Tauberian theorems and stability of solutions of the Cauchy problem - Batty, C; Van Neerven, J; Rabiger, F (1998)
Let f : ℝ + → X be a bounded, strongly measurable function with values in a Banach space X, and let iE be the singular set of the Laplace transform f ̃ in iℝ. Suppose that E is.countable and α ∥ f 0∞ e -(a+il)u/(s + u)du∥ → 0 uniformly for s ≥ 0, as α ↘0, for each η in E. It is shown that ∥ 0t e -iμuf(u)du-f̃(iμ)∥ →0 as t → ∞, for each μ, in ℝ \ E; in particular, ∥f(t)∥ → 0 if f is uniformly continuous. This result is similar to a Tauberian theorem of Arendt and Batty. It is obtained by applying a result of the authors concerning local stability of bounded semigroups to the translation semigroup on BUC(ℝ +, X), and it implies several results concerning stability of solutions of Cauchy problems. ©1998 American Mathematical Society.