Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems
Abstract In this paper, we present the definitions of fractional integrals and fractional derivatives of a Pettis integrable function with respect to another function. This concept follows the idea of Stieltjes-type operators and should allow us to study fractional integrals using methods known from...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2023-05-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13661-023-01745-y |
| _version_ | 1797817880101781504 |
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| author | Hussein A. H. Salem Mieczysław Cichoń Wafa Shammakh |
| author_facet | Hussein A. H. Salem Mieczysław Cichoń Wafa Shammakh |
| author_sort | Hussein A. H. Salem |
| collection | DOAJ |
| description | Abstract In this paper, we present the definitions of fractional integrals and fractional derivatives of a Pettis integrable function with respect to another function. This concept follows the idea of Stieltjes-type operators and should allow us to study fractional integrals using methods known from measure differential equations in abstract spaces. We will show that some of the well-known properties of fractional calculus for the space of Lebesgue integrable functions also hold true in abstract function spaces. In particular, we prove a general Goebel–Rzymowski lemma for the De Blasi measure of weak noncompactness and our fractional integrals. We suggest a new definition of the Caputo fractional derivative with respect to another function, which allows us to investigate the existence of solutions to some Caputo-type fractional boundary value problems. As we deal with some Pettis integrable functions, the main tool utilized in our considerations is based on the technique of measures of weak noncompactness and Mönch’s fixed-point theorem. Finally, to encompass the full scope of this research, some examples illustrating our main results are given. |
| first_indexed | 2024-03-13T08:59:59Z |
| format | Article |
| id | doaj.art-f2e31c2edc334f6395dc7d1c25a37cb6 |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-03-13T08:59:59Z |
| publishDate | 2023-05-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-f2e31c2edc334f6395dc7d1c25a37cb62023-05-28T11:23:18ZengSpringerOpenBoundary Value Problems1687-27702023-05-012023113010.1186/s13661-023-01745-yGeneralized fractional calculus in Banach spaces and applications to existence results for boundary value problemsHussein A. H. Salem0Mieczysław Cichoń1Wafa Shammakh2Department of Mathematics and Computer Science, Faculty of Sciences, Alexandria UniversityFaculty of Mathematics and Computer Science, Adam Mickiewicz UniversityCollege of Science, Department of Mathematics, University of JeddahAbstract In this paper, we present the definitions of fractional integrals and fractional derivatives of a Pettis integrable function with respect to another function. This concept follows the idea of Stieltjes-type operators and should allow us to study fractional integrals using methods known from measure differential equations in abstract spaces. We will show that some of the well-known properties of fractional calculus for the space of Lebesgue integrable functions also hold true in abstract function spaces. In particular, we prove a general Goebel–Rzymowski lemma for the De Blasi measure of weak noncompactness and our fractional integrals. We suggest a new definition of the Caputo fractional derivative with respect to another function, which allows us to investigate the existence of solutions to some Caputo-type fractional boundary value problems. As we deal with some Pettis integrable functions, the main tool utilized in our considerations is based on the technique of measures of weak noncompactness and Mönch’s fixed-point theorem. Finally, to encompass the full scope of this research, some examples illustrating our main results are given.https://doi.org/10.1186/s13661-023-01745-yFractional integralBoundary value problemsOrlicz spacePettis integral |
| spellingShingle | Hussein A. H. Salem Mieczysław Cichoń Wafa Shammakh Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems Boundary Value Problems Fractional integral Boundary value problems Orlicz space Pettis integral |
| title | Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems |
| title_full | Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems |
| title_fullStr | Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems |
| title_full_unstemmed | Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems |
| title_short | Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems |
| title_sort | generalized fractional calculus in banach spaces and applications to existence results for boundary value problems |
| topic | Fractional integral Boundary value problems Orlicz space Pettis integral |
| url | https://doi.org/10.1186/s13661-023-01745-y |
| work_keys_str_mv | AT husseinahsalem generalizedfractionalcalculusinbanachspacesandapplicationstoexistenceresultsforboundaryvalueproblems AT mieczysławcichon generalizedfractionalcalculusinbanachspacesandapplicationstoexistenceresultsforboundaryvalueproblems AT wafashammakh generalizedfractionalcalculusinbanachspacesandapplicationstoexistenceresultsforboundaryvalueproblems |