Bicontrolled Contractive Approach to the Existence of Solutions to the Fractional Aizawa Model

We introduce a novel class of (F,B)-contractions on complete metric spaces by imposing appropriate conditions on the auxiliary functions F and B. A new fixed-point theorem has been established, guaranteeing both the existence and uniqueness of the fixed point. This result brings together and expands upon several traditional contributions in fixed-point theory. The practicality of this theoretical framework is demonstrated through illustrative examples, including the existence and uniqueness of solutions to the fractional-order Aizawa system formulated with the Atangana–Baleanu fractional derivative. Numerical simulations, supported by graphical analysis and Lyapunov exponents, are presented to characterize the system’s dynamical behavior and to validate the effectiveness of the proposed results.