Positive solutions of nabla fractional boundary value problem

Abstract

In this article, we consider the following two-point discrete fractional boundary value problem with constant coefficient associated with Dirichlet boundary conditions.

\begin{align*}\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{align*}

where \(1 < \nu < 2\), \(a,b \in \mathbb{R}\) with \(b-a\in\mathbb{N}_{3}\), \(\mathbb{N}^b_{a+2} = \{a+2,a+3, . . . ,b\}\), \(|\lambda| < 1\), \(\nabla^{\nu}_{\rho(a)}u\) 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'kiÄ­ 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.