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.
Highlights
Over the last few decades, the theory of fractional calculus has been extensively developed due to its properties, generalizing most results of differential calculus and its non-local nature of fractional derivatives
The contributions of several mathematicians over the span of three centuries have resulted in a robust theory of fractional differential equations for real functions
On the other side of the coin, nabla fractional calculus is a branch of mathematics, which deals with arbitrary order differences and sums in the backward sense
Summary
Over the last few decades, the theory of fractional calculus has been extensively developed due to its properties, generalizing most results of differential calculus and its non-local nature of fractional derivatives. The notion of nabla fractional difference and sum can be traced back to the work of Gray and Zhang [5], and Miller and Ross [6] In this line, Atici and Eloe [7] developed nabla fractional Riemann–Liouville difference operator, initiated the study of nabla fractional initial value problem and established exponential law, product rule, and nabla Laplace transform. Atici and Eloe [7] developed nabla fractional Riemann–Liouville difference operator, initiated the study of nabla fractional initial value problem and established exponential law, product rule, and nabla Laplace transform Following their works, the contributions of several mathematicians have made the theory of discrete fractional calculus a fruitful field of research in science and engineering, we refer here a few applications of discrete fractional equations [8,9,10].
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