The contractivity of cone-preserving multilinear mappings

Abstract

With the notion of the mode-j Birkhoff contraction ratio, we prove a multilinear version of the Birkhoff–Hopf and 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 , with xi being an element of a cone Ci in a Banach space . We then consider a family of nonlinear integral operators f i with positive kernel, acting on the 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 operator plays a central role in numerous models and applications.