Tauberian theorems and stability of solutions of the Cauchy problem
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 transform f ̃ in iℝ. Suppose that E is.countable and α ∥ f 0∞ e -(a+il)u/(s + u)du∥ → 0 un...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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1998
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| _version_ | 1797067590674153472 |
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| author | Batty, C Van Neerven, J Rabiger, F |
| author_facet | Batty, C Van Neerven, J Rabiger, F |
| author_sort | Batty, C |
| collection | OXFORD |
| description | 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 transform f ̃ in iℝ. Suppose that E is.countable and α ∥ f 0∞ e -(a+il)u/(s + u)du∥ → 0 uniformly for s ≥ 0, as α ↘0, for each η in E. It is shown that ∥ 0t e -iμuf(u)du-f̃(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. ©1998 American Mathematical Society. |
| first_indexed | 2024-03-06T21:58:31Z |
| format | Journal article |
| id | oxford-uuid:4dc7d823-6019-4e1b-9978-de72b7a51152 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T21:58:31Z |
| publishDate | 1998 |
| record_format | dspace |
| spelling | oxford-uuid:4dc7d823-6019-4e1b-9978-de72b7a511522022-03-26T15:57:19ZTauberian theorems and stability of solutions of the Cauchy problemJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:4dc7d823-6019-4e1b-9978-de72b7a51152EnglishSymplectic Elements at Oxford1998Batty, CVan Neerven, JRabiger, FLet 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 transform f ̃ in iℝ. Suppose that E is.countable and α ∥ f 0∞ e -(a+il)u/(s + u)du∥ → 0 uniformly for s ≥ 0, as α ↘0, for each η in E. It is shown that ∥ 0t e -iμuf(u)du-f̃(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. ©1998 American Mathematical Society. |
| spellingShingle | Batty, C Van Neerven, J Rabiger, F Tauberian theorems and stability of solutions of the Cauchy problem |
| title | Tauberian theorems and stability of solutions of the Cauchy problem |
| title_full | Tauberian theorems and stability of solutions of the Cauchy problem |
| title_fullStr | Tauberian theorems and stability of solutions of the Cauchy problem |
| title_full_unstemmed | Tauberian theorems and stability of solutions of the Cauchy problem |
| title_short | Tauberian theorems and stability of solutions of the Cauchy problem |
| title_sort | tauberian theorems and stability of solutions of the cauchy problem |
| work_keys_str_mv | AT battyc tauberiantheoremsandstabilityofsolutionsofthecauchyproblem AT vanneervenj tauberiantheoremsandstabilityofsolutionsofthecauchyproblem AT rabigerf tauberiantheoremsandstabilityofsolutionsofthecauchyproblem |