On Some Self-Adjoint Fractional Finite Difference Equations

Abstract

Recently, the existence of solution for the fractional self-adjoint equation Delta(nu)(nu-1) (p Delta y)(t) = h(t) for order 0 < nu <= 1 was reported in [9]. In this paper, we investigated the self-adjoint fractional finite difference equation Delta(nu)(nu-2)(p Delta u(t) = j(t,p(t+nu - 2)) via the boundary conditions y(nu - 2) = 0 , such that Delta y(nu - 2) = 0 and Delta y(nu+b) = 0. Also, we analyzed the self-adjoing fractional finite difference equation Delta(nu()(nu-2)p Delta(2)y)(t) = j(t,[(t+nu - 2)Delta(2)y(t+nu-3)) via the boundary conditions y(nu - 2) = 0, Delta y(nu - 2) = 0, Delta(2)y(nu - 2) = 0 and Delta 2y(nu+b) = 0. Finally, we conclude a result about the existence of solution for the general equation Delta(nu()(nu-2)p Delta(m)y)(t) = h(t,p(t+nu - m - 1)Delta(m)y(t+nu - m - 1)) via the boundary conditions y(nu - 2) = Delta y(nu - 2) = Delta(2)y(nu - 2) = center dot center dot center dot Delta(m)y(nu+b) = 0 for oder 1 < nu <= 2.