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Book DetailsMathematical Surveys and MonographsVolume: 23; 1986; 242 ppMSC: Primary 47; Secondary 54; 55; 58;
Since its development by Leray and Schauder in the 1930's, degree theory in Banach spaces has proved to be an important tool in tackling many analytic problems, including boundary value problems in ordinary and partial differential equations, integral equations, and eigenvalue and bifurcation problems. With this volume E. H. Rothe provides a largely selfcontained introduction to topological degree theory, with an emphasis on its functionanalytical aspects. He develops the definition and properties of the degree as much as possible directly in Banach space, without recourse to finitedimensional theory. A basic tool used is a homotopy theorem for certain linear maps in Banach spaces which allows one to generalize the distinction between maps with positive determinant and those with negative determinant in finitedimensional spaces.
Rothe's book is addressed to graduate students who may have only a rudimentary knowledge of Banach space theory. The first chapter on functionanalytic preliminaries provides most of the necessary background. For the benefit of less experienced mathematicians, Rothe introduces the topological tools (subdivision and simplicial approximation, for example) only to the degree of abstraction necessary for the purpose at hand. Readers will gain insight into the various aspects of degree theory, experience in functionanalytic thinking, and a theoretic base for applying degree theory to analysis.
Rothe describes the various approaches that have historically been taken towards degree theory, making the relationships between these approaches clear. He treats the differential method, the simplicial approach introduced by Brouwer in 1911, the LeraySchauder method (which assumes Brouwer's degree theory for the finitedimensional space and then uses a limit process in the dimension), and attempts to establish degree theory in Banach spaces intrinsically, by an application of the differential method in the Banach space case. 
Table of Contents

Chapters

Introduction

1. Functionanalytic preliminaries

2. The LeraySchauder degree for differentiable maps

3. The LeraySchauder degree for not necessarily differentiable maps

4. The PoincaréBohl theorem and some of its applications

5. The product theorem and some of its consequences

6. The finitedimensional case

7. On spheres

8. Some extension and homotopy theorems

9. The Borsuk theorem and some of its consequences


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Since its development by Leray and Schauder in the 1930's, degree theory in Banach spaces has proved to be an important tool in tackling many analytic problems, including boundary value problems in ordinary and partial differential equations, integral equations, and eigenvalue and bifurcation problems. With this volume E. H. Rothe provides a largely selfcontained introduction to topological degree theory, with an emphasis on its functionanalytical aspects. He develops the definition and properties of the degree as much as possible directly in Banach space, without recourse to finitedimensional theory. A basic tool used is a homotopy theorem for certain linear maps in Banach spaces which allows one to generalize the distinction between maps with positive determinant and those with negative determinant in finitedimensional spaces.
Rothe's book is addressed to graduate students who may have only a rudimentary knowledge of Banach space theory. The first chapter on functionanalytic preliminaries provides most of the necessary background. For the benefit of less experienced mathematicians, Rothe introduces the topological tools (subdivision and simplicial approximation, for example) only to the degree of abstraction necessary for the purpose at hand. Readers will gain insight into the various aspects of degree theory, experience in functionanalytic thinking, and a theoretic base for applying degree theory to analysis.
Rothe describes the various approaches that have historically been taken towards degree theory, making the relationships between these approaches clear. He treats the differential method, the simplicial approach introduced by Brouwer in 1911, the LeraySchauder method (which assumes Brouwer's degree theory for the finitedimensional space and then uses a limit process in the dimension), and attempts to establish degree theory in Banach spaces intrinsically, by an application of the differential method in the Banach space case.

Chapters

Introduction

1. Functionanalytic preliminaries

2. The LeraySchauder degree for differentiable maps

3. The LeraySchauder degree for not necessarily differentiable maps

4. The PoincaréBohl theorem and some of its applications

5. The product theorem and some of its consequences

6. The finitedimensional case

7. On spheres

8. Some extension and homotopy theorems

9. The Borsuk theorem and some of its consequences