Studies on Fractional Differential Equations With Functional Boundary Condition by Inverse Operators

Fractional differential equations (FDEs) generalize classical integer-order calculus to noninteger orders, enabling the modeling of complex phenomena that classical equations cannot fully capture. Their study has become essential across science, engineering, and mathematics due to their unique ability to describe systems with nonlocal interactions. In this paper, we study the uniqueness, existence, and stability for a new nonlinear FDE with functional boundary condition (which describes nonlocal properties) based on several well-known fixed-point theorems, the two-parameter Mittag-Leffler function, and an implicit integral equation obtained from inverse operators. Several examples are presented to demonstrate applications of our key theorems. Furthermore, we also indicate that the method used can deal with PDEs, with various initial or boundary conditions.