Existence of a positive solution to a class of fractional differential equations

In this paper, we consider a (continuous) fractional boundary value problem of the form −D0+νy(t)=f(t,y(t)), y(i)(0)=0y(i)(0)=0, [D0+αy(t)]t=1=0, where 0≤i≤n−20≤i≤n−2, 1≤α≤n−21≤α≤n−2, ν>3ν>3 satisfying n−1<ν≤nn−1<ν≤n, n∈Nn∈N, is given, and D0+ν is the standard Riemann–Liouville fractional derivative of order νν. We derive the Green’s function for this problem and show that it satisfies certain properties. We then use cone theoretic techniques to deduce a general existence theorem for this problem. Certain of our results improve on recent work in the literature, and we remark on the consequences of this improvement.