Green's Functions and Existence of Solutions of Nonlinear Fractional Implicit Difference Equations with Dirichlet Boundary Conditions

TL;DR

This paper derives the Green's function for a class of nonlinear implicit fractional difference equations with Dirichlet boundary conditions and establishes existence results for solutions using fixed point theorems.

Contribution

It introduces a method to compute Green's functions for implicit fractional difference equations and proves solution existence under nonlinear conditions.

Findings

Green's function expressed via infinite series using Laplace transform

Existence of solutions proved for nonlinear fractional difference equations

Handles implicit fractional operators with Dirichlet boundary conditions

Abstract

This article is devoted to deduce the expression of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional operators is applied, we are in presence of a implicit fractional difference equation. Such property makes it more complicated to calculate and manage the expression of the Green's function. Such expression, on the contrary to the explicit case where it follows from finite sums, is deduced from series of infinity terms. Such expression will be deduced from the Laplace transform on the time scales of the integers. Finally, we prove two existence results for nonlinear problems, via suitable fixed point theorems.