On a Nabla Fractional Boundary Value Problem with General Boundary Conditions

TL;DR

This paper studies a nabla fractional boundary value problem with general boundary conditions, analyzing its Green's function to establish properties and develop a Lyapunov-type inequality for the problem.

Contribution

It provides an explicit analysis of the Green's function for a nabla fractional boundary value problem and derives a Lyapunov-type inequality using fractional nabla Taylor monomials.

Findings

Green's function is nonnegative

An upper bound for the Green's function's maximum is obtained

A Lyapunov-type inequality is established

Abstract

In this article, we consider a nabla fractional boundary value problem with general boundary conditions. Brackins \& Peterson \cite{Br} gave an explicit expression for the corresponding Green's function. Here, we show that this Green's function is nonnegative and obtain an upper bound for its maximum value. Since the expression for the Green's function is complicated, derivation of its properties may not be straightforward. For this purpose, we use a few properties of fractional nabla Taylor monomials. Using the Green's function, we will then develop a Lyapunov-type inequality for the nabla fractional boundary value problem.