On a discrete fractional three-point boundary value problem

Abstract

In this paper, we analyse a ν-th order, , discrete fractional three-point boundary value problem (BVP). We show that Green's function associated to this problem satisfies certain conditions. We demonstrate that the range of admissible boundary conditions depends upon the order ν of the difference equation, and we give explicit formulae for this dependence. By using both the Brouwer fixed point theorem and the Krasnosel'skiĭ fixed point theorem, we then show that a solution to this problem exists. Our results extend recent results on discrete fractional BVPs (FBVPs), and they also provide an initial set of results on the theory of multipoint FBVPs on the time scale of integers.