Exponential stability of positive semigroups in Banach spaces

Abstract

The paper establishes a link between the stability of the semigroup $e^{(-\Gamma+M)t}$ and the spectral radius of $\Gamma^{-1}M$ in ordered Banach spaces. On the one hand our result allows utilizing simple estimates for the eigenvalues of $-\Gamma+M$ in order to provide general conditions for the convergence of the successive approximation scheme for semilinear operator equations. On the other hand, this paper helps examining the stability of the semigroup $e^{(-\Gamma+M)t}$ for those classes of matrices $-\Gamma$ and $M$, which lead to observable expressions for $\Gamma^{-1}M$, e.g. when $M$ is a coupling applied to disjoint systems representing $\Gamma$. The novelty of the paper is in the development of an infinite-dimensional framework, where an absolute value function induced by a cone is introduced and a way to deal with the lack of global continuity of eigenvalues is presented.