Laplace Transforms and Integrated, Regularized Semigroups in Locally Convex Spaces

In this paper, we consider the vector-valued Laplace transforms,r-times (r∈[0, ∞)) integrated semigroups and regularized semigroups in the context of sequentially complete locally convex spaces. Our theorems develop the corresponding results in [1, 11], including the well known integrated version of the classical Widderʹs representation theorem of Laplace transforms for functions taking values in Banach spaces. Moreover, we study a class of differential operators on certain function spaces. Optimal conditions, making them the generators of integrated or regularized semigroups, are obtained. We finally show some applications to abstract Cauchy problems.