Classification of Solution Operators Semigroups for Abstract Cauchy Problems
The paper is devoted to studying solution operators semigroups and its generators for abstract Cauchy problem in Banach space. It is considered two types of families — "classical” that defined on whole Banach space and possesses the semigroup property, and "regularized" that can be de...
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| Format: | Article |
| Language: | English |
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Irkutsk State University
2014-09-01
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| Series: | Известия Иркутского государственного университета: Серия "Математика" |
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| Online Access: | http://isu.ru/journal/downloadArticle?article=_995c077c9d5c4da0afad58f5b3bf590e&lang=rus |
| _version_ | 1819042294603448320 |
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| author | V.S. Parfenenkova |
| author_facet | V.S. Parfenenkova |
| author_sort | V.S. Parfenenkova |
| collection | DOAJ |
| description | The paper is devoted to studying solution operators semigroups and its generators for abstract Cauchy problem in Banach space. It is considered two types of families — "classical” that defined on whole Banach space and possesses the semigroup property, and "regularized" that can be defined on some subspace, it doesn’t possess the semigroup property but some their transformation possesses. Among the classical semigroups are considered semigroups of class C0, Cesaro-summable and Abel-summable semigroups, semigroups of classes Ck and ℭk, semigroups of growth α. Among the regularized semigroups are considered integrated semigroups, R-semigroups, convoluted semigroups. For each kind of regularized semigroups it’s described the regularization method that allows to consider the amended semigroup property defined on whole Banach space. Also for each kind of regularized semigroups are considered the definition of its generator and in addition the exponentially bounded and local versions of semigroups.
The paper deduces the diagram of solution operators semigroups inclusions. Implications that involve regularized semigroups are by embedding of generators. Implication with pair of classical semigroups are by embedding of semigroups themeselves and as a consequence by embedding of generators too. Particular attention is paid for giving an examples that prove strictness for some embeddings. For the simplicity of the main diagram the relationship between Abel-summable semigroups (i.e. semigroups of classes Ab, (0, Ab), (1,Ab)) and their relationship with semigroups of class Ck are taken out into separate diagram. |
| first_indexed | 2024-12-21T09:38:36Z |
| format | Article |
| id | doaj.art-6be3a7b054f84769a870e7b928fd2e52 |
| institution | Directory Open Access Journal |
| issn | 1997-7670 2541-8785 |
| language | English |
| last_indexed | 2024-12-21T09:38:36Z |
| publishDate | 2014-09-01 |
| publisher | Irkutsk State University |
| record_format | Article |
| series | Известия Иркутского государственного университета: Серия "Математика" |
| spelling | doaj.art-6be3a7b054f84769a870e7b928fd2e522022-12-21T19:08:32ZengIrkutsk State UniversityИзвестия Иркутского государственного университета: Серия "Математика"1997-76702541-87852014-09-0191103117Classification of Solution Operators Semigroups for Abstract Cauchy ProblemsV.S. ParfenenkovaThe paper is devoted to studying solution operators semigroups and its generators for abstract Cauchy problem in Banach space. It is considered two types of families — "classical” that defined on whole Banach space and possesses the semigroup property, and "regularized" that can be defined on some subspace, it doesn’t possess the semigroup property but some their transformation possesses. Among the classical semigroups are considered semigroups of class C0, Cesaro-summable and Abel-summable semigroups, semigroups of classes Ck and ℭk, semigroups of growth α. Among the regularized semigroups are considered integrated semigroups, R-semigroups, convoluted semigroups. For each kind of regularized semigroups it’s described the regularization method that allows to consider the amended semigroup property defined on whole Banach space. Also for each kind of regularized semigroups are considered the definition of its generator and in addition the exponentially bounded and local versions of semigroups. The paper deduces the diagram of solution operators semigroups inclusions. Implications that involve regularized semigroups are by embedding of generators. Implication with pair of classical semigroups are by embedding of semigroups themeselves and as a consequence by embedding of generators too. Particular attention is paid for giving an examples that prove strictness for some embeddings. For the simplicity of the main diagram the relationship between Abel-summable semigroups (i.e. semigroups of classes Ab, (0, Ab), (1,Ab)) and their relationship with semigroups of class Ck are taken out into separate diagram.http://isu.ru/journal/downloadArticle?article=_995c077c9d5c4da0afad58f5b3bf590e&lang=rusabstract Cauchy problemsolution operators Semigroupgenerator of semigroup |
| spellingShingle | V.S. Parfenenkova Classification of Solution Operators Semigroups for Abstract Cauchy Problems Известия Иркутского государственного университета: Серия "Математика" abstract Cauchy problem solution operators Semigroup generator of semigroup |
| title | Classification of Solution Operators Semigroups for Abstract Cauchy Problems |
| title_full | Classification of Solution Operators Semigroups for Abstract Cauchy Problems |
| title_fullStr | Classification of Solution Operators Semigroups for Abstract Cauchy Problems |
| title_full_unstemmed | Classification of Solution Operators Semigroups for Abstract Cauchy Problems |
| title_short | Classification of Solution Operators Semigroups for Abstract Cauchy Problems |
| title_sort | classification of solution operators semigroups for abstract cauchy problems |
| topic | abstract Cauchy problem solution operators Semigroup generator of semigroup |
| url | http://isu.ru/journal/downloadArticle?article=_995c077c9d5c4da0afad58f5b3bf590e&lang=rus |
| work_keys_str_mv | AT vsparfenenkova classificationofsolutionoperatorssemigroupsforabstractcauchyproblems |