This paper develops a functional operator-theoretic framework for nonlinear Erdelyi–Kober (EK) fractional dynamical systems formulated in Banach spaces endowed with the compact-open topology. Within this setting, sufficient conditions for existence, uniqueness, and Ulam–Hyers stability of solutions are established using the Banach and Schaefer fixed-point theorems. The continuity, boundedness, and Lipschitz properties of the associated nonlinear operators are analyzed to ensure well-posedness of the fractional system. As a constructive complement to the theoretical results, a power series iterative method (PSIM) is employed to obtain an explicit fractional series representation of the solution in the case $0<\alpha <1$. The applicability of the theoretical framework is illustrated through a nonlinear fractional dynamical Belousov–Zhabotinsky system (DBZS), where the assumptions of the main theorems are verified and the solution is constructed via the proposed series scheme. The results provide a coherent link between abstract fixed-point analysis and a constructive semi-analytical representation of solutions for EK fractional systems.

This paper proposes and studies a new class of nonlinear nonlocal problem driven by a tempered Caputo-type fractional derivative with respect to an arbitrary smooth kernel. The novelty lies in treating a single nonlocal integro-delay setting that simultaneously couples an arbitrary kernel, exponential tempering, a delayed state, a lower-order distributed fractional memory term, and multipoint nonlocal initial data, rather than introducing a new fractional operator. The resulting problem can be viewed as a rigorous well-posedness prototype for hereditary systems with delayed feedback, tempered memory, and nonlocal initialization. First, an equivalent Volterra integral equation is derived. Then, the existence and uniqueness of solutions are obtained by the Banach contraction principle in a suitable Banach space of continuous functions. Next, a Picard successive approximation procedure is introduced and shown to converge uniformly to the unique solution, together with an explicit a priori error estimate. Moreover, a continuous dependence result is proved with respect to perturbations in the initial constants, the multipoint coefficients, and the nonlinear term. Finally, the main results are illustrated with two examples enhanced by graphs of explicit Picard approximations and convergence tables.

This paper examines the existence and stability of an m-cyclic coupled system of higher-order fractional differential equations with non-singular kernels. Sufficient conditions for the existence and stability of solutions are obtained using fixed-point techniques. Two numerical examples involving coupled and triply coupled systems are presented to validate the theoretical results, and simulations of the triply coupled case illustrate the influence of different fractional orders on the system dynamics.

This paper addresses a class of multi-point initial value problems with impulses and proportional delays. The framework is based on the Atangana–Baleanu–Caputo (ABC) fractional derivative, which allows the model to incorporate hereditary memory effects absent in standard integer-order systems. Using suitable fixed-point arguments, conditions ensuring the existence and uniqueness of solutions are derived. The reliability and robustness of the obtained solutions is analyzed through the Hyers–Ulam (H-U) method and generalized H-U stability. To demonstrate the theoretical findings, a general numerical example model and a fractional-order housefly population model are considered that incorporate impulsive effects and delay terms reflecting real ecological feedback. Numerical simulations illustrate how variations in the fractional order and impulse intensity influence the dynamic behavior of the adult population. The results reveal that impulsive interventions can effectively regulate population oscillations, while the fractional order governs the rate of decay and long-term stability of the system.

Researchers have devised numerous methods to model intricate behaviors in phenomena that unfold in multiple stages. This work focuses on a specific category of piecewise hybrid terminal systems characterized by delay. To account for hereditary memory effects, which are absent in standard integer-order systems, our framework partitions the time interval into two distinct phases. The initial phase employs the classical derivative, while the subsequent phase utilizes the Atangana–Baleanu–Caputo (ABC) fractional derivative. We establish conditions that guarantee both the existence and uniqueness of solutions through the application of suitable fixed-point arguments. Furthermore, Hyers–Ulam (H-U) stability is investigated to ascertain the robustness and reliability of the derived solutions. To exemplify these theoretical findings, we present a fractional-order tuberculosis treatment model that incorporates the development of drug resistance, alongside a general numerical example. Numerical simulations reveal that changes in the fractional order influence the dynamic behavior of the disease.

This paper introduces a new family of q-special polynomials, termed q-general Bell polynomials, and systematically explores their structural and analytical properties. We establish their generating functions, derive explicit series representations, and develop recurrence relations to characterize their combinatorial behavior. Additionally, we characterize their quasi-monomial properties and construct associated differential equations governing these polynomials. To demonstrate the versatility and applicability of this family, we investigate certain examples, including the q-Gould–Hopper–Bell and q-truncated exponential-Bell polynomials, deriving analogous results for each. Further, we employ computational tools in Mathematica to examine zero distributions and produce visualizations, offering numerical and graphical insights into polynomial behavior.