Convergence of iterates in nonlinear Perron-Frobenius theory

Abstract

Let $C$ be a closed cone with nonempty interior $C^{\circ }$ in a Banach space. Let $f:C^{\circ }\to C^{\circ }$ be an order-preserving subhomogeneous function with a fixed point in $C^{\circ }$. We introduce a condition which guarantees that the iterates $f^k(x)$ converge to a fixed point for all $x\in C^{\circ }$. This condition generalizes the notion of type K order-preserving for maps on ${\mathbb{R}}_{>0}^n$. We also prove that when iterates converge to a fixed point, the rate of convergence is always R-linear in two special cases: for piecewise affine maps and also for order-preserving, homogeneous, analytic, multiplicatively convex functions on ${\mathbb{R}}_{>0}^n$. This later category includes the maps associated with the homogeneous eigenvalue problem for nonnegative tensors.

arXiv commentAdded a section with applications. Also includes minor changes and corrections