Nonlinear Perron–Frobenius Theory

Guides the reader through the nonlinear Perron–Frobenius theory, introducing them to recent developments and challenging open problems.

Bas Lemmens (Author), Roger Nussbaum (Author)

9780521898812, Cambridge University Press

Hardback, published 3 May 2012

336 pages, 15 b/w illus.
23.4 x 15.6 x 2.1 cm, 0.62 kg

'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

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.

Preface
1. What is nonlinear Perron–Frobenius theory?
2. Non-expansiveness and nonlinear Perron–Frobenius theory
3. Dynamics of non-expansive maps
4. Sup-norm non-expansive maps
5. Eigenvectors and eigenvalues of nonlinear cone maps
6. Eigenvectors in the interior of the cone
7. Applications to matrix scaling problems
8. Dynamics of subhomogeneous maps
9. Dynamics of integral-preserving maps
Appendix A. The Birkhoff–Hopf theorem
Appendix B. Classical Perron–Frobenius theory
Notes and comments
References
List of symbols
Index.

Subject Areas: Topology [PBP], Geometry [PBM], Integral calculus & equations [PBKL], Differential calculus & equations [PBKJ]