on a self-adjoint fractional nabla finite difference equation with initial value condition

By using discrete fractional calculus, we investigate the existence of solutions for a self-adjoint finite nabla fractional difference equation on the time scale n b a+1 via initial boundary conditions. also, we check some conditions for uniqueness of solution of the problem. for finding the solution, we use the green function which defined by using the cauchy function. the principle of contraction mapping also plays an essential role in the existence of the solution. we provided two examples, a figure and numerical results to illustrate our main result.