On a two-point nonlinear hybrid fractional boundary value problem

Rabab Alghamdi; Bashir Ahmad; Sotiris K. Ntouyas; Rabab Alghamdi; Bashir Ahmad; Sotiris K. Ntouyas

doi:10.3934/math.2026258

AIMS Mathematics

2026, Volume 11, Issue 3: 6251-6268. doi: 10.3934/math.2026258

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On a two-point nonlinear hybrid fractional boundary value problem

1.
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2.
College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
3.
Department of Mathematics, University of Ioannina, 45110, Ioannina, Greece

Received: 17 January 2026 Revised: 13 February 2026 Accepted: 03 March 2026 Published: 12 March 2026
MSC : 34A08, 34A38, 34B15

We study a nonlinear Caputo type hybrid fractional differential equation subject to a two-point boundary condition. The existence of at least one solution is proven by applying a Burton-type modification of Krasnosel'skii's fixed point theorem. To establish uniqueness, we derive a novel form of Grönwall's inequality and employ it to show that the problem admits a unique solution. In addition, the uniqueness of solutions for the proposed hybrid fractional boundary value problem is further analyzed through Banach's contraction principle. Moreover, the Ulam–Hyers stability of the model is investigated. Finally, illustrative examples are provided to demonstrate the applicability of the theoretical results.
- fractional hybrid differential equations,
- existence,
- Grönwall's inequality,
- fixed point,
- Ulam-Hyers stability
Citation: Rabab Alghamdi, Bashir Ahmad, Sotiris K. Ntouyas. On a two-point nonlinear hybrid fractional boundary value problem[J]. AIMS Mathematics, 2026, 11(3): 6251-6268. doi: 10.3934/math.2026258

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Abstract

We study a nonlinear Caputo type hybrid fractional differential equation subject to a two-point boundary condition. The existence of at least one solution is proven by applying a Burton-type modification of Krasnosel'skii's fixed point theorem. To establish uniqueness, we derive a novel form of Grönwall's inequality and employ it to show that the problem admits a unique solution. In addition, the uniqueness of solutions for the proposed hybrid fractional boundary value problem is further analyzed through Banach's contraction principle. Moreover, the Ulam–Hyers stability of the model is investigated. Finally, illustrative examples are provided to demonstrate the applicability of the theoretical results.

References

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