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Topological methods in the study of boundary value problems. (English) Zbl 1319.34001

Universitext. New York, NY: Springer (ISBN 978-1-4614-8892-7/pbk; 978-1-4614-8893-4/ebook). xvi, 226 p. (2014).
This book reviews in a nice and well-written way the most important topological methods in the study of boundary value problems associated to ordinary differential equations. It is understandable even with a minimal knowledge of functional analysis; as a consequence, it is useful for many readers, starting from Master’s degree students and up to experienced researchers, who will appreciate this thorough and self- contained description of the basic concepts in this field. Moreover, the author proposes some less common proofs of well-known results (e.g., the contraction mapping theorem and Brouwer’s theorem). Another remarkable feature of this book consists in the fact that many times the same example is treated in different chapters (with different methods); this happens, e.g., for the “pendulum paradigm”. The book is divided in the following chapters: Shooting type methods, The Banach fixed point theorem, Iterative methods, The Schauder theorem and applications, Topological degree: an introduction, Applications. Hints and solutions to some of the proposed exercises are available as well.

MSC:

34-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
47H11 Degree theory for nonlinear operators
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