Existence and Uniqueness of Solutions for the Bagley–Torvik Problem Involving the Hilfer Fractional Derivative

Abstract

In this paper, we investigate the existence and uniqueness of solutions for the Bagley–Torvik equation involving the Hilfer fractional derivative. By transforming the original fractional integro-differential equation into an equivalent integral equation, the problem is reformulated in a framework suitable for fixed-point analysis. The existence of at least one solution is established by applying the Schauder fixed-point theorem, while the uniqueness of the solution is obtained under an appropriate Lipschitz-type condition using the Banach contraction principle. The proposed approach provides a rigorous theoretical framework for analyzing Bagley–Torvik equations with Hilfer fractional derivatives.