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Book review

Author(s) Brown, R.F.
Title A topological introduction to nonlinear analysis
Publisher Birkhäuser
Year of publication 1993
Reviewed by Radu Precup

This book is intended to help the graduate student or even the young mathematician who face to the broad subject of nonlinear analysis and to the branchy material offered by monumental monographs in the field, asks himself: how do I get started in nonlinear analysis? How do I go in the heart of major problems of nonlinear analysis? This book is indeed of immense help in providing the reader with an understanding of the mathematics of the nonlinear phenomena and with two basic techniques of nonlinear analysis: topological fixed point and degree methods. Although an introduction, the book is not a simple set of topics but is goal-oriented and by this it makes a very exciting reading. Its goal has a name: the Krasnoselskii-Rabinowitz bifurcation theorem.

Part I: Fixed point existence theory (about 40pp) is concerned with solutions to the equation f(x) = x in a Banach space X. The author starts with a topological proof of Cauchy-Peano existence theorem for the initial-value problem and goes on with Ascoli-Arzela theory, Brouwer, Schauder and Schaeffer fixed point theorems. Some applications to the existence of solutions for two-points boundary value problems are then presented. Part II: Degree And bifurcation (about lOOpp) is mainly about solutions to the equations <formula> for <formula>. It is required that G is Fréchet differentiable at 0 and that <formula>, the points <formula> are considered trivial solutions to the equation. Denote by S the set of nontrivial solutions, that is, <formula>. If a point <formula>, then the number <formula> is called a bifurcation point for the solutions to <formula>. The goal of this book is the following Krasnoselskii-Rabinowitz theorem: Let X be a Banach space and let G be completely continuous and Fréchet differentiable at 0 with derivative <formula>. Suppose <formula> is a real number such that <formula> and it is of odd multiplicity. Then there exists a maximal closed connected subset <formula> of <formula> which contains <formula>, so <formula> is a bifurcation point, and either <formula> is unbounded or <formula> contains <formula> for some other bifurcation point <formula>. In order to understand and to apply this beautiful result some additional facts are needed: Leray-Schauder degree, a separation theorem, compact linear operators, a degree calculation formula. All these topics are clearly presented. A classical example is given in the differential equations that models the maximum weight a column can support without buckling.

The book is very well written and essentially self-contained. Upon completing the book, the reader will be able to make rapid progress through the existing literature in topological methods in nonlinear analysis starting with nine essential titles that are recommended by the author.

I highly recommend this book to students, young mathematicians and teachers interested in nonlinear analysis, differential equations and applied mathematics.

Our thanks are due to the Author and to the Publisher for this excellent introductory book in nonlinear analysis.