Initial and Boundary Value Problems for the Caputo Fractional Self-Adjoint Difference Equations

TL;DR

This paper develops a theoretical framework for solving initial and boundary value problems involving Caputo fractional nabla difference equations, including existence, uniqueness, and solution representations.

Contribution

It introduces the nabla fractional calculus with Caputo differences and establishes foundational theorems for self-adjoint fractional difference equations.

Findings

Proved existence and uniqueness theorems for initial and boundary value problems.

Developed a variation of constants formula using the Cauchy function.

Derived inequalities for Green's functions relevant to nonlinear fractional equations.

Abstract

In this paper we develop the theory of initial and boundary value problems for the self-adjoint nabla fractional difference equation containing a Caputo fractional nabla difference that is given by \[ \nabla[p(t+1)\nabla_{a*}^\nu x(t+1)] + q(t)x(t) = h(t), \] where $0 < \nu \leq 1$. We give an introduction to the nabla fractional calculus with Caputo fractional differences. We investigate properties of the specific self-adjoint nabla fractional difference equation given above. We prove existence and uniqueness theorems for both initial and boundary value problems under appropriate conditions. We introduce the definition of a Cauchy function which allows us to give a variation of constants formula for solving initial value problems. We then show that this Cauchy function is important in finding a Green's function for a boundary value problem with Sturm-Liouville type boundary conditions. Several inequalities concerning a certain Green's function are derived. These results are important in using fixed point theorems for proving the existence of solutions to boundary value problems for nonlinear fractional equations related to our linear self-adjoint equation.