This paper investigates the existence and uniqueness of solutions to a fractional (p, q)-difference equation of order \(\alpha \) belonging to the interval (2, 3], subject to separated boundary conditions at three distinct points. By employing fixed-point techniques including Krasnoselskii’s fixed-point theorem, the Leray-Schäuder alternative, and Banach’s contraction principle, we establish the existence of at least one solution and provide conditions guaranteeing uniqueness. These analytical results extend earlier studies on fractional boundary value problems and are motivated by the broad applicability of Caputo (p, q)-fractional derivatives in scientific and engineering contexts. Such operators capture enriched memory effects and arise in models of epidemiology, neural signal propagation, and viscoelastic materials. In particular, the problem studied here may be viewed as describing the transverse deflection of a viscoelastic beam governed by a Caputo (p, q)-fractional differential law. The existence and uniqueness results ensure a well defined and physically meaningful deflection profile, reinforcing the theoretical basis for further analysis and applications. An illustrative example is provided to demonstrate and validate the developed theory.
