On a novel n-tuple variable order q-fractional derivative with respect to ψ function: hybrid difference equation and the well-posedness of the solution

Abstract

This paper introduces a novel generalized fractional operator: the $n$-tuple variable-order $q$-fractional derivative with respect to a $\psi$ function. This operator provides a unified framework that extends several existing $q$-fractional models, including the Riemann–Liouville, Caputo, and Hilfer types, along with their variable-order and $\psi$-dependent variants. Fundamental properties and results are established to analyze the well-posedness of a class of hybrid fractional difference equations. The existence of the solution is established via Krasnoselskii’s fixed point theorem, while uniqueness is demonstrated using the Banach contraction principle. Furthermore, the Ulam–Hyers stability of the solution is rigorously investigated. An illustrative example is provided to demonstrate the applicability of the theoretical results. This work concludes by identifying future research trajectories, specifically the extension of $n$-tuple operators to non-local boundary value problems and the exploration of $(q, \psi)$-difference systems with higher-order derivatives ($n > 1$), offering a definitive foundation for subsequent developments in quantum fractional calculus.