Analysis of a Coupled System of Implicit Fractional Differential Equations of Order α ∈ (1, 2] with Anti-Periodic Boundary Conditions

This paper investigates a coupled system of nonlinear implicit fractional differential equations of order α∈(1,2] subject to anti-periodic boundary conditions. The analysis is conducted using the ψ-Caputo fractional derivative, a generalized operator that incorporates several well-known fractional derivatives. The system features implicit coupling, where each equation depends on both unknown functions and their first derivatives, as well as an implicit dependence on the fractional derivatives themselves. The boundary value problem is transformed into an equivalent system of integral equations. Sufficient conditions for the existence and uniqueness of solutions are established using Banach’s and Krasnoselskii’s fixed-point theorems in an appropriately chosen Banach space. Furthermore, the Ulam–Hyers stability of the system is analyzed. The applicability of the theoretical results is demonstrated through a detailed example of a coupled system where all hypotheses are verified.