Qualitative Analysis and Applications of Fractional Stochastic Systems with Non-Instantaneous Impulses
Muhammad Imran Liaqat, Abdelhamid Mohammed Djaouti
Source abstract
Fractional stochastic differential Equations (FSDEs) with time delays and non-instantaneous impulses describe dynamical systems whose evolution relies not only on their current state but also on their historical context, random fluctuations, and impulsive effects that manifest over finite intervals rather than occurring instantaneously. This combination of features offers a more precise framework for capturing critical aspects of many real-world processes. Recent findings demonstrate the existence, uniqueness, and Ulam–Hyers stability of standard fractional stochastic systems. In this study, we extend these results to include systems characterized by FSDEs that incorporate time delays and non-instantaneous impulses. We prove the existence and uniqueness of the solution for this system using Krasnoselskii’s and Banach’s fixed-point theorems. Additionally, we present findings related to Ulam–Hyers stability. To illustrate the practical application of our results, we develop a population model that incorporates memory effects, randomness, and non-instantaneous impulses. This model is solved numerically via the Euler–Maruyama method, and graphical simulations effectively depict the dynamic behavior of the system.
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