The difference between a class of discrete fractional and integer order boundary value problems

•We give the differences between the fractional-order and integer-order problems.•Multiplicity of solutions for the fractional problem is given by a new theorem.•We obtain the uniqueness theorem for the fractional problem applying a new theorem.•It’s the first time to study the existence for this class of fractional problems.•We study the Green’s function of the fractional boundary value problems in detail.

In this paper, we point out the differences between a class of fractional difference equations and the integer-order ones. We show that under the same boundary conditions, the problem of the fractional order is nonresonant, while the integer-order one is resonant. Then we analyse the discrete fractional boundary value problem in detail. Then the uniqueness and multiplicity of the solutions for the discrete fractional boundary value problem are obtained by two new tools established in 2012, respectively.