The contractivity of cone-preserving multilinear mappings

TL;DR

This paper extends classical theorems to multilinear mappings that preserve cones, providing conditions for solutions to nonlinear systems and explicit formulas for contraction ratios in integral operators with applications.

Contribution

It introduces a multilinear version of the Birkhoff-Hopf and Perron-Frobenius theorems, including explicit formulas for contraction ratios in integral operators.

Findings

Established conditions for existence and uniqueness of solutions to nonlinear systems.

Derived explicit formulas for mode-$j$ contraction ratios.

Applied results to integral operators with positive kernels.

Abstract

With the notion of mode-$j$ Birkhoff contraction ratio, we prove a multilinear version of the Birkhoff-Hopf and the Perron-Fronenius theorems, which provide conditions on the existence and uniqueness of a solution to a large family of systems of nonlinear equations of the type $f_i(x_1,\dots,x_\nu)= \lambda_i x_i$, being $x_i$ and element of a cone $C_i$ in a Banach space $V_i$. We then consider a family of nonlinear integral operators $f_i$ with positive kernel, acting on product of spaces of continuous real valued functions. In this setting we provide an explicit formula for the mode-$j$ contraction ratio which is particularly relevant in practice as this type of operators play a central role in numerous models and applications.