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Tauberian theorems and stability of solutions of the Cauchy problem

Authors:	Charles J K Batty Jan van Neerven Frank Rä biger

Institution:	St. John's College, Oxford OX1 3JP, England ; Department of Mathematics, Delft Technical University, P.O. Box 356, 2600 AJ Delft, The Netherlands ; Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany

Abstract:	Let $f : \mathbb{R}_{+} \to X$ be a bounded, strongly measurable function with values in a Banach space , and let be the singular set of the Laplace transform $\widetilde f$ in $i\mathbb{R}$ . Suppose that is countable and $\alpha \left \\| \int _{0}^{\infty }e^{-(\alpha + i\eta ) u} f(s+u) \, du \right \\| \to 0$ uniformly for $s\ge 0$ , as $\alpha \searrow 0$ , for each $\eta$ in . It is shown that $\begin{displaymath}\left \\| \int _{0}^{t} e^{-i\mu u} f(u) \, du - \widetilde f(i\mu ) \right \\| \to 0\end{displaymath}$ as $t\to \infty$ , for each $\mu$ in $\mathbb{R} \setminus E$ ; in particular, $\\|f(t)\\| \to 0$ if 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 $BUC(\mathbb{R}_{+}, X)$ , and it implies several results concerning stability of solutions of Cauchy problems.

Keywords:	Laplace transform Tauberian theorem singular set countable $C_{0}$-semigroup stability local spectrum orbit Cauchy problem

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