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Under the effect of the Rosenblatt process, the well-posedness and Hyers–Ulam stability of nonlinear fractional stochastic delay systems are considered. First, depending on fixed-point theory, the existence and uniqueness of solutions are proven. Next, utilizing the delayed Mittag–Leffler matrix functions and Grönwall's inequality, sufficient criteria for Hyers–Ulam stability are established. Ultimately, an example is presented to demonstrate the effectiveness of the obtained findings. [ABSTRACT FROM AUTHOR]

Under the effect of the Rosenblatt process, the well-posedness and Hyers–Ulam stability of nonlinear fractional stochastic delay systems are considered. First, depending on fixed-point theory, the existence and uniqueness of solutions are proven. Next, utilizing the delayed Mittag–Leffler matrix functions and Grönwall’s inequality, sufficient criteria for Hyers–Ulam stability are established. Ultimately, an example is presented to demonstrate the effectiveness of the obtained findings.

This paper first finishes off the exact solutions of Riemann‐Liouville fractional differential time‐delay oscillatory system of order ρ∈(1,2)$$ \rho \in \left(1,2\right) $$ by using two newly defined delayed perturbations of Mittag‐Leffler matrix functions and constant variation method. In the light of the exact solutions, we explore the finite time stability of the nonhomogeneous fractional oscillatory differential equations with pure delay. Ultimately, an example is cited to verify the rationality of the results. Through our method, the public problems left by Mahmudov in 2022 were partially solved. [ABSTRACT FROM AUTHOR]

Linear and nonlinear fractional-delay systems are studied. As an application, we derive the controllability and Hyers–Ulam stability results using the representation of solutions of these systems with the help of their delayed Mittag–Leffler matrix functions. We provide some sufficient and necessary conditions for the controllability of linear fractional-delay systems by introducing a fractional delay Gramian matrix. Furthermore, we establish some sufficient conditions of controllability and Hyers–Ulam stability of nonlinear fractional-delay systems by applying Krasnoselskii's fixed-point theorem. Our results improve, extend, and complement some existing ones. Finally, numerical examples of linear and nonlinear fractional-delay systems are presented to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

Nonhomogeneous systems of fractional differential equations with pure delay are considered. As an application, the representation of solutions of these systems and their delayed Mittag-Leffler matrix functions are used to obtain the finite time stability results. Our results improve and extend the previous related results. Finally, to illustrate our theoretical results, we give an example. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to investigate the finite-time stability of Hadamard fractional differential equations (HFDEs). Firstly, the standard definitions of finite-time stability of HFDEs in compatible Banach spaces are proposed. In light of the method of successive approximation and Beesack inequality with weakly singular kernel, the criteria of finite-time stability for linear and nonlinear HFDEs are established, respectively. Then with regard to linear HFDEs with pure delay, a novel fractional delayed matrix function (also called delayed Mittag-Leffler matrix function) is given. Specific to nonlinear HFDEs with constant time delay, both Beesack inequality and Hölder inequality are utilized in the framework of the generalized Lipschitz condition. Finally, several indispensable simulations are implemented to verify the effectiveness and practicability of the main results. [ABSTRACT FROM AUTHOR]

In this work, we consider linear and nonlinear fractional stochastic delay systems driven by the Rosenblatt process. With the aid of the delayed Mittag-Leffler matrix functions and the representation of solutions of these systems, we derive the controllability results as an application. By introducing a fractional delayed Gramian matrix, we provide sufficient and necessary criteria for the controllability of linear fractional stochastic delay systems. Furthermore, by employing Krasnoselskii’s fixed point theorem, we establish sufficient conditions for the controllability of nonlinear fractional stochastic delay systems. Finally, an example is given to illustrate the main results.

Linear and nonlinear fractional-delay systems are studied. As an application, we derive the controllability and Hyers–Ulam stability results using the representation of solutions of these systems with the help of their delayed Mittag–Leffler matrix functions. We provide some sufficient and necessary conditions for the controllability of linear fractional-delay systems by introducing a fractional delay Gramian matrix. Furthermore, we establish some sufficient conditions of controllability and Hyers–Ulam stability of nonlinear fractional-delay systems by applying Krasnoselskii’s fixed-point theorem. Our results improve, extend, and complement some existing ones. Finally, numerical examples of linear and nonlinear fractional-delay systems are presented to demonstrate the theoretical results.

Nonhomogeneous systems of fractional differential equations with pure delay are considered. As an application, the representation of solutions of these systems and their delayed Mittag-Leffler matrix functions are used to obtain the finite time stability results. Our results improve and extend the previous related results. Finally, to illustrate our theoretical results, we give an example.