LINEARIZED STABILITY AND IRREDUCIBILITY FOR A FUNCTIONAL-DIFFERENTIAL EQUATION

A principle of linearized stability is given for the abstract functional differential equation u(t) = Bu(t) + Ku(t), t greater-than-or-equal-to 0, u(O) = f, where B generates a strongly continuous semigroup of bounded linear operators on a Banach space X, and K:E = C([-r(O), 0], X) --> X is a nonlinear, continuously Frechet-differentiable operator. The strong positivity property of irreducibility is also investigated for the semigroup associated with solutions of the linearized equation. The theory is applied to the stability analysis of an equation from population dynamics.