Genericity in Nonlinear Analysis

This book presents an extensive collection of state-of-the-art results and references in nonlinear functional analysis demonstrating how the generic approach proves to be very useful in solving many interesting and important problems. Nonlinear analysis plays an ever-increasing role in theoretical and applied mathematics, as well as in many other areas of science such as engineering, statistics, computer science, economics, finance, and medicine. The text may be used as supplementary material for graduate courses in nonlinear functional analysis, optimization theory and approximation theory, and is a treasure trove for instructors, researchers, and practitioners in mathematics and in the mathematical sciences.

Each chapter is self-contained; proofs are solid and carefully communicated. Genericity in Nonlinear Analysis is the first book to systematically present the generic approach to nonlinear analysis. Topics presented include convergence analysis of powers and infinite products via the Baire Category Theorem, fixed point theory of both single- and set-valued mappings, best approximation problems, discrete and continuous descent methods for minimization in a general Banach space, and the structure of minimal energy configurations with rational numbers in the Aubry–Mather theory.

From the book reviews:

“This book contains an extensive collection of current results with solid proofs and each chapter is self-contained. It can be used as a supplementary material for graduate courses in nonlinear functional analysis, optimization theory and approximation theory. It is also a rich source of problems and results for researchers and practitioners working in mathematics and the mathematical sciences.” (Sehie Park, Mathematical Reviews, August, 2014)

“The book, which is self-contained and very well-written, is the first book to present an extensive collection of generic results in nonlinear analysis. As such, this book should prove useful to those currently doing research in nonlinear analysis and to those interested in beginning research in nonlinear analysis.” (Barry Turett, zbMATH, Vol. 1296, 2014)

“The book … contains a lot of interesting and deep generic existence results for some classes of problems in nonlinear analysis. By bringing together results spread through various journals, it will be of great help for researchers in fixed point theory, optimization, best approximation and dynamical systems. Being carefully written, with complete proofs and illuminating examples, it can serve also as an introductory book to this areas of current research.” (S. Cobzaş, Studia Universitatis Babes-Bolyai, Mathematica, Vol. 59 (1), 2014)

Simeon Reich is the Lord Leonard Wolfson Academic Chair and Professor of Mathematics at Technion-Israel Institute of Technology in Haifa. He has published more than fifty articles in mathematics journals including the Journal of Nonlinear Convex Analysis, SIAM Journal of Optimization, and Journal of Applied Analysis and fifteen books.

Alexander J. Zaslavski is a Senior Researcher in the Department of Mathematics at Technion-Israel Institute of Technology in Haifa. He has published over one hundred journal articles, and has authored books including Optimization on Metric and Normed Spaces (Springer, 2010) and Nonconvex Optimal Control and Variational Problems (Springer, 2013).