On the Existence of Solutions to a Nonlinear Atangana–Baleanu Type Fractional Differential Equation

In this research work, we investigate a nonlinear fractional initial value problem involving the Atangana-Baleanu-Caputo derivative of order 0<α<1. By means of the associated fractional integral operator, the problem is converted into an equivalent nonlinear integral equation. The existence of solutions is established in the context of extended F-metric spaces via a fixed point approach based on an (α,ψ)-contractive condition of rational form. Furthermore, we develop the notion of graphic rational contractions in the setting of extended F-metric spaces and prove new fixed point results. Our results extend and unify several known results in the existing literature as special cases. Nontrivial examples are provided to demonstrate the applicability of the theoretical findings. These results highlight the effectiveness of extended F-metric techniques in the analys.