Existence theory for finite delayed fractional differential equations with nonlinear variable order

Ibraheem M. Alsulami; Ramsha Shafqat; Ibraheem M. Alsulami; Ramsha Shafqat

doi:10.3934/math.2026247

AIMS Mathematics

2026, Volume 11, Issue 3: 5966-5991. doi: 10.3934/math.2026247

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Existence theory for finite delayed fractional differential equations with nonlinear variable order

Ibraheem M. Alsulami ¹,
Ramsha Shafqat ^{2
,
,}

1.
Mathematics Department, Faculty of Science, Umm Al-Qura University, Makkah 21955, Saudi Arabia
2.
Department of Mathematics and Statistics, The University of Lahore, Sargodha 40100, Pakistan

Received: 20 January 2026 Revised: 17 February 2026 Accepted: 03 March 2026 Published: 09 March 2026
MSC : 34D20, 34K20, 34K60, 92C60, 92D45

We investigated a class of finite-delay fractional differential equations in which the differentiation order combined with time and depended explicitly on the past evolution of the state. The proposed model combined a history-dependent variable fractional order with a nonlinear source term involving an integral memory functional, rather than a pointwise delay. This structure provided a flexible framework for describing adaptive memory effects and cumulative nonlocal behavior that cannot be captured by classical constant-order or pointwise variable-order models. By reformulating the problem as an equivalent integral equation, we established rigorous existence and uniqueness results in the Banach space $ L^{1}(\Delta) $ using the Banach contraction principle and the Schauder fixed-point theorem under mild and verifiable assumptions. In addition, Ulam–Hyers stability of the system was proved, ensuring robustness of solutions with respect to perturbations. An illustrative example, together with a numerical stability verification, was presented to support the theoretical findings. The obtained results contributed to the mathematical understanding of delayed variable-order fractional systems and provided a solid foundation for future studies in engineering, economics, and biomedical modeling.
- variable-order fractional differential equations,
- integral memory effects,
- finite delay,
- fixed point theory,
- existence and uniqueness, Ulam-Hyers stability
Citation: Ibraheem M. Alsulami, Ramsha Shafqat. Existence theory for finite delayed fractional differential equations with nonlinear variable order[J]. AIMS Mathematics, 2026, 11(3): 5966-5991. doi: 10.3934/math.2026247

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Abstract

We investigated a class of finite-delay fractional differential equations in which the differentiation order combined with time and depended explicitly on the past evolution of the state. The proposed model combined a history-dependent variable fractional order with a nonlinear source term involving an integral memory functional, rather than a pointwise delay. This structure provided a flexible framework for describing adaptive memory effects and cumulative nonlocal behavior that cannot be captured by classical constant-order or pointwise variable-order models. By reformulating the problem as an equivalent integral equation, we established rigorous existence and uniqueness results in the Banach space $ L^{1}(\Delta) $ using the Banach contraction principle and the Schauder fixed-point theorem under mild and verifiable assumptions. In addition, Ulam–Hyers stability of the system was proved, ensuring robustness of solutions with respect to perturbations. An illustrative example, together with a numerical stability verification, was presented to support the theoretical findings. The obtained results contributed to the mathematical understanding of delayed variable-order fractional systems and provided a solid foundation for future studies in engineering, economics, and biomedical modeling.

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