Existence of Solutions to Fractional Differential Equations with Mixed Caputo–Riemann Derivative
The study of fractional differential equations is gaining increasing significance due to their wide-ranging applications across various fields. Different methods, including fixed-point theory, variational approaches, and the lower and upper solutions method, are employed to analyze the existence and...
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| Language: | English |
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MDPI AG
2025-06-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/9/6/374 |
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| author | Mahir Almatarneh Sonuc Zorlu Nazim I. Mahmudov |
| author_facet | Mahir Almatarneh Sonuc Zorlu Nazim I. Mahmudov |
| author_sort | Mahir Almatarneh |
| collection | DOAJ |
| description | The study of fractional differential equations is gaining increasing significance due to their wide-ranging applications across various fields. Different methods, including fixed-point theory, variational approaches, and the lower and upper solutions method, are employed to analyze the existence and uniqueness of solutions to fractional differential equations. This paper investigates the existence and uniqueness of solutions to a class of nonlinear fractional differential equations involving mixed Caputo–Riemann fractional derivatives with integral initial conditions, set within a Banach space. Sufficient conditions are provided for the existence and uniqueness of solutions based on the problem’s parameters. The results are derived by constructing the Green’s function for the initial value problem. Schauder’s fixed-point theorem is used to prove existence, while Banach’s contraction mapping principle ensures uniqueness. Finally, an example is given to demonstrate the practical application of the results. |
| format | Article |
| id | doaj-art-31cc9cbbc561499291a0991bd5ac7d75 |
| institution | Kabale University |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-31cc9cbbc561499291a0991bd5ac7d752025-08-20T03:26:56ZengMDPI AGFractal and Fractional2504-31102025-06-019637410.3390/fractalfract9060374Existence of Solutions to Fractional Differential Equations with Mixed Caputo–Riemann DerivativeMahir Almatarneh0Sonuc Zorlu1Nazim I. Mahmudov2Department of Mathematics, Eastern Mediterranean University, Mersin 10, Famagusta 99628, North Cyprus, TurkeyDepartment of Mathematics, Eastern Mediterranean University, Mersin 10, Famagusta 99628, North Cyprus, TurkeyDepartment of Mathematics, Eastern Mediterranean University, Mersin 10, Famagusta 99628, North Cyprus, TurkeyThe study of fractional differential equations is gaining increasing significance due to their wide-ranging applications across various fields. Different methods, including fixed-point theory, variational approaches, and the lower and upper solutions method, are employed to analyze the existence and uniqueness of solutions to fractional differential equations. This paper investigates the existence and uniqueness of solutions to a class of nonlinear fractional differential equations involving mixed Caputo–Riemann fractional derivatives with integral initial conditions, set within a Banach space. Sufficient conditions are provided for the existence and uniqueness of solutions based on the problem’s parameters. The results are derived by constructing the Green’s function for the initial value problem. Schauder’s fixed-point theorem is used to prove existence, while Banach’s contraction mapping principle ensures uniqueness. Finally, an example is given to demonstrate the practical application of the results.https://www.mdpi.com/2504-3110/9/6/374Caputo and Riemann fractional derivativesSchauder’s fixed-point theoremBanach contraction mapping principle |
| spellingShingle | Mahir Almatarneh Sonuc Zorlu Nazim I. Mahmudov Existence of Solutions to Fractional Differential Equations with Mixed Caputo–Riemann Derivative Fractal and Fractional Caputo and Riemann fractional derivatives Schauder’s fixed-point theorem Banach contraction mapping principle |
| title | Existence of Solutions to Fractional Differential Equations with Mixed Caputo–Riemann Derivative |
| title_full | Existence of Solutions to Fractional Differential Equations with Mixed Caputo–Riemann Derivative |
| title_fullStr | Existence of Solutions to Fractional Differential Equations with Mixed Caputo–Riemann Derivative |
| title_full_unstemmed | Existence of Solutions to Fractional Differential Equations with Mixed Caputo–Riemann Derivative |
| title_short | Existence of Solutions to Fractional Differential Equations with Mixed Caputo–Riemann Derivative |
| title_sort | existence of solutions to fractional differential equations with mixed caputo riemann derivative |
| topic | Caputo and Riemann fractional derivatives Schauder’s fixed-point theorem Banach contraction mapping principle |
| url | https://www.mdpi.com/2504-3110/9/6/374 |
| work_keys_str_mv | AT mahiralmatarneh existenceofsolutionstofractionaldifferentialequationswithmixedcaputoriemannderivative AT sonuczorlu existenceofsolutionstofractionaldifferentialequationswithmixedcaputoriemannderivative AT nazimimahmudov existenceofsolutionstofractionaldifferentialequationswithmixedcaputoriemannderivative |