Well-Posedness and Ulam-Hyers Stability of Coupled Deformable Fractional Differential Systems via Perov and Leray-Schauder Approaches

In this paper, we investigate the well-posedness and stability of a class of coupled systems of deformable fractional differential equations in Banach spaces. The deformable fractional derivative, which interpolates continuously between a function and its classical derivative through a single scalar parameter, provides a flexible and tractable framework for modeling complex dynamical phenomena with memory effects. By employing Perov’s fixed-point theorem under matrix contractive conditions, we establish the existence and uniqueness of solutions for the considered coupled system. The existence of at least one solution under broader growth conditions is then proved via the nonlinear alternative of Leray–Schauder type. Furthermore, the continuous dependence of solutions on initial data is rigorously established, confirming the well-posedness of the system. Hyers–Ulam stability and generalized Hyers–Ulam–Rassias stability results are also derived, providing quantitative estimates relevant to numerical approximation and applied analysis. Three illustrative examples are presented to demonstrate the applicability and effectiveness of the theoretical results.