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Topological Degree Theory and Applications

By Yeol Je Cho, Yu-Qing Chen

Series Editor: Donal O'Regan, Ravi P. Agarwal

Chapman and Hall/CRC – 2006 – 232 pages

Series: Mathematical Analysis and Applications

Purchasing Options:

  • Add to CartHardback: $109.95
    978-1-58488-648-8
    March 27th 2006

Description

Since the 1960s, many researchers have extended topological degree theory to various non-compact type nonlinear mappings, and it has become a valuable tool in nonlinear analysis. Presenting a survey of advances made in generalizations of degree theory during the past decade, this book focuses on topological degree theory in normed spaces and its applications.

The authors begin by introducing the Brouwer degree theory in Rn, then consider the Leray-Schauder degree for compact mappings in normed spaces. Next, they explore the degree theory for condensing mappings, including applications to ODEs in Banach spaces. This is followed by a study of degree theory for A-proper mappings and its applications to semilinear operator equations with Fredholm mappings and periodic boundary value problems. The focus then turns to construction of Mawhin's coincidence degree for L-compact mappings, followed by a presentation of a degree theory for mappings of class (S+) and its perturbations with other monotone-type mappings. The final chapter studies the fixed point index theory in a cone of a Banach space and presents a notable new fixed point index for countably condensing maps.

Examples and exercises complement each chapter. With its blend of old and new techniques, Topological Degree Theory and Applications forms an outstanding text for self-study or special topics courses and a valuable reference for anyone working in differential equations, analysis, or topology.

Reviews

"The book is very well written, presents essential ideas and results with typical applications, being extremely useful to the beginners in nonlinear analysis . . . This is a really valuable text for self-study and special courses in nonlinear analysis and also a good reference for anyone applying topological methods to integral, ordinary and partial differential equations."

– Radu Precup, in Babes-Bolyal Math, June 2007, Vol. 52, No. 2

Contents

BROUWER DEGREE THEORY

Continuous and Differentiable Functions

Construction of Brouwer Degree

Degree Theory for Functions in VMO

Applications to ODEs

Exercises

LERAY-SCHAUDER DEGREE THEORY

Compact Mappings

Leray-Schauder Degree

Leray-Schauder Degree for Multi-valued Mappings

Applications to Bifurcations

Applications to ODEs and PDEs

Exercises

DEGREE THEORY FOR SET CONTRACTIVE MAPS

Measure of Non-compactness and Set Contractions

Degree Theory for Countably Condensing Mappings

Applications to ODEs in Banach Spaces

Exercises

GENERALIZED DEGREE THEORY FOR A-PROPER MAPS

A-Proper Mappings

Generalized Degree for A-Proper Mappings

Equations with Fredholm Mappings of Index Zero

Equations with Fredholm Mappings of Index Zero Type

Applications of the Generalized Degree

Exercises

COINCIDENCE DEGREE THEORY

Fredholm Mappings

Coincidence Degree for L-Compact Mappings

Existence Theorems for Operator Equations

Applications to ODEs

Exercises

DEGREE THEORY FOR MONOTONE TYPE MAPS

Monotone Type Mappings in Reflexive Banach Spaces

Degree Theory for Mappings of Class (S+)

Degree for Perturbations of Monotone Type Mappings

Degree Theory for Mappings of Class (S+)L

Coincidence Degree for Mappings of Class L - (S+)

Computation of Topological Degree

Applications to PDEs and Evolution Equations

Exercises

FIXED POINT INDEX THEORY

Cone in Normed Spaces

Fixed Point Index Theory

Fixed Point Theorems in Cones

Perturbations of Condensing Mappings

Index Theory for Nonself-Mappings

Applications to Integral and Differential Equations

Exercises

REFERENCES

SUBJECT INDEX

Name: Topological Degree Theory and Applications (Hardback)Chapman and Hall/CRC 
Description: By Yeol Je Cho, Yu-Qing ChenSeries Editor: Donal O'Regan, Ravi P. Agarwal. Since the 1960s, many researchers have extended topological degree theory to various non-compact type nonlinear mappings, and it has become a valuable tool in nonlinear analysis. Presenting a survey of advances made in generalizations of degree theory...
Categories: Mathematical Analysis, Mathematical Physics, Differential Equations