Tauberian theorems and stability of solutions of the Cauchy problem

Batty, Charles; van Neerven, Jan; Räbiger, Frank

doi:10.1090/S0002-9947-98-01920-5

Tauberian theorems and stability of solutions of the Cauchy problem
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by Charles J. K. Batty, Jan van Neerven and Frank Räbiger PDF

Trans. Amer. Math. Soc. 350 (1998), 2087-2103 Request permission

Abstract:

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.

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