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Keywords = hybrid neutral stochastic differential equations

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29 pages, 2409 KB  
Article
Mathematical Perspectives of a Coupled System of Nonlinear Hybrid Stochastic Fractional Differential Equations
by Rabeb Sidaoui, Alnadhief H. A. Alfedeel, Jalil Ahmad, Khaled Aldwoah, Amjad Ali, Osman Osman and Ali H. Tedjani
Fractal Fract. 2025, 9(10), 622; https://doi.org/10.3390/fractalfract9100622 - 24 Sep 2025
Viewed by 43
Abstract
This research develops a novel coupled system of nonlinear hybrid stochastic fractional differential equations that integrates neutral effects, stochastic perturbations, and hybrid switching mechanisms. The system is formulated using the Atangana–Baleanu–Caputo fractional operator with a non-singular Mittag–Leffler kernel, which enables accurate representation of [...] Read more.
This research develops a novel coupled system of nonlinear hybrid stochastic fractional differential equations that integrates neutral effects, stochastic perturbations, and hybrid switching mechanisms. The system is formulated using the Atangana–Baleanu–Caputo fractional operator with a non-singular Mittag–Leffler kernel, which enables accurate representation of memory effects without singularities. Unlike existing approaches, which are limited to either neutral or hybrid stochastic structures, the proposed framework unifies both features within a fractional setting, capturing the joint influence of randomness, history, and abrupt transitions in real-world processes. We establish the existence and uniqueness of mild solutions via the Picard approximation method under generalized Carathéodory-type conditions, allowing for non-Lipschitz nonlinearities. In addition, mean-square Mittag–Leffler stability is analyzed to characterize the boundedness and decay properties of solutions under stochastic fluctuations. Several illustrative examples are provided to validate the theoretical findings and demonstrate their applicability. Full article
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21 pages, 456 KB  
Article
Exponential Stability of Highly Nonlinear Hybrid Differently Structured Neutral Stochastic Differential Equations with Unbounded Delays
by Boliang Lu, Quanxin Zhu and Ping He
Fractal Fract. 2022, 6(7), 385; https://doi.org/10.3390/fractalfract6070385 - 9 Jul 2022
Cited by 5 | Viewed by 1948
Abstract
This paper mainly studies the exponential stability of the highly nonlinear hybrid neutral stochastic differential equations (NSDEs) with multiple unbounded time-dependent delays and different structures. We prove the existence and uniqueness of the exact global solution of the new stochastic system, and then [...] Read more.
This paper mainly studies the exponential stability of the highly nonlinear hybrid neutral stochastic differential equations (NSDEs) with multiple unbounded time-dependent delays and different structures. We prove the existence and uniqueness of the exact global solution of the new stochastic system, and then give several criteria of the exponential stability, including the q1th moment and almost surely exponential stability. Additionally, some numerical examples are given to illustrate the main results. Such systems are widely applied in physics and other fields. For example, a specific case is pantograph dynamics, in which the delay term is a proportional function. These are widely used to determine the motion of a pantograph head on an electric locomotive collecting current from an overhead trolley wire. Compared with the existing works, our results extend the single constant delay of coefficients to multiple unbounded time-dependent delays, which is more general and applicable. Full article
(This article belongs to the Special Issue Fractional Processes and Multidisciplinary Applications)
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