Exponential stability of positive semigroups in Banach spaces

The paper establishes a link between the stability of the semigroup e^(−Γ+M)t and the spectral radius of Γ^(−1)M in ordered Banach spaces. On the one hand our result allows utilizing simple estimates for the eigenvalues of −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)t for those classes of matrices −Γand M, which lead to observable expressions for Γ(−1)M, 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