## Topological Fixed Point Theory of Multivalued MappingsThis 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

1 | |

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 |

381 | |

397 | |

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### Common terms and phrases

acyclic map admissible compact admissible map arbitrary Assume assumption Banach space Carathéodory Cech cohomology closed subset commutative diagram compact map compact set compact space compact subset compact values compact vector field Consequently continuous map convex values Corollary deduce defined as follows Definition Deg(p denote differential inclusions e-approximation exists an open fived point Fix F Fix(p fixed point index fixed point theorem formula function functor graded vector space hence homeomorphic homology functor homotopy joining implies isomorphism k-set contraction Lemma Leray endomorphism Let f limn m-map map f map with compact Math metric space Moreover morphism multivalued map n-admissible nonempty normed space Observe obtain open neighbourhood open set open subset pair p,q polyhedron proof is completed Proposition prove the following retract satisfies sequence singlevalued map solution strongly acyclic supp topological degree tr(f us.c Vietoris map