Topological and approximation methods of degree theory of set-valued maps

Wojciech Kryszewski

GDML_Books, (1994), p.

Résumé

SummaryThe theory of topological degree of set-valued maps determined by morphisms, i.e. maps with values which are continuous images of almost acyclic sets, is presented, together with some of its applications.In the first part, morphisms defined on finite-dimensional Euclidean manifolds are considered and the integer-valued degree is introduced by means of the Eilenberg-Montgomery-Górniewicz method based on the Vietoris-Begle-Sklyarenko theorem and using the approach of Dold in terms of the Alexander-Spanier cohomology theory.In the second part, the degree theory is extended to a wide class of noncompact set-valued maps determined by morphisms acting in infinite-dimensional spaces. This new class, of the so-called A-morphisms, generalizing the compact set-valued vector fields, is studied from the general approximation viewpoint.Topological and approximation methods are also used in the context of another class of set-valued maps whose values satisfy some geometrical, rather than topological, conditions. Moreover, the degree theory is extended to the class of Petryshyn's A-proper maps with nonconvex values. The class of A-proper maps contains many different types of maps considered elsewhere.The degree theory is compared with the notion of essentiality, another useful tool in nonlinear analysis.The paper proposes several results extending the well-known theorems on single-valued maps, such as the Borsuk antipodal theorem, the Bourgin-Yang theorem, nonlinear alternative, invariance of domain and others.

CONTENTS   Introduction................................................................................................................5   Preliminaries...............................................................................................................7A. Elements of homology theory.....................................................................................8   1. Products.................................................................................................................8   2. Orientation of manifolds........................................................................................10I. Topology of morphisms..............................................................................................12   1. Set-valued maps....................................................................................................12   2. Vietoris maps.........................................................................................................14   3. Category of morphisms..........................................................................................18   4. Operations in the category of morphisms..............................................................21   5. Homotopy and extension properties of morphisms................................................23   6. Essentiality of morphisms.......................................................................................28   7. Concluding remarks................................................................................................32II. The topological degree theory of morphisms............................................................33   1. Cohomological properties of morphisms.................................................................34   2. The fundamental cohomology class........................................................................36   3. The topological degree of morphisms.....................................................................38   4. The degree of morphisms of spheres and open subsets of Euclidean space..........43   5. Borsuk type theorems..............................................................................................48   6. Applications.............................................................................................................56III. The class of approximation-admissible morphisms......................................................59   1. Filtrations.................................................................................................................60   2. Approximation-admissible morphisms and maps......................................................63   3. Approximation of A-maps.........................................................................................68IV. Approximation degree theory for A-morphisms...........................................................73   1. The degree of A-morphisms.....................................................................................73   2. Properties of the degree of A-morphisms................................................................75   3. Further properties of the degree. Applications.......................................................78V. Other classes of set-valued maps..............................................................................83   1. Single-valued approximations..................................................................................83   2. Linear filtrations. AP-maps of Petryshyn...................................................................91   References..................................................................................................................97

1991 Mathematics Subject Classification: 54C60, 47H04, 55M25, 47H11, 47H10, 47H17, 54H25, 55M20.

Détail

Zbl 0832.55004

EUDML-ID : urn:eudml:doc:268505