On the Solution Theory of Factored Cauchy Problems and the Abstract d'Alembert's Formula

摘要

We consider mild solutions of n-th order factored abstract Cauchy problems. In the commuting case we develop a new framework leading to the abstract d'Alembert's formula with necessary and sufficient conditions. We obtain a closure extension of the Vandermondian operator matrix which is used for the analysis of the d'Alembert's formula. A sufficient condition for a reduction of the solution formula for inhomogeneous problems is also given.