Approximations of Laplace Transforms and Integrated Semigroups

Let {fm; m∈N} be a sequence of functions from [0, ∞) to a Banach space E. We give a new and essential condition on fm, which is weaker than the usual “local Lipschitz continuity” condition, ensuring that the convergence of fm is equivalent to the convergence of their Laplace transforms. This enables us to establish new approximation theorems for r-times integrated semigroups on E, for all r⩾0. As a consequence, an open problem for the convergence of integrated semigroups on the whole space E, is solved in essence. Moreover, we present an application to nonhomogeneous Cauchy problems.