On a self-adjoint fractional Nabla finite difference equation with initial value condition

Abstract

By using discrete fractional calculus, we investigate the existence of solutions for the initial boundary value problem of self-adjoint finite Nabla fractional difference equation on the time scale $\mathbb{N}_{a+1}^{b}$. Also, we check some conditions for uniqueness of solution of the problem. For finding of the solution of the equation, the Green function will be presented 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.