In this paper, we look at the two-point boundary value problem for a finite nabla fractional difference equation with dual non-local boundary conditions. First, we derive the associated Green’s function and some of its properties. Using the Guo–Krasnoselkii fixed point theorem on a suitable cone and under appropriate conditions on the non-linear part of the difference equation, we establish sufficient requirements for at least one and at least two positive solutions of the boundary value problem. Next, we discuss the existence and uniqueness of solutions to the considered problem. For this purpose, we use Brouwer and Banach fixed point theorem, respectively. Finally, we provide a few examples to illustrate the applicability of established results.

In this article, we consider the following two-point discrete fractional boundary value problem with constant coefficient associated with Dirichlet boundary conditions. \begin{equation*} \begin{cases} -\big{(}\nabla^{\nu}_{\rho(a)}u\big{)}(t) + \lambda u(t) = f(t, u(t)), \quad t \in \mathbb{N}^{b}_{a + 2}, \\ u(a) = u(b) = 0, \end{cases} \end{equation*} where $1 < \nu < 2$, $a,b \in \mathbb{R}$ with $b-a\in\mathbb{N}_{3}$, $\mathbb{N}^b_{a+2} = \{a+2,a+3,\hdots,b\}$, $|\lambda| < 1$, $\nabla^{\nu}_{\rho(a)}u$ denotes the $\nu^{\text{th}}$-order Riemann--Liouville nabla difference of $u$ based at $\rho(a)=a-1$, and $f : \mathbb{N}^{b}_{a + 2} \times \mathbb{R} \rightarrow \mathbb{R}^{+}$. We make use of Guo--Krasnosels'ki\v{\i} and Leggett--Williams fixed-point theorems on suitable cones and under appropriate conditions on the non-linear part of the difference equation. We establish sufficient requirements for at least one, at least two, and at least three positive solutions of the considered boundary value problem. We also provide an example to demonstrate the applicability of established results.

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