Existence and Uniqueness of Solutions to a Nabla Fractional Difference Equation with Dual Nonlocal Boundary Conditions

Abstract

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.

Multiple Positive Solutions of Discrete Fractional Boundary Value Problems

“…The following theorems which are useful for the main results has appeared in [13] and the same has been proved here for the completeness of the article. Lemma 3.1.…”

“…Gholami et al [17] obtained the Green's function for a non-homogeneous Riemann-Liouville nabla boundary value problem with Dirichlet boundary conditions. Jonnalagadda [13,20,21,[23][24][25] analysed some qualitative properties of two-point non-linear Riemann-Liouville nabla boundary value problem associated with various types of boundary conditions.…”

Existence and Uniqueness Results for Fractional (p, q)-Difference Equations with Separated Boundary Conditions

“…Fractional calculus, dealing with the integrals and derivatives of arbitrary order, constitutes an important area of investigation in view of its extensive theoretical development and applications during the last few decades. For some interesting results on fractional differential equations ranging from the existence and uniqueness of solutions to the analytic and numerical methods for finding solutions, we refer the reader to the following articles: [1][2][3][4][5]. Concerning the applications of fractional differential equations in engineering, clinical disciplines, biology, physics, chemistry, economics, signal and image processing, and control theory, for example, see [6][7][8][9][10] for more details.…”

Editorial for the Special Issue of Foundations “Recent Advances in Fractional Differential Equations and Inclusions”

“…In [5], a two-point boundary value problem for a finite Nabla fractional difference equation with dual non-local boundary conditions is studied. The existence of at least one and at least two positive solutions is verified by using Guo-Kranosel'ski ȋ fixed point theorem on cones.…”