Regularized semigroups of bounded semivariation

We use regularized semigroups to consider local linear and semilinear inhomogeneous abstract Cauchy problems on a Banach space in a unified way. We show that the inhomogeneous abstract Cauchy problem {fx43-1} has a unique classical solution, for allf εC(0,T], Im(C)]),x inC(D(A)), if and only ifA generates aC-regularized semigroup of bounded semivariation, and has a strong solution for allf εL ¹ (0,T], Im(C)]),x εC(D(A)) if and only if theC-regularized semigroup is what we call of bounded super semivariation. This includes locally Lipschitz continuousC-regularized semigroups. We give similar simple sufficient conditions for the semilinear abstract Cauchy problem {fx43-2} to have a unique solution. Well-known results for generators of strongly continuous semigroups, as well as more recent results for Hille-Yosida operators, originally due to Da Prato and Sinestrari, regarding (0.1), are immediate corollaries of our results. Results due to Desch, Schappacher and Zhang, on (0.2), for generators of strongly continuous semigroups, are similarly generalized to Hille-Yosida operators with our approach. This article appeared in the last issue of the Forum. However, due to an error by the Journal Secetary, the Abstract was omitted, and with it the equations which are the focus of the article. We therefore are reprinting the article in its entirety. The Journal Secretary regrets the error.