Comparison Results for Generalized Kolmogorov Systems with Respect to Multiplicative Semigroups

Abstract

The group $G(\mathcal{A})$ of the invertible elements of a Banach algebra $\mathcal{A}$ can be pre-ordered by a closed semigroup $S$. We prove a comparison theorem for ODEs of the form $u'=f_1(t,u) u + u f_2(t,u)$ in Banach algebras under the assumption that $S$ is permutation stable. Applications to monotonicity properties of initial value problems and dynamical systems are given.