Series expansions for functional differential equations (Q1892634)

The question is studied whether the solution of a linear functional equation \({d \over dt} Dx_ t = Lx_ t\), \(t \geq 0\), \(x_ 0 = \varphi\), \(\varphi \in {\mathcal C}\) \(({\mathcal C}\) is the Banach space of continuous functions endowed with the supremum norm, \(x_ t\) is the state at time \(t)\) can be represented by a series of elementary solutions. Expansion of the state into a linear combination of eigenvectors and generalized eigenvectors is discussed. Estimates for the inverse of the characteristic matrix \(\Delta (z) = z[I - \int^ h_ 0 e^{- zt} d \mu (t)] - \int^ h_ 0 e^{- zt} d \zeta (t)\) are presented. Series expansions for retarded and neutral equations are discussed. Using Cesàro summation a more general Laplace inversion formula is obtained. The theory of hyperbolic semigroups is used to extend the results to general neutral equations.