Stability Results for Stochastic Delay Differential Equations in the Framework of Conformable Fractional Derivatives

Abstract

Well-Posedness is essential in various scientific and engineering fields, including physics, engineering, biological sciences, economics, and environmental science. Well-posedness ensures that the mathematical model corresponds to a physically meaningful situation. Without well-posedness, the solution might not make sense in the context of the problem being modeled. The concept of well-posedness refers to certain desirable properties that a differential equation must satisfy, which are existence, uniqueness, and continuous dependency. Regularization is an additional feature of the solution of the differential equation, such as the smoothness of the solution. Stability theory is one of the indispensable qualitative concepts of dynamical systems. We established results about the well-posedness, regularity, and Ulam-Hyers stability of solutions to conformable fractional stochastic delay differential equations. First, we discussed the results about the existence and uniqueness of solutions when the global and local Lipschitz conditions of the coefficients are satisfied. Second, we demonstrated the results about continuously depend on the fractional order $\xi$ and initial values under the global Lipschitz condition of coefficients. Thirdly, we constructed results regarding regularity and Ulam-Hyers stability, and two examples that demonstrate our results are presented. The main part of the proof made use of the truncation process, the Banach fixed point theorem, It\^{o} isometry, temporally weighted norm, Gr\"{o}nwall's inequality, and H\"{o}lder's inequality.