This manuscript demonstrates the existence, uniqueness, and different kinds of Ulam stability for a $\mathtt{q}$-Caputo implicit fractional jerk coupled system involving $\mathtt{q}$-fractional Erdélyi-Kober integral conditions. The existence and uniqueness results were investigated by employing the Leray-Schauder alternative and the Banach contraction mapping principle. We also derived various kinds of Ulam stability under specific conditions. We provided an example to verify our main results.

This paper presents four uniqueness criteria for the initial value problem of a differential equation which depends on conformable fractional derivative. Among them is the generalization of Nagumo-type uniqueness theory and Lipschitz conditional theory, and advances its development in proving fractional differential equations. Finally, we verify the main conclusions of this paper by providing four concrete examples.

This paper focuses on investigating the existence of positive solutions to boundary value problems (BVPs) of higher-order nonlinear fractional differential equations (FDES) involving fractional boundary conditions. We examine the characteristics of Green's functions. By utilizing the powerful methodologies of Schauder's fixed-point theorem, upper and lower solutions, and cone theory techniques, we obtain significant existence results for nonsingular boundary value problems. For singular problems, we employ cone theory techniques in conjunction with the Leray-Schauder nonlinear alternative to obtain positive solutions. To further clarify and validate the primary findings, several illustrative examples are provided.