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 ψ function. This operator extends and unifies 
several existing q-fractional operators, including the Riemann–Liouville, Caputo, and Hilfer 
types, together with their variable order variant and ψ-dependent variant. Fundamental 
properties and related results are established to analyze the well-posedness of hybrid 
fractional difference equations. The existence of the solution is proved via Krasnoselskii’s 
fixed point theorem, while uniqueness is demonstrated using the Banach contraction 
principle. Furthermore, the Ulam–Hyers stability of the solution is investigated. Also, an 
example is presented to illustrate the result.