Positive Properties of Green’s Function for Fractional Dirichlet Boundary Value Problem with a Perturbation Term and Its Applications

In this article, we study a fractional lower-order differential equation, −D0+αΥ(ξ)+a(ξ)Υ(ξ)=y(ξ),ξ∈(0,1),α∈(1,2), with a Dirichlet-type boundary condition, where a(ξ)∈L1[0,1] permits singularity. When the coefficient of perturbation term a(ξ) is continuous on [0,1], Graef et al. derived the associated Green’s function under certain conditions on a, but failed to obtain its positivity. We obtain some positive properties of the Green’s function of this problem under certain conditions. The results will fill the gap in the above literature. As for application of the theoretical results, a singular fractional differential equation with a perturbation term is considered. The unique positive solution is obtained by using the fixed point theorem, and an iterative sequence is established to approximate it. In addition, a semipositone fractional differential equation, −D0+αΥ(ξ)=μF(ξ,Υ(ξ)),ξ∈(0,1),α∈(1,2), is also considered. The existence of positive solutions is determined under a more general condition, F(ξ,x)≥−b(ξ)x−e(ξ), where b(ξ),e(ξ)∈L1[0,1] are non-negative functions. Relevant examples are listed to manifest the theoretical results.