Green’s Functions for a Fractional Boundary Value Problem with Three Terms

Abstract

We construct a Green’s function for the three-term fractional differential equation −D0+αu+aD0+μu+f(t)u=h(t), 0<t<b, where α∈(2,3], μ∈(1,2], and f is continuous, satisfying the boundary conditions u(0)=u′(0)=0, D0+βu(b)=0, where β∈[0,2]. To accomplish this, we first construct a Green’s function for the two-term problem −D0+αu+aD0+μu=h(t), 0<t<b, satisfying the same boundary conditions. A lemma from spectral theory is integral to our construction. Some limiting properties of the Green’s function for the two-term problem are also studied. Finally, existence results are given for a nonlinear problem.