Vallée-Poussin theorem for fractional functional differential equations with integral boundary condition

This research paper focuses on the study of a Riemann-Liouville fractional functional differential equation and a linear continuous operator acting from the space of continuous functions to the space of essentially bounded functions with a boundary condition involving integral terms. We investigates the solvability and uniqueness of the equation under certain conditions on the coefficients. The paper utilizes techniques of Vallée-Poussin theorem, and Green’s function sign constancy to establish the main results. Choosing a corresponding function within the context of the Vallée-Poussin theorem results in explicit criteria presented as algebraic inequalities. These inequalities, as we illustrate through examples, cannot be further improved.