On Hadamard fractional operator and three-point fractional boundary value problem in integral-form Hölder spaces

Mohamed M. A. Metwali; Jihan Alahmadi; Fawziah M. Alotaibi; Mohammad Esmael Samei; Mohamed M. A. Metwali; Jihan Alahmadi; Fawziah M. Alotaibi; Mohammad Esmael Samei

doi:10.3934/math.2026289

AIMS Mathematics

2026, Volume 11, Issue 3: 7047-7065. doi: 10.3934/math.2026289

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On Hadamard fractional operator and three-point fractional boundary value problem in integral-form Hölder spaces

1.
Department of Mathematics, College of Science and Humanities in AlKharj, Prince Sattam Bin Abdulaziz University, AlKharj 11942, Saudi Arabia
2.
Department of Mathematics, Turabah University college, Taif University, P.O.Box 11099, Taif 21944, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran

Received: 15 December 2025 Revised: 04 March 2026 Accepted: 12 March 2026 Published: 18 March 2026
MSC : 46E25, 46E30, 47H30, 47N2

The paper aims to put forward and discuss the proper assumptions for the existence, in addition to the uniqueness, of the solutions to the non-local fractional boundary value problem containing a Hadamard-type fractional operator in integral-form Hölder Banach space $ J_{\alpha, \beta} $, which has much better properties than both classical Hölder spaces $ C^\alpha $ and the space of continuous solutions $ C $. Based on these results, we investigate and prove some essential properties of the Hadamard fractional operators, such as the boundedness, acting, and continuity in the studied spaces. Our analysis is based on Darbo's fixed point principle and the measure of noncompactness with fractional calculus. The results are confirmed with a numerical example.
- integral-form Hölder space,
- Darbo's fixed point theorem,
- Hadamard-type fractional operator,
- measure of noncompactness
Citation: Mohamed M. A. Metwali, Jihan Alahmadi, Fawziah M. Alotaibi, Mohammad Esmael Samei. On Hadamard fractional operator and three-point fractional boundary value problem in integral-form Hölder spaces[J]. AIMS Mathematics, 2026, 11(3): 7047-7065. doi: 10.3934/math.2026289

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Abstract

The paper aims to put forward and discuss the proper assumptions for the existence, in addition to the uniqueness, of the solutions to the non-local fractional boundary value problem containing a Hadamard-type fractional operator in integral-form Hölder Banach space $ J_{\alpha, \beta} $, which has much better properties than both classical Hölder spaces $ C^\alpha $ and the space of continuous solutions $ C $. Based on these results, we investigate and prove some essential properties of the Hadamard fractional operators, such as the boundedness, acting, and continuity in the studied spaces. Our analysis is based on Darbo's fixed point principle and the measure of noncompactness with fractional calculus. The results are confirmed with a numerical example.

References

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