The existence of solutions for some fractional finite difference equations via sum boundary conditions

In this manuscript we investigate the existence of the fractional finite difference equation (FFDE) Delta(mu)(mu-2)x(t) = g(t + mu - 1, x(t + mu - 1), Delta x(t + mu - 1)) via the boundary condition x(mu - 2) = 0 and the sum boundary condition x(mu + b + 1) = Sigma(alpha)(k=mu-1) x(k) for order 1 < mu <= 2, where g : N-mu-1(mu+b+1) x R x R -> R, alpha is an element of N-mu-1(mu+b), and t is an element of N-0(b+2). Along the same lines, we discuss the existence of the solutions for the following FFDE: Delta(mu)(mu-3)x(t) = g(t + mu - 2, x(t + mu - 2)) via the boundary conditions x(mu - 3) = 0 and x(mu + b + 1) = 0 and the sum boundary condition x(alpha) = Sigma(beta)(k=gamma)x(k) for order 2 < mu <= 3, where g : N-mu-2(mu+b+1) x R -> R, b is an element of N-0, t is an element of N-0(b+3), and alpha, beta,gamma N-mu-2(mu+b) with gamma < beta < alpha.