New Results for Existence, Uniqueness, and Ulam Stable Theorem to Caputo–Fabrizio Fractional Differential Equations with Periodic Boundary Conditions

The research is focused on establishing the existence and uniqueness of solutions for a specific set of Caputo–Fabrizio fractional differential equations under periodic boundary conditions (PBCs). To achieve this, the study employs the fractional derivative within the Caputo–Fabrizio framework and utilizes the proposed variation of the parameter method. This method is utilized to simplify the second-order fractional differential equation, transforming it into a second-order nonlinear ordinary differential equation. The research findings are rooted in the fixed points of the Schauder fixed point and Banach fixed point theorems. Furthermore, the study delves into stability analysis using the Hyers–Ulam stability concept. Theoretical examples are included to illustrate and demonstrate the implications of the established theorems.