A Comparison Result for the Fractional Difference Operator

In this paper, we deduce the Green’s function for a -th order, 1 < 2, discrete fractional boundary value problem with boundary conditions of the type y ( 2) y( 2) = 0,y ( +b) + y( +b) = 0, for , , , 0 and + + 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.

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