Perturbed Dual Semigroups and Delay Equations

Abstract

The variation — of — constants formula is a very convenient starting point for the derivation of many results in the local stability and bifurcation theory of ordinary and partial differential equations. This statement is equally valid for retarded functional differential equations, but here the variation — of — constants formula shows a peculiar feature. Indeed, in the book of Hale [6] we find that the formula involves the so-called fundamental matrix solution X which is, by definition, the solution corresponding to the special discontinuous initial condition X(t) = 0 for t < 0 and X(0) = I (the identity matrix) and which, therefore, does not “live” in the state space C. As a consequence one has to interpret the convolution integral which figures in the variation — of — constants formula as a family (parametrized by the independent variable of the functions in C ) of integrals in Euclidean space. Thus the formula becomes symbolic rather than functional analytic (it does not fit into the standard semigroup framework).

The theory of dual semigroups on non-reflexive Banach spaces can be used to define a natural generalization of the notion of a bounded perturbation of the generator and a new version of the variation — of — constants formula. This approach was developed in joint work with Ph. Clément, M. Gyllenberg, H.J.A.M. Heijmans and H.R. Thieme, motivated by some applications to physiologically structured population growth models. In this paper it is shown that delay differential equations fit very well into exactly the same functional analytic framework.