Nonlinear Perron-Frobenius theory (Q2884326)

The book deals with the generalizations and modifications of the classical Perron-Frobenius theory for nonnegative matrices to nonlinear mappings in finite-dimensional spaces. The authors themselves write: ``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''. The reviewer completely agrees with Hans Schneider's opinion that the authors excellently achieved this aim. The main subject of this book is an analysis of order-preserving (monotone) maps \(F:\;K \to K\), where \(K\) is a solid closed cone in a finite-dimensional vector space. The main problems about such maps are the following: the existence of nontrivial fixed points and eigenvectors, the existence and properties of periodic points, the behaviour of iterates. The authors study all these problems for different classes of order-preserving maps and consider some concrete order preserving maps that appear in the theory of matrices, the theory of Markov decision processes and of stochastic games, nonlinear diffusion processes and so on.NEWLINENEWLINEThe book contains nine chapters and two appendices. Ch. 1: \textsc{What is nonlinear Perron-Frobenius theory?} presents some account of the classical (linear) Perron-Frobenius theory, gives the main notions in the theory of cones in finite-dimensional vector spaces (in particular, polyhedral cones) and describes some important classes of order-preserving operators, such as homogeneous, subhomogeneous (concave in Krasnosel'skiĭ's sense), topical and subtopical (\(=\) order preserving additively (sub)homogeneous), integral preserving, and so on. In this chapter, the authors also consider some problems that are tightly connected with nonlinear order preserving maps. Ch. 2: \textsc{Non-expansiveness and nonlinear Perron-Frobenius theory} deals with definitions of the Hilbert and Thompson metrics in spaces with cones and the corresponding properties of completeness and convexity; in particular, the authors consider polyhedral cones, Lorentz cones, the cone of positive-semidefinite symmetric matrices. The second part of that chapter develops the relation between order-preserving maps and their nonexpansiveness property in Hilbert and Thompson metrics. Ch. 3: \textsc{Dynamics of non-expansive maps} is concerned with the iterative behavior of nonexpansive maps, including \(\omega\)-limit sets, fixed point theorems, horofunctions, nonexpansive retractions, and so on. NEWLINENEWLINENEWLINENEWLINECh. 4: \textsc{Eigenvectors and eigenvalues of nonlinear cone maps} focuses on the dynamics of sup-norm nonexpansive maps; orbits of such mappings are either unbounded or converge to periodic orbits whose period can be \textit{a priori} estimated in terms of the dimension of the underlying space. The corresponding results allow the author to give an analysis of the iterative behavior of order preserving subhomogeneous maps on polyhedral cones and some topical maps. Ch. 5: \textsc{Eigenvectors in the interior of the cone} deals with eigenvectors and the cone spectral radius; in particular, the authors consider the continuity property of the cone spectral radius and a ``nonlinear'' modification of the classical Collatz-Wielandt ``minimax'' formula for the spectral radius of nonnegative matrices. Ch. 6: \textsc{Eigenvectors in the interior of the cone} deals with the problem of the existence of an eigenvector in the interior of the cone for a given order-preserving homogeneous operator. The corresponding results are illustrated by particular homogeneous maps. Ch. 7: \textsc{Application to matrix scaling problem} is completely devoted to the well-known DAD-problem for matrices; due to M. V. Menon, this problem is reduced to the analysis of a nonlinear order-preserving map and then are used the results from previous chapters. NEWLINENEWLINENEWLINENEWLINECh. 8: \textsc{Dynamics of subhomogeneous maps} deals with the detailed analysis of the behavior iterates of order-preserving subhomogeneous maps; here one finds greatly curious results about periodic orbits for mappings on polyhedral cones and theorems about such mappings without eigenvectors in the interior of the cone. Ch. 9: \textsc{Dynamics of integral-preserving maps} is devoted to nonlinear Perron-Frobenius theorems for order- and integral-preserving maps on the standard positive cone; such mappings are non-expansive with respect the \(\ell_1\)-norm. The dynamics of these mappings is tightly connected with the dynamics of lower-lattice diffeomorphisms and complete a characterization of the set of possible periods of periodic orbits in terms of so-called admissible arrays. Appendix A: \textsc{The Birkhoff-Hopf theorem} presents a refined (and new) account of the well-known Birkhoff-Hopf theorem about the existence of the leading eigenvalue and eigenvectors for order-preserving linear mappings. Appendix B: \textsc{Classical Perron-Frobenius theory} presents a short account of the classical Perron-Frobenius theory for positive and irreducible matrices. The book ends with \textsc{Notes and comments}, \textsc{References} (230 items), \textsc{List of symbols}, and \textsc{Index}. NEWLINENEWLINENEWLINE NEWLINEThe only deficit of this book I have found is the absence of the Krasnosel'skiĭ theory of concave (order-preserving subhomogeneous) operators. Undoubtedly, this remarkable book will be of interest to all specialists in Nonlinear Analysis and its Applications. Certainly, every mathematical library ought to carry this book.