Topological Degree Theory and Applications
By Yeol Je Cho, Yu-Qing Chen
Series Editor: Donal O'Regan, Ravi P. Agarwal
Published March 27th 2006 by Chapman and Hall/CRC – 232 pages
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
Related Subjects
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
