TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 350, Number 5, May 1998, Pages 2087–2103 S 0002-9947(98)01920-5 TAUBERIAN THEOREMS AND STABILITY OF SOLUTIONS OF THE CAUCHY PROBLEM CHARLES J. K. BATTY, JAN VAN NEERVEN, AND FRANK R ¨ ABIGER Abstract. 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 trans- form  f in iR. Suppose that E is countable and α    ∞ 0 e −(α+iη)u f (s + u) du   → 0 uniformly for s ≥ 0, as α ց 0, for each η in E. It is shown that      t 0 e −iμu f (u) du −  f (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. 1. Introduction Let T = {T (t): t ≥ 0} be a C 0 -semigroup, with generator A, on a Banach space X . For x in X , the orbit f (t) := T (t)x has Laplace transform given by the resolvent  f (λ)= R(λ, A)x for Re λ>ω, where ω is the growth bound of the semigroup. Thus Laplace transform theory often has implications for semigroup theory. On the other hand, one can sometimes apply semigroup results to translations on suitable function spaces to recover information about Laplace transforms. In this situation, Tauberian theorems relating Abel, Ces` aro and standard con- vergence of f produce results about the long-time asymptotic behaviour of orbits of semigroups. A simple example of this is a special case of a Tauberian theorem of Ingham [17], for which Korevaar [20] has given an elegant proof by contour integra- tion, and which immediately gives the result that if T is a bounded semigroup and σ(A) ∩ iR is empty, then ‖T (t)A −1 ‖→ 0 as t →∞. A more complicated version of this, initiated in [1], provides an estimate which was exploited in [2] both to establish a more general Tauberian theorem and to show that if T is a bounded semigroup, σ(A) ∩ iR is countable and σ p (A ∗ ) ∩ iR is empty, then ‖T (t)x‖→ 0 as Received by the editors February 12, 1996 and, in revised form, September 6, 1996. 1991 Mathematics Subject Classification. Primary 44A10; Secondary 47D06, 47D03. Key words and phrases. Laplace transform, Tauberian theorem, singular set, countable, C 0 - semigroup, stability, local spectrum, orbit, Cauchy problem. The work on this paper was done during a two-year stay at the University of T¨ ubingen. Support by an Individual Fellowship from the Human Capital and Mobility Programme of the European Community is gratefully acknowledged. I warmly thank Professor Rainer Nagel and the members of his group for their hospitality (second author). It is part of a research project supported by the Deutsche Forschungsgemeinschaft DFG (third author). Work in Oxford was also supported by an EPSRC Visiting Fellowship Research Grant (first and third authors). c 1998 American Mathematical Society 2087 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use