The contractivity of cone-preserving multilinear mappings

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 $$f_i(x_1,…,x_ν)= λ_i x_i, ,$$ 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.

Please cite this work as:

@article{gautier2019contractivity,
  title={The contractivity of cone-preserving multilinear mappings},
  author={Gautier, Antoine and Tudisco, Francesco},
  journal={Nonlinearity},
  volume={32},
  pages={4713},
  year={2019},
  publisher={IOP Publishing Ltd \& London Mathematical Society}
}

Keywords: Perron-Frobenius theory nonlinear eigenvalues nonnegative tensors Hilbert metric power method multi-homogeneous maps multilinear maps cone-preserving maps