Analyzing Uniqueness of Solutions in Nonlinear Fractional Differential Equations with Discontinuities Using Lebesgue Spaces
We explore the existence and uniqueness of solutions to nonlinear fractional differential equations (FDEs), defined in the sense of RL-fractional derivatives of order <inline-formula><math display="inline"><semantics><mrow><mi>η</mi><mo>∈</mo>...
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MDPI AG
2024-12-01
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| author | Farva Hafeez Mdi Begum Jeelani Nouf Abdulrahman Alqahtani |
| author_facet | Farva Hafeez Mdi Begum Jeelani Nouf Abdulrahman Alqahtani |
| author_sort | Farva Hafeez |
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| description | We explore the existence and uniqueness of solutions to nonlinear fractional differential equations (FDEs), defined in the sense of RL-fractional derivatives of order <inline-formula><math display="inline"><semantics><mrow><mi>η</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>. The nonlinear term is assumed to have a discontinuity at zero. By employing techniques from Lebesgue spaces, including Holder’s inequality, we establish uniqueness theorems for this problem, analogous to Nagumo, Krasnoselskii–Krein, and Osgood-type results. These findings provide a fundamental framework for understanding the properties of solutions to nonlinear FDEs with discontinuous nonlinearities. |
| format | Article |
| id | doaj-art-ad5cf437e2b040419f25547a2bae94d6 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-ad5cf437e2b040419f25547a2bae94d62025-01-24T13:22:11ZengMDPI AGAxioms2075-16802024-12-011412610.3390/axioms14010026Analyzing Uniqueness of Solutions in Nonlinear Fractional Differential Equations with Discontinuities Using Lebesgue SpacesFarva Hafeez0Mdi Begum Jeelani1Nouf Abdulrahman Alqahtani2Department of Mathematics and Statistics, University of Lahore, Sargodha 40100, PakistanDepartment of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13314, Saudi ArabiaDepartment of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13314, Saudi ArabiaWe explore the existence and uniqueness of solutions to nonlinear fractional differential equations (FDEs), defined in the sense of RL-fractional derivatives of order <inline-formula><math display="inline"><semantics><mrow><mi>η</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>. The nonlinear term is assumed to have a discontinuity at zero. By employing techniques from Lebesgue spaces, including Holder’s inequality, we establish uniqueness theorems for this problem, analogous to Nagumo, Krasnoselskii–Krein, and Osgood-type results. These findings provide a fundamental framework for understanding the properties of solutions to nonlinear FDEs with discontinuous nonlinearities.https://www.mdpi.com/2075-1680/14/1/26FDEsRL-derivativesHolder’s inequalityLebesgue spaces |
| spellingShingle | Farva Hafeez Mdi Begum Jeelani Nouf Abdulrahman Alqahtani Analyzing Uniqueness of Solutions in Nonlinear Fractional Differential Equations with Discontinuities Using Lebesgue Spaces Axioms FDEs RL-derivatives Holder’s inequality Lebesgue spaces |
| title | Analyzing Uniqueness of Solutions in Nonlinear Fractional Differential Equations with Discontinuities Using Lebesgue Spaces |
| title_full | Analyzing Uniqueness of Solutions in Nonlinear Fractional Differential Equations with Discontinuities Using Lebesgue Spaces |
| title_fullStr | Analyzing Uniqueness of Solutions in Nonlinear Fractional Differential Equations with Discontinuities Using Lebesgue Spaces |
| title_full_unstemmed | Analyzing Uniqueness of Solutions in Nonlinear Fractional Differential Equations with Discontinuities Using Lebesgue Spaces |
| title_short | Analyzing Uniqueness of Solutions in Nonlinear Fractional Differential Equations with Discontinuities Using Lebesgue Spaces |
| title_sort | analyzing uniqueness of solutions in nonlinear fractional differential equations with discontinuities using lebesgue spaces |
| topic | FDEs RL-derivatives Holder’s inequality Lebesgue spaces |
| url | https://www.mdpi.com/2075-1680/14/1/26 |
| work_keys_str_mv | AT farvahafeez analyzinguniquenessofsolutionsinnonlinearfractionaldifferentialequationswithdiscontinuitiesusinglebesguespaces AT mdibegumjeelani analyzinguniquenessofsolutionsinnonlinearfractionaldifferentialequationswithdiscontinuitiesusinglebesguespaces AT noufabdulrahmanalqahtani analyzinguniquenessofsolutionsinnonlinearfractionaldifferentialequationswithdiscontinuitiesusinglebesguespaces |