Abstract It has been shown that, under suitable hypotheses, boundary value problems of the form, Ly + λy = f, BCy = 0 where L is a linear ordinary or partial differential operator and BC denotes a linear boundary operator, then there exists Λ > 0 such that f ≥ 0 implies λy ≥ 0 for λ ∈ [−Λ, Λ] \ {0}, where y is the unique solution of Ly + λy = f, BCy = 0. So, the boundary value problem satisfies a maximum principle for λ ∈ [−Λ, 0) and the boundary value problem satisfies an anti-maximum principle for λ ∈ (0, Λ]. In an abstract result, we shall provide suitable hypotheses such that boundary value problems of the form, Dα 0 y + βDα−1 0 y = f, BCy = 0 where Dα 0 is a Riemann-Liouville fractional differentiable operator of order α, 1 < α ≤ 2, and BC denotes a linear boundary operator, then there exists B > 0 such that f ≥ 0 implies βDα−1 0 y ≥ 0 for β ∈ [−B, B] \ {0}, where y is the unique solution of Dα 0 y+βDα−1 0y = f, BCy = 0. Two examples are provided in which the hypotheses of the abstract theorem are satisfied to obtain the sign property of βDα−1 0y. The boundary conditions are chosen so that with further analysis a sign property of βy is also obtained. One application of monotone methods is developed to illustrate the utility of the abstract result.