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

100%
Kryszewski W.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS    Introduction................................................................................................................5    Preliminaries...............................................................................................................7 A. Elements of homology theory.....................................................................................8    1. Products.................................................................................................................8    2. Orientation of manifolds........................................................................................10 I. 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................................................................................................32 II. 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.............................................................................................................56 III. The class of approximation-admissible morphisms......................................................59    1. Filtrations.................................................................................................................60    2. Approximation-admissible morphisms and maps......................................................63    3. Approximation of A-maps.........................................................................................68 IV. 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.......................................................78 V. Other classes of set-valued maps..............................................................................83    1. Single-valued approximations..................................................................................83    2. Linear filtrations. AP-maps of Petryshyn...................................................................91    References..................................................................................................................97
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Editors’ preface for the topical issue “Topics in Nonlinear Analysis”

61%
Kryszewski W., Skiba R.
Open Mathematics
|
2012
|
tom 10
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nr 6
1915-1919
3
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Perron-Frobenius and Krein-Rutman theorems for tangentially positive operators

61%
Kanigowski A., Kryszewski W.
Open Mathematics
|
2012
|
tom 10
|
nr 6
2240-2263
EN
We study several aspects of a generalized Perron-Frobenius and Krein-Rutman theorems concerning spectral properties of a (possibly unbounded) linear operator on a cone in a Banach space. The operator is subject to the so-called tangency or weak range assumptions implying the resolvent invariance of the cone. The further assumptions rely on relations between the spectral and essential spectral bounds of the operator. In general we do not assume that the cone induces the Banach lattice structure into the underlying space.
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