Abstract

The authors study the equation \((*)\) \({d}/{dt}\) \(u(t) = Au(t) + f(t)\), where \(A\) is a generator of a \(C_{0}\)-semigroup on a Banach space \(X\), and are interested in which properties of the function \(f\) are inherited by the solution \(u\). To that purpose they consider the generator \(G\) of the evolution semigroup \(T(t)g(s) = e^{tA}g(s-t)\) on \(X\)-valued function spaces on \(\mathbb{R}\). Formally this generator is the sum of \({-d}/{dt}\) and the multiplication operator given by \(A\). They use spectral theory to find criteria for the solvability of \((*)\). The method and the results are also applied to higher-order and functional-differential equations.