Existence and stability results of f-Caputo modified proportional fractional delay differential systems with boundary conditions

This paper undertakes a detailed investigation into the fundamental properties of a novel class of f-Caputo modified proportional fractional delay differential equations. These equations are uniquely characterized by the inclusion of two distinct fractional orders. We begin by exploring the inherent characteristics and behaviors of the generalized proportional operator, which is a crucial component of our proposed system. Employing the robust tools of the Schaefer fixed-point theorem and the Banach contraction principle, we rigorously establish theoretical guarantees for both the existence and uniqueness of solutions. Furthermore, we delve into the important aspect of solution stability, presenting a range of Ulam-Hyers-type stability results that underscore the resilience of the solutions under small perturbations. To provide tangible evidence and practical illustration of our theoretical contributions, we conclude with a carefully constructed numerical example that serves to validate and reinforce the analytical findings presented throughout the paper.