Existence results for fractional q-difference equations of order with three-point boundary conditions

The quantum calculus deals with quantum derivatives and integrals, and has proven to be relevant for quantum mechanics. It allows to deal with continuous functions, which do not need to be smooth. Fractional calculus, on the other hand, generalizes integer-order analysis, by considering derivatives and integrals of non-integer order, and found many applications e.g., in physics and signal processing. The natural extension, which we investigate here, is to consider a quantum fractional calculus, which unifies these two theories by considering quantum derivatives of non-integer order. In this paper we present several results on existence of solutions for a fractional q-difference equation of order α ∈ ] 2 , 3 [ with three-point boundary conditions that involves quantum derivatives and quantum integrals.