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This research paper presented a novel investigation into an implicit coupled system of fractional variable order, which has not been previously studied in the existing literature. The study focused on establishing and developing sufficient conditions for the existence and uniqueness of solutions, as well as the Ulam-Hyers stability, for the proposed coupled system without using semigroup property. By extending the existing conclusions examined for the Atangana-Baleanu-Caputo (ABC) operator, we contributed to advancing the understanding of variable-order fractional differential equations. The paper provided a solid theoretical foundation for further analysis, numerical simulations, and practical applications. The obtained results have implications for designing and controlling systems modeled using fractional variable order equations and serve as a basis for addressing a wide range of dynamical problems. The transformation techniques, qualitative analysis, and illustrative examples presented in this work highlight its unique contributions and potential to serve as a foundation for future research. [ABSTRACT FROM AUTHOR]

A new class of nonlocal boundary value problems consisting of multi-term delay fractional differential equations and multipoint-integral boundary conditions is studied in this paper. We derive a more general form of the solution for the given problem by applying a fractional integral operator of an arbitrary order βξ instead of β1; for details, see Lemma 2.2. The given problem is converted into an equivalent fixed-point problem to apply the tools of fixed-point theory. The existence of solutions for the given problem is established through the use of a nonlinear alternative of the Leray-Schauder theorem, while the uniqueness of its solutions is shown with the aid of Banach's fixed-point theorem. We also discuss the stability criteria, icluding Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias, and generalized Ulam-Hyers-Rassias stability, for solutions of the problem at hand. For illustration of the abstract results, we present examples. Our results are new and useful for the discipline of multi-term fractional differential equations related to hydrodynamics. The paper concludes with some interesting observations. [ABSTRACT FROM AUTHOR]

We introduce and study a new kind of nonlocal boundary value problems of multi-term fractional differential equations. The existence and uniqueness results for the given problem are obtained by applying standard fixed point theorems. We also construct some examples for demonstrating the application of the main results.

We discuss the existence and uniqueness of solutions for a coupled system of nonlinear multi-term fractional differential equations complemented with coupled nonlocal boundary conditions by applying the methods of modern functional analysis. An example illustrating the uniqueness result is presented. Some interesting observations are also described. [ABSTRACT FROM AUTHOR]

In this paper, we study a new boundary value problem of coupled nonlinear multi‐term fractional differential equations supplemented with integral multi‐strip coupled boundary conditions. The modern tools of fractional analysis are applied to derive the existence and uniqueness results for the given problem. We emphasize that our results are new and enrich the literature on the topic. [ABSTRACT FROM AUTHOR]

In this paper we study a new kind of boundary value problems of multi-term fractional differential inclusions and three-point nonlocal boundary conditions. The existence of solutions is established for convex and non-convex multivalued maps by using standard theorems from the fixed point theory. We also construct some examples for demonstrating the application of the main results. [ABSTRACT FROM AUTHOR]

In this article, we introduce a class of multi-term fractional-order boundary-value problems involving nonlocal integral boundary conditions. Existence results for the given problem are obtained by means of standard tools of fixed point theory. The results are illustrated with the aid of examples and make a useful contribution to the existing literature on the topic. [ABSTRACT FROM AUTHOR]