A Comparison Result for the Fractional Difference Operator

In this paper, we deduce the Green’s function for a ν-th order, 1 < ν ≤ 2, discrete fractional boundary value problem with boundary conditions of the type αy(ν−2)−β∆y(ν−2) = 0, γy(ν+ b)+δ∆y(ν+ b) = 0, for α, β, γ, δ ≥ 0 and αγ+αδ+βγ 6= 0. This extends and generalizes the results of some recent papers. We then show that this Green’s function satisfies a positivity property. From this we deduce a relatively general comparison result for boundary value problems of this sort. In particular, this shows that the fractional difference operator satisfies the same sort of comparison principle that is well-known in the integer-order case. AMS Subject Classifications: Primary: 26A33, 39A12, 39A70.