In this work, we present some analytical and topological framework for fractional nonlinear systems on compact‐open Banach spaces. By using the locally compact property of these spaces, the continuity and compactness of nonlinear operators are rigorously established. Using the Atangana–Baleanu–Caputo (ABC) fractional derivative in conjunction with the β ‐Laplace transform, new transformation properties are derived to guarantee the boundedness and Lipschitz continuity of the associated operators. The Schaefer and Banach fixed‐point theorems are applied to investigate the existence, uniqueness, and Ulam–Hyers stability of the obtained solutions of these nonlinear systems. Furthermore, a hybrid β ‐Laplace homotopy perturbation method ( β ‐LHPM) is developed to construct rapidly convergent analytical series solutions. For work applications, the proposed framework is employed to analyze the fractional Belousov–Zhabotinsky system (BZS), a canonical model of chemical wave propagation in excitable media. The results reveal how fractional order dynamics and memory effects influence the speed, attenuation, and stability of propagating chemical waves. Numerical simulations confirm the accuracy and convergence of the β ‐LHPM and demonstrate that the fractional parameter significantly alters the nature of oscillatory wave fronts.
Damag et al. (Thu,) studied this question.