  In this note we study strong contractivity of monotone systems and
equilibrium-contractivity of positive systems with respect to non-Euclidean
norms. We first introduce the notion of conic matrix measure and study its
properties. Using conic matrix measures and weak semi-inner products, we
characterize strongly contracting monotone systems in non-Euclidean spaces. We
apply this framework to i) study stability of monotone separable systems, ii)
establish strong contractivity of excitatory Hopfield neural networks, and iii)
obtain a novel version of the Matrosov-Bellman comparison lemma. We also
characterize equilibrium-contracting positive systems in non-Euclidean spaces
and provide a sufficient condition for equilibrium-contractivity using conic
measures. We apply this framework to i) study contractivity of positive
separable systems, and ii) develop a novel comparison-based framework for
studying interconnected systems.
