LINEAR-THEORY OF ABSTRACT FUNCTIONAL-DIFFERENTIAL EQUATIONS OF RETARDED TYPE

The purpose of this paper is to provide an extension of the linear the ory of functional differential equations of retarded type to abstract equations. Such equations include examples borrowed from population dy namics to which the theory applies. An application will be given elsew here. Our main effort in this work consists in providing a suitable ex tension of the formal adjoint equation and the formal duality. The sol utions of the linear autonomous retarded functional differential equat ion (1) x(1)(t) = L(x(t)), where L is a bounded linear operator mappin g the space C([-r, 0]; E) into the Banach space E, define a strongly c ontinuous translation semigroup. We show the existence of a direct sum decomposition of C([-r, 0]; E) into two subspaces which are semigroup invariants. The how induced by the solutions of Eq. (1) can be interp reted as the flow induced by an ordinary differential equation in a fi nite-dimensional space. We explicitly characterize this decomposition by an orthogonality relation associated to a certain definition of for mal duality. The existence of an integral representation for the opera tor L leads to an equation formally adjoint to (1) characterizing the projection operator defined by the above decomposition of C([-r, 0]; E ). (C) 1995 Academic Press, Inc.