Positive Solutions of a Fractional BVP and the Reach of the Omitted Ray Fixed Point Theorem

In this paper, we study the problem D0+αu+f(t,u)=0 on (0,1), u(0)=0, D0+α−1u(1)=∫01udA, where 1<α≤2 and the measure dA may change sign. For a measure made of two point masses of opposite sign, we show that the function KA(s)=∫01G(τ,s)dA(τ), through which dA enters the Green function, is nonnegative exactly when the positive weight is at least θ* times the size of the negative one, where θ* is given in closed form. Together with a simple condition Δ>0, this makes the Green function nonnegative. The threshold θ* decreases in α and tends to infinity as α→1+. For α<2 the Green function has no uniform Harnack estimate, so we work in a cone weighted by Ψ(t)=∫01G(t,s)ds, where Schauder’s theorem gives a positive solution with explicit bounds whenever f lies between two positive constants. For the omitted ray fixed point theorem of Avery, Henderson and Liu, we describe exactly when its key inequality holds for the two-point functional used here, and we show that the hypotheses under which we can verify it already imply the classical inward condition. Examples with α=3/2 and α=1.08 illustrate the results.