We investigate a class of delayed fractional dynamical systems motivated by neurodynamic models with memory effects. The main contribution of this work is the development of a novel multivariate fractional Grönwall--Wendroff type inequality that accommodates matrix-valued coefficients, discrete time delays, and implicit fractional dynamics. Unlike existing Wendroff-type inequalities, our result is not restricted to scalar or delay-free settings and is specifically designed to handle coupled systems arising in fractional-order models. Based on this inequality, we establish well-posedness results for a delayed fractional FitzHugh--Nagumo system, including global existence, uniqueness, and continuous dependence on initial data. We further derive Ulam--Hyers stability estimates and show that the fractional delay system generates a dissipative semiflow in an appropriate phase space with memory. As a consequence, we prove the existence of nontrivial periodic solutions and provide sufficient conditions for the stability of limit cycles in the fractional-order sense. Numerical simulations are presented to illustrate the theoretical findings and to highlight the role of fractional memory and delay in shaping the long-term dynamics. The results reveal that fractional effects significantly influence the amplitude, stability, and persistence of oscillatory patterns. Overall, this work provides new analytical tools for the study of delayed fractional systems and offers a rigorous framework for understanding complex oscillatory behavior in neurodynamic and related applications.
Rômulo Damasclin Santos (Sun,) studied this question.