Overview

In the past several decades the classical Perron–Frobenius theory for nonnegative matrices has been extended to obtain remarkably precise and beautiful results for classes of nonlinear maps. This nonlinear Perron–Frobenius theory has found significant uses in computer science, mathematical biology, game theory and the study of dynamical systems. This is the first comprehensive and unified introduction to nonlinear Perron–Frobenius theory suitable for graduate students and researchers entering the field for the first time. It acquaints the reader with recent developments and provides a guide to challenging open problems. To enhance accessibility, the focus is on finite dimensional nonlinear Perron–Frobenius theory, but pointers are provided to infinite dimensional results. Prerequisites are little more than basic real analysis and topology.

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9781139026079

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Reviews

In their introduction the authors state that The main purpose of this book is to give a systematic self-contained introduction to nonlinear Perron-Frobenius theory and to provide a guide to various challenging open problems. They have achieved their aim excellently. After a discussion of the linear theory of cone preserving maps, the book turns to its main theme, nonlinear Perron-Frobenius theory in finite dimension. Of particular importance is the linking of this theory to that of non-expansive maps in various metrics. Applications are presented, for example to dynamical systems and diagonal scaling of matrices. In its various incarnations, Perron-Frobenius theory has had a deep influence over 100 years on many parts of pure and applied mathematics. An exposition of the finite-dimensional nonlinear theory from a specific point of view is a valuable and timely addition to the literature. Hans Schneider, University of Wisconsin, Madison

'In their introduction the authors state that 'the main purpose of this book is to give a systematic self-contained introduction to nonlinear Perron-Frobenius theory and to provide a guide to various challenging open problems'. They have achieved their aim excellently.' Hans Schneider, University of Wisconsin, Madison

'In their introduction the authors state that 'the main purpose of this book is to give a systematic self-contained introduction to nonlinear Perron-Frobenius theory and to provide a guide to various challenging open problems'. They have achieved their aim excellently.' Hans Schneider, University of Wisconsin, Madison 'Undoubtedly, this remarkable book will be of interest to all specialists in nonlinear analysis and its applications. Certainly, any mathematical library ought to carry this book.' Peter Zabreiko, Zentralblatt MATH 'This textbook is a carefully arranged journey through large parts of this beautiful theory, which has seen various contributions by the authors in the past. The material is accessible with little more than a basic knowledge of linear algebra, real analysis and some topology. The book is self-contained, all results are proven very rigorously, and where appropriate, the evolution of results is explained and framed in the historical context. I recommend this book very warmly and without any reservations to anyone interested in nonlinear Perron-Frobenius theory.' Bjorn S. Ruffer, Mathematical Reviews

Author Information

Bas Lemmens is a Lecturer in Mathematics at the University of Kent, Canterbury. His research interests lie in nonlinear operator theory, dynamical systems theory and metric geometry. He is one of the key developers of nonlinear Perron–Frobenius theory. Roger Nussbaum is a Professor of Mathematics at Rutgers University. His research interests include nonlinear differential-delay equations, the theory of nonlinear positive operators and fixed point theory and its applications. He has published extensively on nonlinear Perron–Frobenius theory.

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