Applied Nonlinear Functional Analysis: An Introduction

This book presents a gentle introduction to the fundamentals of nonlinear functional analysis. The topics covered are carefully chosen from a vast number of hard-to-pick interesting topics in the subject, including connections between topological and measure-theoretic structures of spaces.  The book gives a feeling of being invited for a seamless entry to the subject. The exposition is concise, smooth, and elegant.  Although the presentation is traditional – definitions, lemmas, propositions, theorems, remarks, and problems – it does lack examples that would help better understand definitions and illustrate the main results. However, the problems at the end of each chapter are extensive and would certainly help fine-tune the understanding of the definitions and main results presented in that chapter.  Each chapter contains a section for an extensive remark about historical developments, generalizations and/or further directions of main results presented in it.

 

Starting with the basics of topology, measure theory and Banach space theory in earlier chapters, the book embarks on a beautiful area of nonsmooth analysis and multivalued functions in the context of optimization theory.  Although the materials of all other chapters have set foundations for nonlinear functional analysis, the last chapter on Nonlinear Analysis deals with theories and tools which have wide applications to the study of boundary value problems in differential equations and problems in calculus of variations, optimization, and optimal control. Monotone operators, topological degree theory, metric and topological fixed-point theories, Lax-Milgram theorem and Ekeland variational principle are some of the highly important topics that are covered.  Also included in the chapter is a notion of \(\Gamma\)-convergence, which is used for examining the stability properties of variational problems.

 

The authors have been able to fit together a wide variety of topics in a single volume in a very welcoming format that seemingly invites the reader to the beautiful area of nonlinear analysis. The book is suitable for graduate students of applied mathematics and is an excellent resource for anyone interested in stepping into nonlinear functional analysis and its applications.

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Dhruba R. Adhikari is Associate Professor of Mathematics, Kennesaw State University, Marietta, Georgia.  His web page is http://facultyweb.kennesaw.edu/dadhikar/