## Topological Fixed Point Theory of Multivalued MappingsThis 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 |

381 | |

397 | |

### Common terms and phrases

acyclic map admissible compact admissible map approximative retract arbitrary Assume Banach space Caratheodory Cech cohomology closed subset commutative diagram compact map compact set compact space compact subset compact values compact vector field consider continuous map contraction map convex values Corollary deduce defined as follows Definition denote differential equations differential inclusions e-approximation endomorphism exists an open fixed point index fixed point theorem formula function functor graded vector space hence homeomorphic homology functor homotopy joining implies ip(x ISBN isomorphism Kn(r Lefschetz fixed point Lemma Leray endomorphism Let F Let ip map ip Math metric space morphism multivalued map n-admissible nonempty Nonlinear normed space Observe obtain open neighbourhood open set open subset Polish Acad polyhedron proof is completed Proposition prove the following Rn+1 Rn+l satisfies semicontinuous sequence singlevalued map solution strongly acyclic theory topological degree tp(x Vietoris map