Exponential stability of positive semigroups in Banach spaces

The paper establishes a link between the stability of the semigroup e(−Γ+M)te(−Γ+M)t and the spectral radius of Γ−1MΓ−1M in ordered Banach spaces. On the one hand our result allows utilizing simple estimates for the eigenvalues of −Γ+M−Γ+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(−Γ+M)te(−Γ+M)t for those classes of matrices −Γ and M  , which lead to observable expressions for Γ−1MΓ−1M, e.g. when M is a coupling applied to disjoint systems representing Γ. 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.