Extremal Solutions for a Caputo-Type Fractional-Order Initial Value Problem
In this paper, we study the existence of extremal solutions for a Caputo-type fractional-order initial value problem. By using the monotone iteration technique and the upper–lower solution method, we obtain our existence theorem when the nonlinearity satisfies a reverse-type Lipschitz condition. Not...
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| Format: | Article |
| Language: | English |
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MDPI AG
2025-05-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/9/5/308 |
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| _version_ | 1850126899980795904 |
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| author | Keyu Zhang Tian Wang Donal O’Regan Jiafa Xu |
| author_facet | Keyu Zhang Tian Wang Donal O’Regan Jiafa Xu |
| author_sort | Keyu Zhang |
| collection | DOAJ |
| description | In this paper, we study the existence of extremal solutions for a Caputo-type fractional-order initial value problem. By using the monotone iteration technique and the upper–lower solution method, we obtain our existence theorem when the nonlinearity satisfies a reverse-type Lipschitz condition. Note that our nonlinearity depends on the unknown function and its fractional-order derivative. |
| format | Article |
| id | doaj-art-5ce82c2d1317476db3faa026fda9384f |
| institution | OA Journals |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-5ce82c2d1317476db3faa026fda9384f2025-08-20T02:33:48ZengMDPI AGFractal and Fractional2504-31102025-05-019530810.3390/fractalfract9050308Extremal Solutions for a Caputo-Type Fractional-Order Initial Value ProblemKeyu Zhang0Tian Wang1Donal O’Regan2Jiafa Xu3School of Mathematics, Qilu Normal University, Jinan 250013, ChinaSchool of Mathematics, Qilu Normal University, Jinan 250013, ChinaSchool of Mathematical and Statistical Sciences, University of Galway, H91 TK33 Galway, IrelandSchool of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, ChinaIn this paper, we study the existence of extremal solutions for a Caputo-type fractional-order initial value problem. By using the monotone iteration technique and the upper–lower solution method, we obtain our existence theorem when the nonlinearity satisfies a reverse-type Lipschitz condition. Note that our nonlinearity depends on the unknown function and its fractional-order derivative.https://www.mdpi.com/2504-3110/9/5/308Caputo-type fractional-order differential equationsinitial value problemsextremal solutionsupper–lower solution methods |
| spellingShingle | Keyu Zhang Tian Wang Donal O’Regan Jiafa Xu Extremal Solutions for a Caputo-Type Fractional-Order Initial Value Problem Fractal and Fractional Caputo-type fractional-order differential equations initial value problems extremal solutions upper–lower solution methods |
| title | Extremal Solutions for a Caputo-Type Fractional-Order Initial Value Problem |
| title_full | Extremal Solutions for a Caputo-Type Fractional-Order Initial Value Problem |
| title_fullStr | Extremal Solutions for a Caputo-Type Fractional-Order Initial Value Problem |
| title_full_unstemmed | Extremal Solutions for a Caputo-Type Fractional-Order Initial Value Problem |
| title_short | Extremal Solutions for a Caputo-Type Fractional-Order Initial Value Problem |
| title_sort | extremal solutions for a caputo type fractional order initial value problem |
| topic | Caputo-type fractional-order differential equations initial value problems extremal solutions upper–lower solution methods |
| url | https://www.mdpi.com/2504-3110/9/5/308 |
| work_keys_str_mv | AT keyuzhang extremalsolutionsforacaputotypefractionalorderinitialvalueproblem AT tianwang extremalsolutionsforacaputotypefractionalorderinitialvalueproblem AT donaloregan extremalsolutionsforacaputotypefractionalorderinitialvalueproblem AT jiafaxu extremalsolutionsforacaputotypefractionalorderinitialvalueproblem |