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- W1538134127 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 $imathbb {R}$. Suppose that $E$ is countable and $alpha left | int _{0}^{infty }e^{-(alpha + ieta ) u} f(s+u) du right | to 0$ uniformly for $sge 0$, as $alpha searrow 0$, for each $eta$ in $E$. It is shown that [ left | int _{0}^{t} e^{-imu u} f(u) du - widetilde f(imu ) right | to 0] as $tto infty$, 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." @default.
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- W1538134127 date "1998-01-01" @default.
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- W1538134127 title "Tauberian theorems and stability of solutions of the Cauchy problem" @default.
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- W1538134127 doi "https://doi.org/10.1090/s0002-9947-98-01920-5" @default.
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