BOUNDARY VALUE PROBLEMS OF NABLA FRACTIONAL DIFFERENCE EQUATIONS

In this dissertation we develop the theory of the nabla fractional self-adjoint difference equation, ∇νa(p∇y)(t) + q(t)y(ρ(t)) = f(t) where 0 < ν < 1. We begin with an introduction to the nabla fractional calculus. In the second chapter, we show existence and uniqueness of the solution to a frac-tional self-adjoint initial value problem. We find a variation of constants formula for this fractional initial value problem, and use the variation of constants formula to derive the Green’s function for a related boundary value problem. We study the Green’s function and its properties in several settings. For a simplified boundary value problem with p ≡ 1, we show that the Green’s function is nonnegative and we find its maximum and the maximum of its integral. For a boundary value problem with generalized boundary conditions, we find the Green’s function and show that it is a generalization of the first Green’s function. In the third chapter, we use the Contraction Mapping Theorem to prove existence and uniqueness of a positive so-lution to a forced self-adjoint fractional difference equation with a finite limit. We explore modifications to the forcing term and modifications to the space of functions in which the solution exists, and we provide examples to demonstrate the use of these theorems. iii DEDICATION This dissertation is dedicated to my father, Steven Dennis Brackins, who taught me to dream big, and to my mother, Donna Mae Brackins, who taught me the tenacity to realize those dreams. iv ACKNOWLEDGMENT