# The contractivity of cone-preserving multilinear mappings

Antoine Gautier,
Francesco Tudisco,

**Nonlinearity (IOP and London Mathematical Society)**,
32 :
4713
(2019)

### 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}
}
```

**Links:**
doi
arxiv

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