Decompositions of Weakly Compact Valued Integrable Multifunctions
We give a short overview on the decomposition property for integrable multifunctions, i.e., when an “integrable in a certain sense” multifunction can be represented as a sum of one of its integrable selections and a multifunction integrable in a narrower sense. The decomposition theorems are importa...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/6/863 |
| _version_ | 1797566960049848320 |
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| author | Luisa Di Piazza Kazimierz Musiał |
| author_facet | Luisa Di Piazza Kazimierz Musiał |
| author_sort | Luisa Di Piazza |
| collection | DOAJ |
| description | We give a short overview on the decomposition property for integrable multifunctions, i.e., when an “integrable in a certain sense” multifunction can be represented as a sum of one of its integrable selections and a multifunction integrable in a narrower sense. The decomposition theorems are important tools of the theory of multivalued integration since they allow us to see an integrable multifunction as a translation of a multifunction with better properties. Consequently, they provide better characterization of integrable multifunctions under consideration. There is a large literature on it starting from the seminal paper of the authors in 2006, where the property was proved for Henstock integrable multifunctions taking compact convex values in a separable Banach space <i>X</i>. In this paper, we summarize the earlier results, we prove further results and present tables which show the state of art in this topic. |
| first_indexed | 2024-03-10T19:34:52Z |
| format | Article |
| id | doaj.art-925127ad22b84db88caf40040546f27c |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T19:34:52Z |
| publishDate | 2020-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-925127ad22b84db88caf40040546f27c2023-11-20T01:47:47ZengMDPI AGMathematics2227-73902020-05-018686310.3390/math8060863Decompositions of Weakly Compact Valued Integrable MultifunctionsLuisa Di Piazza0Kazimierz Musiał1Department of Mathematics, University of Palermo, Via Archirafi 34, 90123 Palermo, ItalyInstitut of Mathematics, Wrocław University, Pl. Grunwaldzki 2/4, 50-384 Wrocław, PolandWe give a short overview on the decomposition property for integrable multifunctions, i.e., when an “integrable in a certain sense” multifunction can be represented as a sum of one of its integrable selections and a multifunction integrable in a narrower sense. The decomposition theorems are important tools of the theory of multivalued integration since they allow us to see an integrable multifunction as a translation of a multifunction with better properties. Consequently, they provide better characterization of integrable multifunctions under consideration. There is a large literature on it starting from the seminal paper of the authors in 2006, where the property was proved for Henstock integrable multifunctions taking compact convex values in a separable Banach space <i>X</i>. In this paper, we summarize the earlier results, we prove further results and present tables which show the state of art in this topic.https://www.mdpi.com/2227-7390/8/6/863gauge multivalued integralscalarly defined multivalued integraldecomposition of a multifunction |
| spellingShingle | Luisa Di Piazza Kazimierz Musiał Decompositions of Weakly Compact Valued Integrable Multifunctions Mathematics gauge multivalued integral scalarly defined multivalued integral decomposition of a multifunction |
| title | Decompositions of Weakly Compact Valued Integrable Multifunctions |
| title_full | Decompositions of Weakly Compact Valued Integrable Multifunctions |
| title_fullStr | Decompositions of Weakly Compact Valued Integrable Multifunctions |
| title_full_unstemmed | Decompositions of Weakly Compact Valued Integrable Multifunctions |
| title_short | Decompositions of Weakly Compact Valued Integrable Multifunctions |
| title_sort | decompositions of weakly compact valued integrable multifunctions |
| topic | gauge multivalued integral scalarly defined multivalued integral decomposition of a multifunction |
| url | https://www.mdpi.com/2227-7390/8/6/863 |
| work_keys_str_mv | AT luisadipiazza decompositionsofweaklycompactvaluedintegrablemultifunctions AT kazimierzmusiał decompositionsofweaklycompactvaluedintegrablemultifunctions |