  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.
