Tauberian theorems and stability of solutions of the Cauchy problem

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