Nabla fractional boundary value problem with a non-local boundary condition

Abstract

In this work, we deal with the following two-point boundary value problem for a finite fractional nabla difference equation with non-local boundary condition: (?(???(e) u(z) = p(z, u(z)), z ? Nfe+2, u(e) = g(u), u(f) = 0. Here e, f ? R, with f ?e ? N3, 1 < ? < 2, p : Nfe+2 ×R ? R is a continuous function, the functional g ? C[Nfe ? R] and ???(e) denotes the ?th- order Riemann–Liouville backward (nabla) difference operator. First, we derive the associated Green’s function and some of its properties. Using the Guo–Krasnoselskii fixed point theorem on a suitable cone and under appropriate conditions on the non-linear part of the difference equation, we establish sufficient conditions for the existence of at least one positive solution to the boundary value problem. Next, we discuss the uniqueness of the solution to the considered problem. For this purpose, we use Brouwer and Banach fixed point theorem respectively. Finally, we provide an example to illustrate the applicability of established results.