Existence theory for finite delayed fractional differential equations with nonlinear variable order

We investigated a class of finite-delay fractional differential equations in which the differentiation order combined with time and depended explicitly on the past evolution of the state. The proposed model combined a history-dependent variable fractional order with a nonlinear source term involving an integral memory functional, rather than a pointwise delay. This structure provided a flexible framework for describing adaptive memory effects and cumulative nonlocal behavior that cannot be captured by classical constant-order or pointwise variable-order models. By reformulating the problem as an equivalent integral equation, we established rigorous existence and uniqueness results in the Banach space $L^1(\mathrm{\Delta })$ using the Banach contraction principle and the Schauder fixed-point theorem under mild and verifiable assumptions. In addition, Ulam–Hyers stability of the system was proved, ensuring robustness of solutions with respect to perturbations. An illustrative example, together with a numerical stability verification, was presented to support the theoretical findings. The obtained results contributed to the mathematical understanding of delayed variable-order fractional systems and provided a solid foundation for future studies in engineering, economics, and biomedical modeling.