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

100%

Kryszewski W.

Instytut Matematyczny Polskiej Akademii Nauk

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

Multiplicity of positive solutions for a nonlinear differential equation with nonlinear boundary conditions

98%

R. D., Wang H.

Annales Polonici Mathematici

1998

tom 69

nr 2

155-165

We study the existence and multiplicity of positive solutions of the nonlinear equation u''(x) + λh(x)f(u(x)) = 0 subject to nonlinear boundary conditions. The method of upper and lower solutions and degree theory arguments are used.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Kryszewski W.

1 R. D.

1 Wang H.

1 1998

1 1994

Wyniki wyszukiwania

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

Multiplicity of positive solutions for a nonlinear differential equation with nonlinear boundary conditions