
378
REFERENCES
381
INDEX
397
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