Linear Functional Differential Equations as Semigroups on Product Spaces

In this paper we consider the well-posedness of linear functional differential equations on product spaces. Let L and D be linear Rⁿ -valued functions with domains D (L) and D (D) subspaces of the Lebesgue measurable Rⁿ -valued functions on - r, 0 and such that W^1, p (- r, 0;Rⁿ) D (L) D (D). Under weak conditions on D and L we establish the equivalence between generalized solutions to the functional differential equation \ d{dt}Dxₜ = Lxₜ + f (t) \ and mild solutions to the Cauchy problem in Rⁿ Lₚ (- r, 0;Rⁿ) \ z (t) = a z (t) + (f (t), 0), \ where a is the operator defined on \ D (a) = \ { (, ) Rⁿ Lₚ {{ ({[ - r, 0;Rⁿ) } / } W^1, p (- r, 0;Rⁿ), D = } \}, \] by \ a (, ) = (L, ). \ The results are applicable to neutral functional differential equations and certain singular integral equations.