Nonlinear Perron–Frobenius Theory
Cambridge Tracts in Mathematics Series

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.

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.