On some self-adjoint fractional finite difference equations

Abstract

Recently, the existence of solution for the fractional self-adjoint equation Δν ν-1(pΔy)(t) = h(t) for order 0 < ν ≤ 1 was reported in [9]. In thispaper, we investigated the self-adjoint fractional finite difference equation Δν ν-2((pΔy)(t) = h(t, p(t + ν - 2)Δy(t + ν - 2)) via the boundary conditions y(ν - 2) = 0, such that Δy (ν - 2) = 0 and Δy(ν + b) = 0. Also, we analyzed the self-adjoint fractional finite difference equation Δν ν-2(pΔ2y)(t) = h(t, p(t + ν - 3) Δ2y(t + ν - 3)) via the boundary conditions y(ν - 2) = 0, Δy(ν - 2) = 0, Δ2y(ν - 2) = 0 and Δ2y(ν + b) = 0. Finally, we conclude a result about the existence of solution for the general equation Δν-2937(pΔmy)(t) = h(t, p(t + ν - m - 1)Δmy(t + ν - m - 1)) via the boundary conditions y(ν - 2) = Δy(ν - 2) = Δ2y(ν - 2) = … = Δmy(ν - 2) = 0 and Δmy(ν + b) = 0 for order 1 < ν ≤ 2. © 2015, Eudoxus Press, LLC. All rights reserved.