About the Resolvent Kernel of Neutral Linear Fractional System with Distributed Delays
The present work considers the initial problem (IP) for a linear neutral system with derivatives in Caputo’s sense of incommensurate order, distributed delay and various kinds of initial functions. For the considered IP, the studied problem of existence and uniqueness of a resolvent kernel under som...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/23/4573 |
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| author | Hristo Kiskinov Mariyan Milev Andrey Zahariev |
| author_facet | Hristo Kiskinov Mariyan Milev Andrey Zahariev |
| author_sort | Hristo Kiskinov |
| collection | DOAJ |
| description | The present work considers the initial problem (IP) for a linear neutral system with derivatives in Caputo’s sense of incommensurate order, distributed delay and various kinds of initial functions. For the considered IP, the studied problem of existence and uniqueness of a resolvent kernel under some natural assumptions of boundedness type. In the case when, in the system, the term which describes the outer forces is a locally Lebesgue integrable function and the initial function is continuous, it is proved that the studied IP has a unique solution, which has an integral representation via the corresponding resolvent kernel. Applying the obtained results, we establish that, from the existence and uniqueness of a resolvent kernel, the existence and uniqueness of a fundamental matrix of the homogeneous system and vice versa follows. An explicit formula describing the relationship between the resolvent kernel and the fundamental matrix is proved as well. |
| first_indexed | 2024-03-09T17:40:52Z |
| format | Article |
| id | doaj.art-9be48bba8f084f588c95c339f8a8c280 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T17:40:52Z |
| publishDate | 2022-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-9be48bba8f084f588c95c339f8a8c2802023-11-24T11:35:58ZengMDPI AGMathematics2227-73902022-12-011023457310.3390/math10234573About the Resolvent Kernel of Neutral Linear Fractional System with Distributed DelaysHristo Kiskinov0Mariyan Milev1Andrey Zahariev2Faculty of Mathematics and Informatics, University of Plovdiv, 4000 Plovdiv, BulgariaDepartment of Mathematics, Physics and Information Technologies, University of Food Technology, 4002 Plovdiv, BulgariaFaculty of Mathematics and Informatics, University of Plovdiv, 4000 Plovdiv, BulgariaThe present work considers the initial problem (IP) for a linear neutral system with derivatives in Caputo’s sense of incommensurate order, distributed delay and various kinds of initial functions. For the considered IP, the studied problem of existence and uniqueness of a resolvent kernel under some natural assumptions of boundedness type. In the case when, in the system, the term which describes the outer forces is a locally Lebesgue integrable function and the initial function is continuous, it is proved that the studied IP has a unique solution, which has an integral representation via the corresponding resolvent kernel. Applying the obtained results, we establish that, from the existence and uniqueness of a resolvent kernel, the existence and uniqueness of a fundamental matrix of the homogeneous system and vice versa follows. An explicit formula describing the relationship between the resolvent kernel and the fundamental matrix is proved as well.https://www.mdpi.com/2227-7390/10/23/4573fractional derivativesneutral fractional systemsdistributed delayintegral representationresolvent kernel |
| spellingShingle | Hristo Kiskinov Mariyan Milev Andrey Zahariev About the Resolvent Kernel of Neutral Linear Fractional System with Distributed Delays Mathematics fractional derivatives neutral fractional systems distributed delay integral representation resolvent kernel |
| title | About the Resolvent Kernel of Neutral Linear Fractional System with Distributed Delays |
| title_full | About the Resolvent Kernel of Neutral Linear Fractional System with Distributed Delays |
| title_fullStr | About the Resolvent Kernel of Neutral Linear Fractional System with Distributed Delays |
| title_full_unstemmed | About the Resolvent Kernel of Neutral Linear Fractional System with Distributed Delays |
| title_short | About the Resolvent Kernel of Neutral Linear Fractional System with Distributed Delays |
| title_sort | about the resolvent kernel of neutral linear fractional system with distributed delays |
| topic | fractional derivatives neutral fractional systems distributed delay integral representation resolvent kernel |
| url | https://www.mdpi.com/2227-7390/10/23/4573 |
| work_keys_str_mv | AT hristokiskinov abouttheresolventkernelofneutrallinearfractionalsystemwithdistributeddelays AT mariyanmilev abouttheresolventkernelofneutrallinearfractionalsystemwithdistributeddelays AT andreyzahariev abouttheresolventkernelofneutrallinearfractionalsystemwithdistributeddelays |