In this article, our main aim is to explore the existence and uniqueness of solutions for a fractional-order $q$-equation under a $q$-integral boundary condition. Employing well-known fixed point theorems, we derive theoretical findings that advance the theory of $q$-fractional calculus. An example is provided to show the effectiveness of the results, and the stability of the solutions is discussed.

In this paper, using the Schauder and the Banach fixed point (FP) theorems, we examine the existence and uniqueness of solutions in the Banach space $C([0,1])$ for a boundary value problem (BVP) of non-linear third-order $q$-difference equations with the $q$-integral boundary condition. Then, we impose the sufficient condition that allows us to deduce a nonexistence result. Furthermore, we offer some examples to support our main outcomes.