Sometimes in mathematics a simple-looking observation opens up a new road to a fertile field. Such an observation was made independently by Garrett Birkhoff [25] and Hans Samelson [192], who remarked that one can use Hilbert's (projective) metric and the contraction mapping principle to prove some of the theorems of Perron and Frobenius concerning eigenvectors and eigenvalues of nonnegative matrices. This idea has been pivotal for the development of nonlinear Perron–Frobenius theory.

In the past few decades a number of strikingly detailed nonlinear extensions of Perron–Frobenius theory have been obtained. These results provide an extensive analysis of the eigenvectors and eigenvalues of various classes of order-preserving (monotone) nonlinear maps and give information about their iterative behavior and periodic orbits. Particular classes of order-preserving maps for which there exist nonlinear Perron–Frobenius theorems include subhomogeneous maps, topical maps, and integral-preserving maps. The latter class of order-preserving maps can be regarded as a nonlinear generalization of column stochastic matrices, whereas topical maps generalize row stochastic matrices

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. We hope that it will be a stimulating source for non-experts to learn and appreciate this subject. To keep our task manageable, we restrict ourselves to finite-dimensional vector spaces, which allows us to avoid the use of sophisticated fixed-point theorems, the fixed-point index, and topological degree theory.